Localized Wave and Vortical Solutions to Linear Hyperbolic Systems ...

c Pleiades Publishing, Ltd., 2008. ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 15, No. 2, 2008, pp. 192–221.

In memoriam L. R. Volevich (1934–2007)

Localized Wave and Vortical Solutions to Linear Hyperbolic Systems and Their Application to Linear Shallow Water Equations S. Yu. Dobrokhotov*, A. I. Shafarevich*, and B. Tirozzi Institute for Problems in Mechanics, RAS, Moscow, Russia E-mail: [email protected], [email protected] Department of Physics, University “La Sapienza,” Rome, Italy E-mail: [email protected] Received March 19, 2007

Abstract. The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients. DOI: 10.1134/S1061920808020052

1. INTRODUCTION By a fast-decaying function or (m-D vector function) we mean a function of the form f (x/µ), where x ∈ Rn , µ is a small positive parameter, and f (z) is a smooth function (or m-D vector function) decaying as |z| → ∞ more rapidly than const/|z|β with β > 2 and whose derivatives ∂ |ν| f (z)/∂z ν , where ν = (ν1 , ν2 , . . . , νn ) and |ν| = ν1 + ν2 + · · · = νn , decay at infinity more rapidly than const/|z|β+|α| . Let A1 (x), . . . , An (x), H1 (x) be smooth m × m matrix-valued functions with real entries. Introduce the matrix differential operator ∂ ∂ ∂ A1 (x) + A2 (x) + · · · + An (x) + H1 (x) ∂x1 ∂x2 ∂xn and a linear system of differential equations for an m-D vector function Ψ(x, t) with components Ψ1 (x, t), . . . , Ψm (x, t), ∂Ψ ∂Ψ ∂Ψ ∂Ψ + A1 (x) + A2 (x) + · · · + An (x) + H1 (x)Ψ = 0. (1.1) ∂t ∂x1 ∂x2 ∂xn Our aim is to study the asymptotics (µ ≪ 1) of solutions to the Cauchy problem for this system with fast-decaying or localized initial data Ψ t=0 = V (x/µ) (1.2)

for a given m-D vector function V (z) whose components satisfy the above conditions. In this case, we shall use notation coming from quantum mechanics. Multiply (1.1) by −iµ and define the operators pˆj = −iµ∂/∂xj . Introduce the matrix-valued function H0 (p, x) = A1 (x)p1 + A2 (x)p2 + · · · + An (x)pn and the operator Hˆ0 = A1 (x)ˆ p1 + A2 (x)ˆ p2 + . . . An (x)ˆ pn . In this case, one can represent (1.1) as

∂Ψ = (Hˆ0 − iµH1 )Ψ. (1.3) ∂t Denote by R2n p,x the 2n-D Euclidian space with coordinates (p, x) and refer to it as the phase space. Consider the spectral problem for the matrix (family of matrices) H0 (p, x) and denote the corresponding eigenvalues by H σ (p, x). As is well known (and can readily be verified), every function H σ (p, x) is first-order homogeneous1 with respect to p. Thus, the eigenvalues coincide at the point p = 0. Assume that the following conditions hold. iµ

1 In

the physical language, one can say that we consider the dispersion-free situation.

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(i) The matrix H0 is diagonalizable; for p 6= 0, all eigenvalues are real-valued smooth functions, and their multiplicity does not depend on (p, x). We use the symbol s(σ) for the multiplicity of H σ (p, x). (ii) One can choose a base in each eigensubspace of the matrix H0 (p, x) and take the bi-orthogonal base of the conjugate matrix H0 (p, x) in the form of smooth vector functions ξsk (p, x) and ξsk† (p, x), s = 1, . . . , S(k), outside the point p = 0. In this case, (1.1) is a Petrovskii well-posed hyperbolic system (strictly hyperbolic if S(σ) = 1 ∀σ). The class of systems of this kind includes many important physical problems, like the propagation of electro-magnetic and acoustic waves, etc. The main example in this paper is the two-dimensional linearized shallow water equation. We recall it below. Let us now briefly explain our objective, difficulties, and the main results. Our aim is to construct formulas for the asymptotic behavior of the Cauchy problem (1.1)–(1.2) that are as explicit as possible in the case of arbitrarily varying coefficients of system (1.1). One of the main difficulties is that the dependence of the effective Hamiltonians H σ and/or the corresponding eigenvectors ξsσ on the momentum p at the point p = 0 is not smooth since the H σ ’s coincide at this point. Namely, this fact implies the effects of metamorphosis of the (wave) part of the solutions to problem (1.1)–(1.2): for t = 0, the solution is localized in a neighborhood of the point x = 0, whereas, later on, it can be localized in a neighborhood of some closed surfaces (curves for n = 2) which are well known in geometrical optics as fronts. A similar phenomena is well known in the theory of parametrix (the theory of propagation of singularities), but only the propagation of fronts was considered, rather than that of the profiles of the solutions (the waves). The parametrix theory enables one to obtain some asymptotic formulas for the solution of problem (1.1)–(1.2). Namely, one can first construct the asymptotic with respect to smoothness of the fundamental solution to (1.1), and then the convolution of this solution with the initial function (1.2) gives an asymptotic solution to (1.1)–(1.2). Actually, this scheme (generalizing the scheme of [27] used for hyperbolic systems with constant coefficients) was realized in [10], but the asymptotics thus obtained was complicated, and its simplification seemed to be a nontrivial problem. Here we use another idea. Roughly speaking, we apply a good global ansatz for the asymptotics of (1.1)–(1.2) (guessed in [11–13]) and then simplify it in different situations. Our starting points are as follows: 1) using the Fourier transform, we pass from fast-decaying solutions to fast-oscillating ones, and 2) we use the ideas of boundary-layer expansions near the front since the solution is localized in a neighborhood of the front. To construct the oscillating solutions, we can use the semiclassical approximation, whereas, outside a neighborhood of focal points and caustics, the solutions can be presented in WKB form. However, for the situation in question, the focal point occurs from the very beginning, and this is the point x = 0. For this reason, one must use an analog of WKB asymptotics, taking into account the existence of focal points, and this analog is the Maslov canonical operator [23] generalized to the case of localized initial data. Recall that the construction of the Maslov canonical operator is based on the Lagrangian manifold and on functions defined on these manifolds. In the 2-D case, for t = 0, we suggest the Lagrangian manifold cos(ψ) 0 0 0 0 0 , (1.4) Λ0 = {p = P ≡ n (ψ), x = X ≡ n (ψ)α | ψ ∈ [0, 2π), α ∈ R}, n (ψ) = sin(ψ) and the function V˜ (ρn0 (ψ)), where

V˜ (k) = V˜ (k1 , k2 ) = (1/(2π))

Z

eihk,zi V (z) dz

(1.5)

R2

depending on the coordinate ψ and also on an additional parameter ρ = |k|. Denote by KΛh0 the Maslov canonical operator on Λ0 with the “semiclassical parameter” h and set h = µ/ρ. In this case, Z ∞ p µ/ρ V (x/µ) = µ/(2πi) KΛ0 ρ1/2 V˜ (ρn0 (ψ))dρ. (1.6) √

0

(Here and below, we set i = eiπ/4 .) Formula (1.6) first appeared in [11–13] and was proved in [15]; for completeness, we discuss it below in detail, where we also explain the reason for which the additional parameter ρ occurs. For now, we only say that, using commutation with the original RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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hyperbolic operator, we can first construct an asymptotic solution of (1.1) with the initial data µ/ρ KΛ0 ρ1/2 V˜ (ρn0 (ψ)) without integrating with respect to ρ and then obtain asymptotic formulas after integrating the result against dρ. Note that, if the initial data are presented by using the Maslov canonical operator, then the solution can be constructed by applying a well-developed procedure. Let us briefly recall this procedure. The above eigenvalues (functions) H σ (p, x) are referred to as effective Hamiltonians or modes; they determine the Hamiltonian systems in the phase space, t which in turn define phase flows or the canonical transforms, which we denote by gH σ . Roughly speaking, the construction of an asymptotic solution can be described by the diagram solution to (1.1), (1.2) =⇒ Ψ(x, t) Ψ t=0 ⇓ ⇑ r r Z ∞ Z µ µ X ∞ √ µ/ρ σ σ µ/ρ √ ˜ 0 KΛ0 ρ V ρn (ψ) dρ ρKΛσ A ξ dρ t 2πi 0 2πi σ 0 ⇓ Λ0 , Aσ0 = V˜

t gH σ

=⇒

[ ⇑ {Λσt , Aσ } σ

Diagram 1.

To pass along the upper line (i.e., to construct an asymptotic solution to the original Cauchy problem), one must 1) go down and construct the initial Lagrangian manifold Λ0 and the appropriate function Aσ0 on it (it is already done), 2) go along the bottom line to shift Λ0 along the trajectories of the Hamiltonian systems with the Hamiltonians H σ (this gives Lagrangian manifolds Λσt ) and also to express the functions (amplitudes) Aσ via the initial function V˜ , and 3) go up and, using the Maslov canonical operator, recover the solution Ψ from the manifolds Λσt and the functions Aσ . The realization of this scheme uses the solution of some systems of ordinary differential equations and also the evaluation of some fast oscillating integrals. The new point in this scheme (with respect to the standard Maslov one) is the presence of an additional integral with respect to ρ, which makes it possible to simplify the resulting formulas by using the ideas of boundary-layer expansions or the Maslov complex germ. In this paper, we mainly restrict ourselves to the 2-D case and treat the system of linearized shallow water equations as the main example. These equations are important in different applications to hydrodynamics and plasma physics and also have interesting mathematical properties, in particular, in connection with the problem in question. Namely, they possess both vortices and waves (vortical and wave solutions); in the phase space, the vortices correspond to moving focal points, whereas the creation of waves is equivalent to phenomena known as “metamorphosis of discontinuity”. The paper is organized as follows. The shallow water equation and its linearization, together with its properties, are considered in Section 2. Section 3 is devoted to the representation of rapidly decaying functions via the Maslov canonical operator. The solutions of hyperbolic systems with localized initial data in general form and their general properties are presented in Section 4 (Theorem 1). The main result of that section, which is one of the main results of the paper, is given in Theorem 2. In that section, we also discuss some properties of asymptotic solutions of the linearized shallow water equation. A simplification of the asymptotic solution for different parts (modes) of solution is given in Section 5 (in Theorem 3 for the vortical mode, and in Theorems 4 and 5 for the wave modes). The derivation of the transport equations for various modes (based on the Feynman–Maslov operator calculus) is given in the appendix. In the paper, we use some facts related to semiclassical asymptotics [23–26, 36, 18] and ray expansions [3, 7, 21], classical mechanics and symplectic geometry [1, 2, 18], operator theory [24, 35], boundary layer expansions [37, 30, 25, 28], etc. Our definition of Maslov canonical operator follows [16]. We cite facts and definitions in the appropriate parts of the paper, sometimes in a form which slightly differs from that used in the books listed above. Note that there are many papers and monographs devoted to shallow water equation and its applications. We refer to [17], to the books [29, 32], and to [34], where one can also find a rich bibliography. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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2. LINEARIZED SHALLOW WATER EQUATIONS 2.1. Shallow Water Equation and Its linearization Let us discuss some aspects of our main example. The shallow water equations on the β-plane over a nonuniform bottom is of the form ∂η ∂u + div (η + D(x))u = 0, + (u, ∇)u + ωT u + ∇η = 0. (2.1) ∂t ∂t Here x = (x1 , x2 ) ∈ R2 , u(x, t) = (u1 (x, t), u2 (x, t)) is the velocity vector, the function η > 0 describes the elevation of free surface in the basin with nonuniform bottom whose depth is given by the smooth function D(x), ∇ is the formal vector (∂/∂x1 , ∂/∂x2 ), ω = ω0 + βx2 , where ω0 and β 0 1 are constants, and T = −1 0 . Recall [17] that system (2.1) possesses the so-called Lagrangian invariant: Ω = (curl3 u − ω)/(η + D(x)), (2.2) namely, d Ω ≡ ∂/∂t + (u, ∇) Ω = 0. (2.3) dt Here and below, we write curl3 u = ∂u2 /∂x1 − ∂u1 /∂x2 . Let u = V + u,

η = R + η,

and R(x, t) + D(x) = c2 (x, t),

where (V (x, t), R(x, t)) stands for the background and (u(x, t), η(x, t)) for a small perturbation. For the background, we have: ∂R ∂V + div (R + D)V = 0, + (V, ∇)V + ωT V + ∇R = 0. (2.4) ∂t ∂t For u and η, we have a linear system of equations giving the main example of the paper, ∂u ∂η ∂V + div(c2 u + ηV ) = 0, + (V, ∇)u + u + ωT u + ∇η = 0. (2.5) ∂t ∂t ∂x ∂V /∂x ∂V /∂x Here ∂V /∂x = ∂V12 /∂x11 ∂V12 /∂x22 . The relations ∂ curl3 V − ω + (V, ∇) Ω0 = 0, Ω0 = , ∂t R+D ∂ curl u ∂ η 3 + (u, ∇)Ω − Ω =0 + (V, ∇) + (V, ∇) 0 0 ∂t c2 ∂t c2 also follow from (2.3). The Cauchy problem for (2.5) with localized initial data (1.2) is u01 (x/µ) 0 u = u (x/µ) = , η = η 0 (x/µ), 0 (x/µ) u t=0 t=0 2

(2.6)

(2.7)

where u0j (z), η 0 (z) are smooth functions decaying faster than 1/|z|β as |z| → ∞ with β > 2. We assume also that the k-th derivatives decay faster than 1/|z|β+|k| . 2.2. Effective Hamiltonians (Modes) of the Linearized Shallow Water Equation The matrices H0 (p, x), H1 (x) corresponding to (2.5) are   hV, pi 0 p1 p   0 hV, pi p2 , p = p1 , H0 (p, x) = 2 p1 R p2 R hV, pi   ∂V1 /∂x1 ∂V1 /∂x2 0 0 1 H1 (x) =  ∂V2 /∂x1 ∂V2 /∂x2 0  + ω −1 0 0 0 ∂R/∂x1 ∂R/∂x1 div V u −iµ∂/∂x Write Ψ = η , pˆ = −iµ∇x ≡ −iµ∂/∂x12 .Then Eq.(2.5) can be represented RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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(2.8) 0 0 0

!

.

(2.9)

in the form (1.3). The 2008

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2-D vector p is the momentum, the 2-D vector x is the position, and the pair (p, x) is a 4-D vector in the 4-D phase space R4p,x . Write r = (p, x). The eigenvalues (“effective Hamiltonians” or “modes”) and the corresponding eigenvectors of the matrix H0 and the transposed matrix H0∗ in example (2.5), (2.7) are well known and can be found by means of elementary algebra. This gives three eigenvalues H 0 ,H ± and corresponding † † eigenvectors ξ 0 , ξ 0 and ξ ± , ξ ± , 1 † n(p) n(p) H ± (p, x, t) = hV (x), pi ± |p|c(x, t), ξ ± = , ξ± = , (2.10) ±c(x, t) 2 ±1/c(x, t) n n (p) (p) 0 0 0† ⊥ ⊥ H (p, x) = hV (x), pi, ξ = , ξ = . (2.11) 0 0 p /|p| −p2 /|p| . Here n(p) = p12 /|p| and n⊥ (p) = p1 /|p| It is convenient to introduce the indices σ = 0, +, −. Obviously, H σ coincide at the point p = 0 ′ ′ only, and the vectors ξ σ † , ξ σ † form a biorthogonal system, hξ σ † , ξ σ † i = δσ,σ′ . In this example, at least one of the two objects, the effective Hamiltonian and the corresponding eigenvector, is a nonsmooth function with respect to the momentum p at the point p = 0. 3. REPRESENTATION OF FAST DECAYING FUNCTIONS USING THE MASLOV CANONICAL OPERATOR The most important step in the realization of Diagram 1 is a representation of localized initial data using a generalized canonical Maslov operator. As was noted above, this representation first occurred in the papers [11–13] devoted to the wave equation; however, it does not depend on the application and was described in [15] independently. We briefly recall some formulas from [15] for completeness of our presentation and give useful information on objects treated below. Let f (z), z ∈ R2 , be a smooth function decaying at infinity as 1/|y|β , where β > 2, and let the derivatives ∂f /∂zj decay at infinity as 1/|z|β+1 . Thus, the function f (x/µ) is localized in a neighborhood of the origin x = 0. The central result of this section is the observation that the function f (x/µ) can be presented in the form (1.6) using the Maslov canonical operator on the Lagrangian manifold Λ0 (1.4). 3.1. Some Properties of Λ0 It is clear that Λ0 is a 2-D Lagrangian manifold. Obviously, it is diffeomorphic to 2-D cylinder. We also note that Λ0 can be obtained by shifting the circle S 1 = {p = n0 (ψ), x = 0} along the trajectories of the Hamiltonian system with the Hamiltonian |p|, i.e., [ α Λ0 = g|p| S1. −∞ 0 at the points r 0 (ψ, α) with positive coordinate α. The Maslov index m(r ∗ ) of the distinguished (nonsingular) point r(ψ ∗ , +0) is set to be zero. Since the Jacobian J (1,2) is positive for any positive α, the Maslov index m(r) vanishes at any nonsingular point r(ψ, α) with positive coordinate α. This immediately implies the following proposition. Lemma 2. The Maslov index of any closed path on Λ0 is equal to zero. Proof. It suffices to prove the assertion for any basic cycle on Λ0 . Let us take it in the form {p = n0 (ψ), x = α0 n0 (ψ)|α0 = const > 0, ψ ∈ [0, 2π]}. However, J = J (1,2) 6= 0 on this cycle. Thus, the Maslov index is preserved during the motion along this cycle. This shows that the action function on Λ0 does not depend on ψ and also means that the Bohr–Sommerfeld quantization rule is absent on Λ0 . To find the Maslov index m(r) of nonsingular points r(ψ, α) with negative α, one must consider the matrix C − iεB (see [24, 16]). It is nondegenerate for ε > 0, and we can define a continuous branch of the function Arg Jε , Jε = det(C − iεB)(ψ, α) = α − iε. Choose a branch by setting limε→+0 Arg Jε α=+0 = 0. By Lemma 2, the Maslov index of any nonsingular point r 0 (ψ, α) is (ψ,α) m(r 0 (ψ, α)) = m(r ∗ ) + (1/π) lim Arg Jε ∗ . ε→+0

(3.7)

(ψ ,+0)

Using the explicit formula for Jε , we readily see (directly from (3.7)) that m(r) = Ind r(ψ, α) = −1 for negative α. Using this example, here we present a simple method of evaluating the Maslov index, which is suitable for computer-aided calculations. Let u = Re Jε and v = Im Jε . Then Z (ψ,α) (ψ,α) (ψ,α) udv − vdu Arg Jε (ψ∗ ,+0) = Im{Log Jε } (ψ∗ ,+0) = . (3.8) 2 2 ∗ (ψ ,+0) u + v RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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In the case under consideration, we have u = α and v = −ε, and hence Z (ψ,α) Z +0 (ψ,α) 1 1 εdα 1 dα Ind r(ψ, α) = lim Arg Jε lim lim = = − ε = −1 (3.9) 2 2 2 π ε→+0 π ε→+0 α α + ε2 (ψ∗ ,+0) π ε→+0 (ψ,+0) α + ε

if α < 0, and the Maslov index of the points with negative coordinate α is −1. Relation (3.9) holds because the function ε/(α2 + ε2 ) looks like “smeared” δ-function localized in a neighborhood of the focal point α = 0. The function (3.7)–(3.8) has similar behavior in the general case. A change (a jump) of the Maslov index during the motion along some path γ|r0 →r on Λ0 (connecting the points r 0 and r) is possible only when crossing the focal point. This jump can be only ±1 and 0. At the focal point, we have limε→+0 u = 0 and can usually omit the term udv in the integral (3.8). Thus, the sign of the jump depends on −vdu and the jump of the index is +1 if the sign is positive and −1 if it is negative. The case in which −vdu is 0 needs additional consideration. Anyway, formulas (3.7)–(3.8) are always true and can be realized in computer calculations. (1,0)

(0,2)

Let us now evaluate the indices m(Ωj ) and m(Ωj ) of the singular maps. To this end, we must find the determinants of the following matrices: C12 − iεB12 C11 − iεB11 C(1,0) = ε (C21 − iεB21 ) cos η + (B21 + iεC21 ) sin η (C22 − iεB22 ) cos η + (B22 + iεC22 ) sin η , (C11 − iεB11 ) cos η + (B11 + iεC11 ) sin η (C12 − iεB12 ) cos η + (B12 + iεC12 ) sin η . C(0,2) = ε C21 − iεB21 C22 − iεB22

The general statement claims that the determinants of these matrices are nonzero for ε > 0, any α and ψ, and any η ∈ [0, π/2]. One can readily see that in our situation, for ε = 0, we have (1,0) (0,2) det C0 = α cos η + cos2 ψ sin η and det C0 = α cos η + sin2 ψ sin η. Thus, the above property (1,0) (1,0) (1,0) (0,2) holds for det C0 for any α > 0 and ε = 0 in the mappings Ω1 and Ω3 and for det C0 in (0,2) (0,4) and Ω4 . By [24, 16], the mappings Ω2 η=π/2 η=π/2 1 1 (1,0) (1,0) m(Ωj ) = lim Arg det C(1,0) (ψ, α , η) + m(r(ψ, α )) = 0 + Arg det C = 0. 0 0 ε 0 π ε→0 π η=0 η=0 (0,2)

Similarly, m(Ωj

) = 0.

3.5. Canonical Maslov Operator on Λ0 The objects introduced above enable one to construct the canonical Maslov operator acting on the function f (ψ, ρ). We first define the partial inverse h-Fourier transforms, √ Z ∞ i (x1 , x2 ) = √ χ(p1 , x2 )eip1 x1 /h dp1 , 2πh −∞ √ Z ∞ −h i Fp2 →x2 χ(x1 , p2 ) (x1 , x2 ) = √ χ(x1 , p2 )eip2 x2 /h dp2 . 2πh −∞

Fp−h χ(p1 , x2 ) 1 →x1

Since we have only four mappings, we obtain Ψ(ρ, x1 , x2 , h) = KΛh0 [f ]

(1,0) (ψ,α)+x2 p2 ) exp i(s h p ≡ e f (ρ, ψ)ej (ψ)|ψ=ψj (x1 ,p2 ),α=αj (x1 ,p2 ) |J (1,0) (ψ, α)| j=1,3 (0,2) (ψ,α)+x1 p1 ) X − iπ m(Ω(0,2) ) exp i(s −h h 2 j q + e Fp1 →x1 f (ρ, ψ)ej (ψ) (0,2) ψ=ψj (p1 ,x2 ),α=α(p1 ,x2 ) j=2,4 |Jj (ψ, α)|

X

(1,0)

− iπ 2 m(Ωj

)

Fp−h 2 →x2

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(1,0) √ Z +∞ (ψ,α)+x2 p2 ) exp i(s i h √ p ≡ e f (ρ, ψ)ej (ψ) dp2 (1,0) ψ=ψj (x1 ,p2 ),α=αj (x1 ,p2 ) 2πh |J (ψ, α)| −∞ j=1,3 (0,2) √ Z +∞ (ψ,α)+x1 p1 ) X − iπ m(Ω(0,2) ) exp i(s i h j √ q + e 2 f (ρ, ψ)ej (ψ) dp1 (0,2) ψ=ψj (p1 ,x2 ),α=α(p1 ,x2 ) 2πh −∞ j=2,4 |Jj (ψ, α)| 2 X Z +∞ exp i(α cos hψ+x2 p2 ) i f (ρ, ψ)ej (ψ) ≡√ dp2 | cos ψ| ψ=ψj (x1 ,p2 ),α=αj (x1 ,p2 ) 2πh j=1,3 −∞

X

+

(1,0)

− iπ 2 m(Ωj

X Z

j=2,4

+∞

−∞

)

2 exp i(α sin hψ+x1 p1 ) f (ρ, ψ)ej (ψ) dp1 . | sin ψ| ψ=ψj (p1 ,x2 ),α=α(p1 ,x2 )

(3.10)

Let us replace the variables p2 , p1 in the last integrals by setting p2 = sin ψ and p1 = cos ψ, respectively. With regard to decreasing and increasing of the functions sin ψ and cos ψ in the corresponding mappings and the related signs, we obtain √ X Z 2π i i(α cos2 ψ + x2 sin ψ) √ Ψ(ρ, x1 , x2 , h) = exp f (ρ, ψ)ej (ψ)|α=α(1,0) (x1 ,ψ) dψ h 2πh j=1,3 0 X Z 2π i(α sin2 ψ + x1 cos ψ) + exp f (ρ, ψ)ej (ψ)|α=α(0,2) (x2 ,ψ) dψ h j=2,4 0 √ Z 2π 4 X i i(x1 cos ψ + x2 sin ψ) = (according to (3.4)) = √ exp f (ρ, ψ) ej (ψ)dψ h 2πh 0 j=1 √ Z 2π i(x1 cos ψ + x2 sin ψ) i =√ exp f (ρ, ψ)dψ . (3.11) h 2πh 0 We set h = µ/ρ and use formula (3.6). This gives √ Z 2π iρ Ψ ρ, x1 , x2 , µ/ρ = √ exp (iρhx, n(ψ)i/µ)V˜ (ρn(ψ))dψ (3.12) 2πµ 0 and, finally, √ Z ∞ Z 2π √ Z ∞ i 2πi Ψ ρ, x1 , x2 , µ/ρ dρ = √ exp (iρhx, n(ψ)i/µ)V˜ (ρn(ψ))dψdρ = √ V (x/µ). µ 2πµ 0 0 0 Let us summarize the above consideration. Theorem 1. Let Λ0 be the Lagrangian manifold (1.4) and f (ρ, ψ) be defined in (3.6). Then r Z ∞ µ µ/ρ f (x/µ) = KΛ0 ρ1/2 f˜(ρn0 (ψ))dρ. (3.13) 2πi 0 3.6. Localization of the Source Function and “Lagrangian Bands” The function V (x/µ) is localized in a neighborhood of the origin. Thus, we can use ideas of boundary layer theory [37, 30, 25, 28]. Lemma 3. Let V (z) be a smooth function decaying at infinity together with the first derivatives faster than |z|−2 and |z|−3 , respectively. Let σ(x) be a smooth 2-D vector function vanishing at x = 0 only, let ∂σ/∂x(0) = Q be a nondegenerate 2 × 2 matrix, and let χ(x) be smooth and bounded together with the derivatives. Then σ(x) Qx χ(x)V = χ(0)V + ψ(x, µ), where |ψ(x, µ)| 6 Kµ, K = const . µ µ RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Proof. One has |σ(x)|/µ > Const µ−1/2 outside an O(µ1/2 )-neighborhood of the point x = 0 and, since the function V (z) decays as |z|−2 , it follows that the product χ(x)V (σ(x)/µ) is O(µ) outside this neighborhood. By Taylor’s formula, σ(x) = Qx + σ2 (x) + σ3 (x) + O(µ2 ) in some O(µ1/2 )-neighborhood of x = 0, where the components of 2-D vector functions σk (x) are homogenous polynomials of degree k and χ(x) = χ(0) + hb, xi + O(µ) with b = ∂χ/∂x(0). We now apply Taylor’s formula to the function χ(x)V (σ(x)/µ), use the fact that V decays together with the first derivatives, and obtain σ(x) Qx D ∂V Qx σ (x) + σ (x) E 2 3 = (χ(0) + hb, xi) V + , + O(µ) χ(x)V µ µ ∂z µ µ h D ∂V Ei (Qy), σ2 (y) + µσ3 (y) = χ(0)V (Qy) + µ hb, yi V (Qy) + µ + O(µ), ∂z where the O(µ) estimate is valid in the norm of C(R2x ) and x = µy. Since the function V (z) decays at least as |z|−2 and the derivatives ∂V (z)/∂zj decay as |z|−3 , it follows that the expression in square brackets in the last relation is bounded for all y. This proves the lemma. Remark 1. Two functions χ(x)V (σ(x)/µ) and χ(x)V ˜ (˜ σ (x)/µ) in the above class coincide modulo O(µ) if χ(0) = χ(0), ˜ σ(0) = σ ˜ (0) = 0, and ∂σ/∂x(0) = ∂ σ ˜ /∂x(0). Remark 2. The assertion of the lemma and the preceding remarks remain valid if the functions χ(x) and σ(x) smoothly depend on parameters ranging on a compact set. As was noted above, taking into account this property and the asymptotic character of our considerations, we can truncate the full manifold Λ0 by preserving only its part in a neighborhood of the curve (wave front) Γ0 = {p = n(ψ), x = 0|ψ ∈ S1 }. This means that we can replace the function f (ρ, ψ) with noncompact support on Λ0 by the function f (ρ, ψ)e(|α|), where e is a smooth nonincreasing cut-off function, e(z) = {1 for 0 6 z 6 δ/2 and 0 for z > δ}. (3.14) Lemma 4. Let Λ0 be the Lagrangian manifold (1.4), let f (ρ, ψ) and e(z) be defined in (3.6) and (3.14), and let V (y) decay faster than |y|−2 as |y| → ∞. Then r Z ∞ µ KΛh0 f (ρ, ψ)e(|α|) dρ + ǫ(x, µ); (3.15) f (x/µ) = 2πi 0 h=µ/ρ here ǫ(x, µ) = O(µ2 ) in the norm of C(R2x ). Thus, using the representation (3.15), we can replace the unbounded manifold Λ0 by a Lagrangian manifold with boundary or a “Lagrangian band” Λ0,δ = {p = n0 (ψ), x = n0 (ψ)α | ψ ∈ [0, 2π), −δ < α < δ}. (3.16) Later on, we consider Λ0,δ instead Λ0 ; however, to simplify the notation, we omit the second subscript δ and write Λ0 . Let a(x, p) be a smooth function in R4 growing in p not faster than a polynomial. Lemma 5. The following relation holds: r Z ∞ h i ∂ x µ a x, −iµ f = KΛh0 f (ρ, ψ)a(X0 (αn0 (ψ)), n0 (ψ))e(|α|) dρ + ǫ(x, µ). ∂x µ 2πi 0 h=µ/ρ (3.17) Here a(x, −iµ∂/∂x) is a µ-pseudodifferential operator [26, 35] corresponding to the symbol a, ǫ → 0 as µ → 0 in the norm of C(R2x ) and in the Sobolev spaces Hs (R2x ), s = 1, 2. The ordering of x and −iµ∂/∂x is not essential. 3.7. Helmholtz-Type Decomposition of the Initial Velocity Vector Perturbation into Potential and Solenoidal Parts 3.7.1. Formulas for the decomposition.. The function f˜(k) can be nonsmooth because the function f (z) can decay not very rapidly at infinity. Moreover, even if the initial perturbation in the original problem decays very rapidly, the solution need not preserve this decay. This fact is related to the nonsmooth behavior of the vectors ξ σ of the matrix H0 (p, x) in a neighborhood of RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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the value p = 0 (see (2.10), (2.11)). As applied to the shallow water equation, this fact is related to the Helmholtz decomposition of the velocity vector into the potential and solenoidal parts. Let us treat the role of this decomposition in formulas (1.6), (3.15) in detail. Let the components of 2-D vector function (velocity-vector) u0 (z) satisfy the above conditions for the function f (z). Let u˜0 (k) be the Fourier transform of u0 (z). We have

e e + n⊥ (k)θ(k) = k⊥ k⊥ , u0 (k)k−2 + k k, u0 (k)k−2 , u e0 (k) = n(k)φ(k) (3.18) where

e = hn⊥ (k), u θ(k) e0 (k)i,

e φ(k) = hn(k), u e0 (k)i,

k=

k1 , k2

k⊥ =

−k2 . k1

(3.19)

Taking into account the correspondence k ↔ −i∇z or kj ↔ −i∂/∂zj , we can formally write (the Helmholtz decomposition) u0 (z) = u0v (z) + u0w (z), D ∂ D ∂ 0 E 1 0 E u0v (z) = n⊥ − i n⊥ − i , u (z) = ∇z⊥ ∇z⊥ , u (z) , ∂z ∂z △z D ∂ D ∂ 0 E 1 0 E n −i , u (z) = ∇z ∇z , u (z) . u0w (z) = n − i ∂z ∂z △z Using formulas (3.20) and (3.18), we can write, at least formally, x x x u0 = u0v + u0w , µ µ µ r µ Z ∞ µ/ρ 0 x ˜ 0 (ψ))n0 (ψ)dρ, uv KΛ0 ρ1/2 θ(ρn = ⊥ µ 2πi 0 r Z ∞ x µ µ/ρ 0 ˜ u0w = KΛ0 ρ1/2 φ(ρn (ψ))n0 (ψ)dρ. µ 2πi 0

(3.20) (3.21) (3.22)

(3.23) (3.24) (3.25)

3.7.2. Behavior at infinity. Let us discuss the usage of the last formulas. The vector function u e0 (k) belongs to the class of continuous functions decaying at infinity, whereas u0 (z) is a smooth function decaying faster than 1/|z|β , β > 2. The decomposition (3.18) is unique in this class. However, in formula (3.20), we see the inversion of the Laplace operator, which is not unique in general and can lead to functions with different decay as |z| → ∞ with respect to the vector function u0 (z). This fact is known in fluid dynamics [20, 38], and we discuss it here not very rigorously, giving elementary “physical” explanations clarifying the behavior of u0w (z) and u0v (z) at infinity. Let r, ϕ be polar coordinates on the 2-D plane R2 . We have Z 2π ∞ X 1 0 ikϕ u (z) = w0 (r) + wk e , wk (r) = u0 (rn0 )e−ikϕ dϕ. 2π 0 k=1

0 Obviously, the functions wk (r) decay at infinity at least as r −β . To find △−1 z u (z), we must solve the equations 1 ∂ ∂ k2 r − 2 Wk = wk (r). r ∂r ∂r r The solutions to these equations with bounded derivatives on the semi-axes r ∈ [0, ∞) for k 6= 0 are unique and can be presented in the form Z ∞ Z r′ Wk (r) = −r |k| (r ′ )−2|k|−1 (r ′′ )|k|+1 wk (r ′′ )dr ′′ , k 6= 0. r

0

Note that we can put

Z 1 ∆ u (z) = ln |z − z ′ |u0 (z ′ )dz ′ 2π R2 in (3.21), (3.22). The Fourier expansion with respect to the polar angle ϕ leads mod const to the same functions Wk (r). −1 0


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One can readily see that the functions Wk (r) are smooth and decay at infinity as r 2−β and, similarly to u0 (z), ∂2 ∂3 (Wk (r)e−ikϕ ) = r −β , (Wk (r)e−ikϕ ) = r −β−1 . (3.26) ∂zj ∂zk ∂zj ∂zk ∂zl A bounded solution at k = 0 need not exist in general. However, the solution with bounded derivatives on the semi-axes r ∈ [0, ∞) for k = 0 is unique up to constant and can be written as Z r Z r′ ′ −1 W0 = r r ′′ w0 (r ′′ )dr ′′ . 0

0

This solution is bounded for any finite r and W0 = M log r + O(1),

M=

Z

∞

r ′ w0 (r ′ )dr, 0

It is also clear that 1 1 ∂2 ∂2 W0 (r) = M log r + O β = O 2 , ∂zj ∂zk ∂zj ∂zk r r

as

r → ∞.

1 ∂2 W0 (r) = O 3 . ∂zj ∂zk ∂zl r

(3.27)

Summing the functions (∂ 2 /∂zj ∂zk )(Wk (r)e−ikϕ ) and applying estimates (3.26)–(3.27), we obtain u0v (z) = O(|z|−2 ), u0w (z) = O(|z|−2 ), (∂u0v /∂zj )(z) = O(|z|−3 ), (∂u0w /∂zj )(z) = O(|z|−3 ) as |z| → ∞. (3.28) Thus, the assertions in the previous subsections hold for the functions (3.24) and (3.25). 4. SOLUTION USING THE GENERALIZED MASLOV CANONICAL OPERATOR 4.1. Hamiltonian Systems, Phase Flows, and the Manifolds Λσt σ In the phase space R2n p,x , the effective Hamiltonians (modes) H generate the Hamiltonian systems p˙ = −Hpσ , x˙ = Hxσ . (4.1)

t 2n In turn, these Hamiltonian systems define the phase flows (canonical transforms) gH σ in Rp,x , t 0 0 2n 0 0 0 0 namely, gH σ transfers a point (p , x ) in Rp,x to the point (p(p , x , t), x(p , x , t)), where the pair (p(p0 , x0 , t), x(p0 , x0 , t)) is the solution of (4.1) satisfying the Cauchy condition p t=0 = p0 , x t=0 = x0 . Under our assumptions (i)–(ii), the phase flows are well defined everywhere except for a neighborhood of the point p = 0. Let us choose p0 = n0 (ψ), x0 = αn0 (ψ) and denote by (P σ (ψ, α, t), X σ (ψ, α, t)) the solutions to system (4.1) for the Cauchy problem p = n0 (ψ), x = αn0 (ψ). (4.2) t=0

t=0

Along with the Hamiltonian system (4.1), consider the linear variational system σ σ σ σ w˙ = −Hxp w − Hxx z, z˙ = Hpp w + Hpx z, σ σ Hxp , Hxx ,

σ

(4.3)

σ

where the arguments of the derivatives etc., are (P (ψ, α, t), X (ψ, α, t), t). As is well known, the vector functions ∂P σ ∂P σ ∂X σ ∂X σ (ψ, α, t), (ψ, α, t) and Pψσ , Xψσ = (ψ, α, t), (ψ, α, t) Pασ , Xασ = ∂α ∂α ∂ψ ∂α form a solution to system (4.3). The Hamiltonians H σ are first-order homogenous functions. Thus, σ σ hHpσ , pi = H σ , which implies the relations Hpp p = 0 and Hxp p = Hxσ . Using these relations, we immediately see that the vector function P σ (ψ, α, t) is also a solution to system (4.3). Lemma 6. P σ (ψ, α, t) 6= 0 for any finite time t.

Proof. Assume the contrary, namely, let P σ (ψ0 , α0 , t0 ) = 0 for some ψ0 , α0 , t 0 . Then the solution (w, z) = (P σ , 0) to the linear system (4.3) satisfies the initial condition (w, z) t=t = 0. Hence, it 0 vanishes for any time t, which fails at t = 0. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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By this lemma, the functions (P, X σ ) smoothly depend on the parameters ψ and α. For any chosen t, these functions define Lagrangian manifolds (bands), Λσt = p = P σ (ψ, α, t), x = X σ (ψ, α, t) | ψ ∈ [0, 2π), |α| < δ . (4.4)

One can say that the manifolds Λσt are obtained by shifting the manifold Λ0 by using the phase t σ t t 0 flows gH σ . The parameters ψ and α are coordinates on Λt . Write rσ = gH σ r We can introduce matrices BσI , CσI and Jacobians JσI on Λσt similarly to (3.1); however, we actually (1,2) (1,2) need Bσ , Cσ only, ∂P1σ /∂α ∂P1σ /∂ψ ∂X1σ /∂α ∂X1σ /∂ψ (1,2) (1,2) , Cσ (ψ, α, t) = Cσ = , Bσ (ψ, α, t) = Bσ = ∂P2σ /∂α ∂P2σ /∂ψ ∂X2σ /∂α ∂X2σ /∂ψ ∂X1σ /∂α ∂X1σ /∂ψ ∂P1σ /∂α ∂P1σ /∂ψ (1,0) (0,1) Cσ = , Cσ = . ∂P2σ /∂α ∂P2σ /∂ψ ∂X2σ /∂α ∂X2σ /∂ψ (1,2)

= det(Xασ , Xψσ ). Recall that, as above, a point Write JσI = det CσI and Jσ = det Cσ ≡ det Cσ (P σ (ψ, α, t), X σ (ψ, α, t)) on Λσt is nonsingular if Jσ (ψ, α, t) 6= 0. Otherwise it is singular or focal. The special structure of the manifold Λσt enables one to evaluate the Jacobian Jσ . Recall that (4.3) is a linear Hamiltonian system, and thus the skew-inner product hw1 , z 2 i − hw2 , z 1 i of two any solutions (w1 , z 1 ), (w2 , z 2 ) to system (4.3) is constant in time. Hence hXασ , P σ i = hn0 (ψ), n0 (ψ)i = 1,

hXψσ , P σ i = hαn0ψ (ψ), n0 (ψ)i = 0,

and

Xασ 6= 0.

Lemma 7. |Jσ | = |Xψσ |/|P σ |. Proof. Assume that Xψσ 6= 0. We have Xασ = hXασ , P σ iP σ |P σ |−2 + hXασ , Xψσ iXψσ |Xψσ |−2 = P σ |P σ |−2 + hXασ , Xψσ iXψσ |Xψσ |−2 .

Therefore

Jσ = det(P σ |P σ |−2 + hXασ , Xψσ iXψσ |Xψσ |−2 , Xψσ ) det(P σ |P σ |−2 , Xψσ ) = |P σ |−2 det(P σ , Xψσ )

and (according to the orthogonality of P σ and Xψσ ), and also Jσ = 0 if Xψσ = 0.

Jσ = |Xψσ |/|P σ | sign det(P σ , Xψσ ),

Corollary. Jσ = 0 if and only if

∂X σ /∂ψ = 0. This means that the focal points on Λσt are defined by the last equation. t

(4.5) (0,2)

Lemma 8. Let rσf (ψf , αf ) be a focal point. In this case, either det Cσ (1,0) det Cσ (ψf , αf , tf ) 6= 0. (0,2)

(ψf , αf , tf ) 6= 0 or

(1,0)

Proof. Assume that det Cσ (ψf , αf , tf ) = det Cσ (ψf , αf , tf ) = 0. Since Xψσ (ψf , αf , tf ) = 0, σ σ σ σ we have X1α P2ψ = X2α P1ψ = 0 at the point. We also have hXασ , Pψσ , i = 0 at the focal point due to Lagrangian property of Λσt . Thus, the vector Pψσ is normal to the nonzero vectors Xασ and (Xασ )⊥ . It follows that Pψσ = 0. This, together with Xψσ = 0, contradicts the relation dim Λσt = 2. Recall that s(ψ, α, 0) = α on Λ0 . Lemma 9. The action s(ψ, α, t) on Λσt is equal to α. Proof. Since the Hamiltonians H σ are first-order functions, for Lagrangian we have Lσ = hHpσ , pi − H σ = 0.

Thus, the increment of the action along any trajectory (P σ , X σ ) from t = 0 until t is equal to zero. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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4.2. Maslov Index We need the Maslov index of nonsingular points rσt (ψ, α) = (P σ (ψ, α, t), X σ (ψ, α, t)) and singular mappings. Due to general properties of the Maslov index and properties of the initial manifold Λ0 , we can use a formula similar to (3.7) for the calculation of the Maslov index. Namely, introduce the matrix Cσ − iεBσ and the function Jσε (ψ, α, t) = det(Cσ − iεBσ ), where ε is a small positive number. We have Jεσ 6= 0, and thus there is a continuous branch of Arg Jσε . Recall that we choose a branch by the condition Arg Jεσ r ∗ )|ε=0 = 0, r ∗ = r 0 (ψ ∗ , +0). Write r(ψ,α,t) r(ψ,α,t) 1 1 lim Arg Jεσ ∗ lim Arg Jεσ Ind rσt (ψ, α) = ≡ . π ε→+0 π ε→+0 r r(ψ,α,0) Consider the points with positive α. Then we can write r(ψ,α,t) r0 (ψ,α) 1 rt (ψ,α) 1 rt (ψ,α) 1 ε ε σ ε σ ε σ lim Arg Jσ ∗ = lim Arg Jσ ∗ + lim Arg Jσ 0 = lim Arg Jσ 0 . (4.6) ε→+0 π ε→+0 π ε→+0 r r rσ (ψ,α)) π ε→+0 rσ (ψ,α) Thus, for the problems in question, the calculation of the Maslov index is related to focal points crossed by the trajectory (P(ψ, α, t), X (ψ, α, t)). One can find the Maslov index of singular mappings by using the ideas presented in Subsection 3.4. The realization of this idea, as well as that of (4.6), depends on a concrete situation, and we shall treat it below. 4.3. Transport Equations for the Amplitudes 4.3.1. Nondegenerate case. Consider first the equations for the amplitudes by assuming that the effective Hamiltonians H σ are nondegenerate and ξ σ , ξ σ† are the eigenvectors of H0 , H0∗ (see above). The Lagrangian manifolds Λσt , the effective Hamiltonians H σ , and the perturbation matrix H1 induce also the transport equations for the amplitudes Aσ (ρ, ψ, α, t), 1 σ dAσ /dtσ + Gσ σ Aσ = 0, Gσ = iLσ1 − tr Hpx . (4.7) 2 Λt Here d/dtσ stands for the derivative along the trajectories of system (4.1), 2 D X ∂H σ ∂ξ σ ∂ξ σ E ∂H0 ∂ξ σ Lσ1 (p, x, t) = −i ξ σ † , − , (4.8) + H1 ξ σ + ∂pj ∂xj ∂xj ∂pj ∂t j=1

and the symbol |Λσt means that G(p, x, t) is restricted to Λσt with p = P σ (ψ, α, t), x = X σ (ψ, α, t). The initial value Aσ0 for Aσ is the projection of the (initial) vector function V˜ (ρn0 (ψ))e(|α|) to the D E vector ξ σ |Λ0 , Aσ0 = ξ σ† Λ ,t=0 , V˜ (ρn0 (ψ)) e(|α|). (4.9) 0

The role of the factor e(|α|) is auxiliary. It shows that only small values of α are of importance. Obviously, the transport equations canZbe integrated if the Hamiltonians are nondegenerate, t n o Aσ = Aσ0 exp − G P σ (ψ, α, τ ), X σ (ψ, α, τ ), τ dτ . (4.10) 0

The last integral can often be found explicitly in physically reasonable cases, see the example below.

4.3.2. Degenerate case. Let us return to the general situation. As was said above, we assume that the effective Hamiltonians H σ do not coincide for any p, x with p 6= 0. However, some of them can be degenerate with multiplicity s(σ). In this case, we assume that a basis in the corresponding eigensubspace of the matrices H0 and H0 ∗ contains s(σ) vectors ξkσ (p, x, t) and biorthogonal vectors ξ σ † (p, x, t), k = 1, . . . , s(σ). We can arrange these vectors in the matrices ξ σ = (ξ1σ , . . . , ξ s(σ) ) and † ξ σ † = (ξ1σ † , . . . , ξ s(σ) ), and by the inner product hA, Bi of two s(σ) × k matrices A and B we mean the product A∗ B. By Aσ we mean an s(σ)-dimensional vector. With these changes, appropriate solutions of (4.1)–(4.8) define the asymptotics of the original problem (1.1), (1.2) as above; however, (4.7) is now an s(σ)-dimensional system of differential equations, whereas formula (4.9) defines the initial s(σ)-dimensional vector. σ 4.3.3. Decomposition of initial data. The definition of A0 is simply the decomposition of 0 σ the vector V˜ (ρn ) with respect to the basis {ξ Λ0 ,t=0 }, X V˜ (ρn0 ) = ξσ Aσ0 . (4.11) σ

Λ0 ,t=0


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4.4. Hamiltonian Systems and Solutions of the Transport Equations for the Linearized Shallow Water Equations 4.4.1. Hamiltonian systems. The effective Hamiltonians in this case are (2.10), (2.11), and thus the Hamiltonian systems are of the following form: if σ = ±, then p˙ = −t Vx p ∓ |p|∇c, x˙ = V ± (p/|p|)c (4.12) and, for σ = 0, t x˙ = V. (4.13) p˙ = − Vx p,

± 4.4.2. Explicit formulas for L± 1 and G . Note first that, for any σ, D ∂ξ σ E 1 D ∂n E ∂ n, hn, ni = 0. ξσ †, = = ∂pj 2 ∂pj ∂pj Consider the cases σ = ±. Elementary manipulations give

D

2 X ∂H0 ∂ξ σ E

(4.14)

1 1 1 † ± h∇c, ni + hV, ∇ci , hξ ± , H1 ξ ± i = (hn, Vx ni ± 2h∇c, ni + div V ), ∂pj ∂xj 2 c 2 j=1 D ±E 2 ± ∂ξ 1 ∂c ∂ H † ξ± , = , tr = divV ± h∇c, ni. ∂t 2c ∂t ∂p∂x Hence i 1 ∂c ∂2H ± ± + hV, ∇ci ± chn, ∇ci + hn, Vx ni ± hn, ∇ci + tr . (4.15) L1 (p, x, t) = − 2 c ∂t ∂p∂x Consider the Hamiltonian systems (4.12) and denote the derivatives along the trajectories of systems (4.12) by d/dt± . Taking the inner product of the first equation in (4.12) by p and dividing the result by |p|2 , we obtain 1 d|p|2 d = −(hn, Vx ni ± hn, ∇ci) ⇐⇒ log |p| = − hn, Vx ni ± hn, ∇ci . 2 ± ± 2|p| dt dt Let us now find the derivative of the function c(x, t) along the trajectories (4.12), dc ∂c ∂c = + hx, ˙ ∇ci = + hV, ∇ci ± chn, ∇ci. (4.16) ± dt ∂t ∂t Combining these equalities gives the relations c c i d 1 d ± L± = − log + tr H and G = log . px 1 2 dt± |p| 2 dt± |p| We can now integrate (4.7), p A± = A± |p|c(αn0 (ψ), 0)/c(x, t), (4.17) 0 ±†

ξ ,

=

where A± are constant on the trajectories of systems (4.12)). Here 0 are the initial amplitudes (that we used the relations |p| Λ0 = 1 and c Λ0 ,t=0 = c(αn0 (ψ), 0). We can now readily find the initial amplitudes A± 0, A± 0

1e η˜0 (k) = φ(k) ± , 2 2c(αn0 (ψ), 0)

0

n (ψ) ≡ n(k) =

e where φ(k) is defined in (3.19).

cos ψ sin ψ

,

(4.18)

4.4.3. Explicit formulas for L01 and G0 . Consider now the case σ = 0. Obviously, 1 L01 = −ihξ σ ∗ , H1 ξ 0 i = −ihn⊥ , Vx n⊥ i = −i(div V − hn, Vx ni), G0 = div V − hn, Vx ni. 2 The first equation in (4.13) gives 1 d d log |p| 1 d |p|2 = −hp, Vx pi ⇐⇒ |p| ≡ = −hn, Vx ni. 0 0 2 dt |p| dt dt0 Moreover, (2.4) yields 1 ∂R 1 dR d log R d log c div V = − + hV, ∇Ri = − =− = −2 . R ∂t R dt0 dt0 dt0 RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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These relations imply

207

d |p| log . 0 dt c σ Substituting L1 into (4.7) and integrating gives 1 c A0 = A00 , (4.19) 0 c(αn (ψ), 0) |p| e e where A00 = θ(k) is constant on the trajectories of system (4.13) and θ(k) is defined k=ρn0 (ψ) in (3.19). G0 =

σ and ξ σ nor Lσ1 depend on the Coriolis frequency ω. Remark 3. Let us stress that neither Heff Thus, the Coriolis force does not influence on the leading term of the asymptotics under assumption that ω = O(1) as µ ≪ 1. 4.5. Asymptotic Formulas for the Solution Now we can state an important intermediate result of the paper. We now deal with the 2-D case. t In the 4-D phase space R4p,x , we need 1) the 2-D Lagrangian manifolds Λσt = gH σ Λ0 (4.4) obtained by the shifts of the manifold (3.16) along the trajectories of the Hamiltonian systems (4.1), 2) the set (arranged into matrices in the degenerate cases) of eigenvectors ξ σ (p, x, t) of the matrix H0 , and 3) the amplitudes Aσ , which are the solutions to the Cauchy problems (4.7)–(4.9). Let KΛhσ be t the Maslov canonical operator on the families of Lagrangian manifolds Λσt . Set h = µ/ρ and write r Z ∞ X √ µ √ µ/ρ σ σ σ σ Ψas , Ψas = ρKΛσ A ξ dρ, i = eiπ/4 . (4.20) Ψas = t 2πi 0 σ

Theorem 2. Under assumptions (i)–(iii), the representation √ Ψ = Ψas + o( µ) (4.21) √ holds for the solution to the Cauchy problem (2.5), (2.7); here o( µ) denotes estimate in the C-norm. For the solution to the Cauchy problem (2.5), (2.7) in the case of linearized shallow water equa tions, we must set u uas as Ψ = η , Ψ = η as . (4.22) Remark 4. The sum in formulas (4.20), (4.21) reflects the reasonable (and well known) decomposition of the full asymptotic solution into different modes for each time t; making this once at t = 0, we have it asymptotically always. Remark 5. Let us mention an important property of the canonical Maslov operator [23]. We mean its local invariance with respect to changes of variables in the configuration space. Namely, consider a neighborhood of the point x. Assume that new coordinates x′ are introduced and x = x(x′ ) is a diffeomorphism in this neighborhood which preserves the orientation. Then the (same) solution can be represented in the form of the canonical operator in the new coordinates x′ p by including the factor 1/ det(∂x/∂x′ ).

Proof. Let us first regularize formulas for the solutions to avoid singularities in the integral as ρ → 0. Consider the cut-off function χ(τ ) which is smooth on the semiaxis [0, ∞], vanishes in the vicinity of 0, and is equal to 1 in the vicinity of ∞. We represent the initial conditions (1.2) in the form (3.14), x r µ Z ∞ µ/ρ Ψ0 = KΛ0 ρ1/2 f˜(ρn0 (ψ))dρ = Ψ01 + Ψ02 , µ 2πi 0 with the right-hand side r µ Z ∞ ρ µ/ρ 0 x Ψ1 = χ KΛ0 ρ1/2 f˜(ρn0 (ψ))dρ, µ 2πi 0 µ r µ Z ∞ ρ µ/ρ 0 x Ψ2 = 1−χ KΛ0 ρ1/2 f˜(ρn0 (ψ))dρ. µ 2πi 0 µ Let us begin with estimating the function Ψ20 .


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Lemma 10. The estimate kΨ02 ks = O(µ2−s ) holds, where k ◦ ks stands for the norm in the Sobolev space H s . Proof. It suffices to prove the assertion for s = 0. We shall prove the stronger estimate |(Ψ02 , w)| 6 Ckwk0 µ2 for arbitrary w ∈ C0∞ (to obtain the assertion of the lemma, we can then set w = Ψ02 ). Here ( , ) stands for the inner product in L2 . Note that the function in question can be represented in the form Z ∞ √ ρ 1 µ/ρ u Ψ02 = µ v(x, ρ, µ)dρ, where u(τ ) = τ (1 − χ(τ )) and v = √ KΛ0 f˜(ρn0 (ψ)). µ 2πi 0 Z Z ∞ ρ We have (Ψ02 , w) = µ dx dρ u v(x, ρ, µ)w(x) µ R2 0 Z ∞ Z ∞ Z ρ ρ v(x, ρ, µ)w(x) = µ (w, v)dρ. =µ dρ dx u u µ µ 0 R2 0 Z ∞ By the Schwartz inequality, ρ |(Ψ02 , w)| 6 µ||w||0 u ||v||0 dρ. µ 0 Since the Maslov canonical operator is unitary, we have kvk0 6 C1 and Z ∞ ρ 0 |(Ψ2 , w)| 6 C1 µ||w||0 u dρ = Cµ2 ||w||0 . µ 0 √ Remark 6. Using the Sobolev embedding theorem, we obtain |Ψ2 | = o( µ) as µ → 0. Remark 7. The resolving operator for the Cauchy problem for the strictly hyperbolic system (2.5) is bounded in arbitrary Sobolev space (see, e.g., [19]). Thus, the solution Ψ2 of the Cauchy √ problem with the initial function Ψ02 satisfies the estimates kΨ2 ks = O(µ2−s ) and |Ψ2 | = o( µ). Now consider the Cauchy problem (2.5)–(2.7) with the initial function Ψ01 . We claim that the main term of the asymptotic solution of the problem with the function Ψ1,as , where r coincides Z ∞ X µ √ ρ µ/ρ σ σ Ψ1,as = Ψσ1,as , and Ψσ1,as = ρχ KΛσ A ξ dρ. t 2πi µ 0 σ Namely, the following assertion holds. Lemma 11. The solution Ψ1 √ of the Cauchy problem with the initial function Ψ01 is of the form Ψ1 = Ψ1,as + ψ1 , where |ψ1 | = o( µ) as µ → 0. Proof. By the definition of the function Ψ1,as , this function satisfies (2.5) up to “O(µ/ρ);” namely, it follows immediately from the commutation formula for the Maslov canonical operator Z ∞ ∂ ([23, 26]) that µ ρ µ/ρ √ √ ˆ + H Ψ1,as = µ χ KΛσ (y) ρdρ, t ∂t ρ µ 0 ˆ is the operator given in (2.5) and y is a function on Λt depending on ρ and rapidly where H decaying as ρ → ∞. Using the same construction, we can obtain corrections to Ψ1,as such that the corrected function satisfies (2.5) up to an arbitrary power of µ/ρ. Let us use the second-order correction, namely, the function U with the following properties: Z ∂ ∞ 2 µ ρ √ √ µ/ρ ˆ U = F = √µ U = Ψ1,as + U1 , |U1 | = o( µ), +H χ KΛσ (z) ρdρ, 2 t ∂t ρ µ 0 where the properties of z are similar to those of y. Now let us estimate the function F . Since the Z ∞ Maslov operator is unitary, we obtain ρ s/2 5/2−s ||(−∆) F ||0 = µ ρs−3/2 χ κ(ρ)dρ, µ 0 where κ = kz|p|s k (in the last formula, we use the L2 -norm on Λσt ). Now we have Z 1 Z ∞ ||(−∆)s/2 F ||0 = µ5/2−s κdρ + κ(ρ)ρs−3/2 χ(ρ/µ)dρ = O(µ5/2−s + µ2 ). 1

0

This estimate yields kF k3/2 = O(µ). Using the boundedness of the resolving operator, we obtain the same estimate for Ψ1 − U ; finally, the Sobolev theorem implies |Ψ1 − U | = O(µ). RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Let us now complete the proof of the theorem. Using the last two lemmas, we obtain the equalities √ √ √ Ψ = Ψ1 + Ψ2 = Ψ1 + o( µ) = Ψ1,as + o( µ) = Ψas + o( µ) (the last relation can be verified similarly to Lemma 10). 5. SIMPLIFICATION OF ASYMPTOTIC SOLUTIONS 5.1. General Remarks: Wavefronts in the Phase and Configuration Spaces Formulas (4.20), (4.21) show that, due to general properties of continuous media equations, the original problem is divided into some terms corresponding to different Hamiltonians or modes H σ . The behavior of the corresponding terms can be very different. For instance, on the plane R2x , in the case of linearized shallow water equations (see below), the part of the solution corresponding to the term H 0 is localized in a neighborhood of some point, whereas the parts corresponding to the terms H ± are localized (in some time) in neighborhoods of curves, which are known as fronts. Thus, in the case of H ± , one has the phenomenon of profile metamorphosis. Let us make the corresponding simplification and discuss the above behavior in detail for the linearized shallow water equations in the next sections. We begin with general remarks. Consider the solution r Z ∞ √ µ √ µ/ρ σ σ σ Ψas = ρKΛσ A ξ dρ, i = eiπ/4 , (5.1) t 2πi 0 in a neighborhood of some nonsingular point x = X(ψ ′ , α′ , t′ ). By the definition of the Maslov canonical operator, formula (5.1) becomes s r Z µ |P σ (ψ, α, t)| −i π Ind(r′ ) σ ∞√ i µρ α σ† ˜ σ 0 2 Ψas = e ξ ρe hξ , V (ρn (ψ))ie(|α|)dρ , 0 2πi |Xψσ (ψ, α, t)| α=α(x,t),ψ(x,t) (5.2) 0 where ξ0σ† (ψ, α) = ξ σ† |Λ0 ,t=0 and ψ(x, t), α(x, t) is the solution to the system X σ (ψ, α, t) = x. ′

′

′

′

(5.3)

′

Lemma 12. Let x = X(ψ , α , t ) be a nonsingular point and let |α | > δ > 0. Then in this neighborhood, Ψσas = O(µ2 ). (5.4) Proof. Let us change the variable in the integral by setting ρ = z 2 , integrate by parts, and apply the stationary phase method. This immediately gives (5.4). Now consider a focal point x∗ = X(ψ ∗ , α∗ , t∗ ), where we have |α∗ | > δ > 0. In that case, (1,0) (0,2) (∂X/∂ψ)(ψ ∗ , α∗ , t∗ ) = 0, and one of the matrices Cσ and Cσ is nondegenerate. Assume that (1,0) (1,0) Jσ = det Cσ 6= 0. In this case, the part of asymptotic solution in a neighborhood of x∗ is defined by the formula r s Z ∞Z ∞ n ρ I ∗ 1 1 σ∗ −i π Ind (r ) Ψas = e 2 ρei µ (α−P2 (ψ,α,t)X2 (ψ,α,t)+p2 x2 ) ξ σ (1,0) 2π | det Cσ | 0 −∞ o × hξ0σ† , V˜ (ρn0 (ψ))ie(1,0) (ψ, α) dp2 dρ, (5.5) (1,0) (1,0) α=α

(x1 ,p2 ,t),ψ=ψ

(x1 ,p2 ,t)

where the support of the cut-off function belongs to some neighborhood of the point (ψ ∗ , α∗ ) and α = α(1,0) (x1 , p2 , t), ψ = ψ (1,0) (x1 , p2 , t) is the solution to the system X1 (ψ, α, t) = x1 , P2 (ψ, α, t) = p2 . (5.6) Lemma 13. If |α∗ | > δ > 0, then

3/2 Ψσ∗ ). as = O(µ

(5.7)

Proof. We use the stationary phase method. The phase in the integral (5.5) is given by Φ = ρ(α − P2 (ψ, α, t)X2 (ψ, α, t) + p2 x2 ) = ρ(α + p2 (x2 − X2 (ψ, α, t))), where (ψ, α) is the solution to (5.6). According to properties of Lagrangian manifolds (and principles of classical mechanics), ∂Φ/∂p2 = ρ(x2 − X2 (ψ, α, t)), and then ∂Φ/∂ρ = α + p2 (x2 − X2 (ψ, α, t)), ∂ 2 Φ/∂ρ2 = 0, and RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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∂ 2 Φ/∂ρ∂p2 = x2 − X2 (ψ, α, t). The equations for stationary points are X2 (ψ, α, t) = x2 , α = 0, and ρ = 0, X2 (ψ, α, t) = x2 , α + p2 (x2 − X2 (ψ, α, t)) = 0. The first case cannot be realized by the assumption of the lemma. In the second case, the stationary point is nondegenerate because det Hess Φ = (∂ 2 Φ/∂ρ∂p2 )2 = (x2 − X2 (ψ, α, t))2 = α2 at the stationary point, and by the assumption of the lemma again, the determinant is nonzero on the support of the function e(1,0) (ψ, α). Using now the stationary phase method (with the stationary point on the boundary) and at the same time taking into account the factor ρ in the integral, we obtain estimate (5.7). √ Later on, we shall show that the functions Ψσas are at least O( µ) at the points corresponding to α = 0. Thus, the functions Ψσas are localized in a neighborhood of a set formed by the projections of closed curves Γσt = {p = P σ (t, ψ), x = X σ (t, ψ)} in the phase space R4p,x into the configuration space R2x . Here P σ = P|α=0 , X σ = X |α=0 is the 1-D family of solutions to (4.1) satisfying the initial conditions = n0 (ψ), x = 0. (5.8) p t=0

Γσt

t=0

R4p,x

For each fixed t, the curves in the phase space are always smooth and closed. These curves are known as the wave fronts in the phase space. Their projections γtσ = {(x = X(t, ψ)} to the configuration space R2x , on the contrary, can be nonsmooth closed curves or even points (which holds at t = 0 for all cases and for all times in our example, whenever σ = 0). Note that this fact leads to the great difference between solutions corresponding to the modes H 0 and H ± in the example of shallow water equations. The curves γtσ = {(x = X(t, ψ)} are referred to as the wave fronts in the configuration space. In all cases, it is possible to simplify the resulting terms in formula (4.20) by using a localization property and ideas used in boundary layer expansions. This simplification gives the opportunity for applications in real geophysical problems. Now let us discuss some simplifications and the behavior of asymptotic solutions for the shallow water equations. 5.2. Slow or Vortical Mode on the Shallow Water

We start from the mode H 0 . It is often called the “slow” mode or the vortical mode. This part of the solution was first studied with the help of another method in [14]. Let X(t) ∈ R2x be the trajectory of the “basic” vector field V (x, t) starting from the point x = 0, i.e., the solution to the problem x˙ = V (x, t), x t=0 = 0. (5.9) Obviously, (5.9) is the first part of system (4.13) for solutions with zero momentum p. The trajectory X(t) and (5.9) imply the variational system for the 2-D vector function z, z˙ = Vx (X(t), t)z. (5.10) Denote by Z(t) the Cauchy matrix of (4.3), i.e., the matrix solution satisfying the initial condition Z t=0 = E, (5.11) where E is the 2 × 2 identity matrix. Denote by P (t) the momentum along the trajectory X(t) with P (0) = k; according to (4.1), for σ = 0, we have P˙ = −(∂V /∂x)∗ P. (5.12) Equation (5.12) is known as the “conjugate” system to (4.3), and its Cauchy matrix can be represented in the form (Z ∗ )−1 (t). Thus, P (t) = (Z ∗ )−1 (t)k. Theorem 3. The term in (4.20) corresponding to σ = 0 is of the form r Z ∞ µ c2 (X(t), t) Z(t)−1 (x − X(t)) √ µ/ρ 0 as as η0 = O(µ), u0 = ρKΛ0 A n⊥ dρ = g + O(µ) t 2πi 0 c2 (0, 0) µ Z(t)−1 (x − X(t)) c2 (X(t), t) ≡µ 2 ∇⊥ G + O(µ), (5.13) c (0, 0) µ where Z Z 1 |k| 0 1 1 0 ihk,yi g(y, t) = n⊥ (P ) he u (k), n⊥ (k)ie dk = P⊥ 2 he u (k), k⊥ ieihk,yi dk, 2π R2 |P | 2π R2 P Z 1 1 ^0 G(y, t) = (curl3 u )(k)eihk,yi dk. (5.14) 2π R2 P 2 RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Proof. Let us first study the structure of the Lagrangian manifold Λ0t . Denote by X0 (y, t) the family of solutions to the Cauchy problem x˙ = V (x, t), x t=0 = y ∈ R2 . (5.15) For any given t, the mapping

x = X0 (y, t) (5.16) 2 2 0 implies a diffeomorphism R → R . The diffeomorphism X (t, y) generates a linear transformation defined by the 2 × 2 matrix Z(t, y) = (∂X0 /∂y)(t, y), which is the Cauchy matrix of the variational system for Z˙ = Vx (X0 (y, t), t)Z, Z|t=0 = E. (5.17) By the Liouville theorem, Z t div V (Xσ (y, t), t)dt = c2 (Xσ (y, t), t)/c2 (y, 0) 6= 0. det Z = exp 0

In turn, the transformation Z(y, t) induces a linear transformation of the momentum,

p = P0 (y, t, q) = (Z∗ )−1 (y, t)q. The vector function Pσ (y, t, q) is the solution to the Cauchy problem (5.18) p˙ = −(∂V /∂x)∗ (Xσ (y, t), t)p, p t=0 = q. 0 0 4 The pair P (y, t, q), X (y, t) defines a canonical transform in the 4-D phase space R with the coordinates (p, x). Setting y = αn0 (ψ) and q = αn0 (ψ), we obtain (Z∗ )−1 (αn0 (ψ), t)n0 (ψ) = P 0 (ψ, α, t), X0 (αn0 (ψ), t) = X 0 (ψ, α, t). Thus, the Lagrangian manifold Λ0t corresponding to the slow (vortical) mode is given by Λ0t = {p = (Z∗ )−1 (αn0 (ψ), t)n0 (ψ),

(5.19)

x = Xσ (αn0 (ψ), t)}.

Let us change the independent variables in formula (5.13) and pass from the variables x to variables y by means of formula (5.13). The canonical Maslov p operator is invariant with respect to such a change of variables if we introduce the factor 1/ det(∂x/∂y), see Remark 5. In the coordinates y, the Lagrangian manifold has the form Λ0t = {q = n0 (ψ), y = αn0 (ψ)} and coincides with the manifold Λ0 in the phase space with the coordinates q (momenta) and y (positions). We have the chain of relations ∂X σ (t, αn0 (ψ)) ∂X σ ∂x0 c2 (X σ (t, x0 ), t) det = det det =α . ∂(α, ψ) ∂x0 ∂(α, ψ) c2 (x0 , 0) This means that (5.13) becomes Z ∞ r µ √ µ/ρ h c(X0 (αn0 (ψ), t), t) as −iπ/4 u0 = e dρ ρK 0 Λt 2π det(∂Xσ /∂y) c(αn0 (ψ), 0)| Z∗ )−1 (αn0 (ψ), t)n0 (ψ)| 0 i × h˜ u0 (ρn0 (ψ)), n0⊥ (ψ)i Z∗ )−1 (αn0 (ψ), t)n0 (ψ) ⊥ , (5.20) y=y(x,t)

0

where y = y(x, t) is the solution to the equation X (y, t) = x. For the solution to this equation, we have y = Z −1 (t)(x − X(t)) + (x − X(t))2 . Using the results of Subsection 3.6, we can first replace ) + O(µ), where U (y, z) decays like |z|−2 αn0 (ψ) by y. This gives the vector function U (y(x, t), y(x,t) µ as |z| → ∞. Thus, we can replace z = y(x, t)/µ by Z −1 (t)(x − X(t))/µ and y by 0. This procedure is equivalent to replacing α by 0 in (5.20). Comparing this formula with (5.14) completes the proof. Remark 8. Using the formal notation, we can rewrite the last formula as follows: h Z(t)−1 (x − X(t)) i 1 c2 (X(t), t) −1 0 uas = ∇ △ curl u + O(µ). (5.21) x⊥ 3 0 x µ c2 (0, 0) µ Acting on this relation by the operator µ∇⊥ and dividing by c2 (X(t), t), we finally obtain curl3 uas h∇⊥ , uas curl3 u0 (y) 0 0 i µ 2 ≡µ 2 = + O(µ). (5.22) c (X(t), t) c (X(t), t) c2 (0, 0) y= Z(t)−1 (x−X(t)) µ RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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This equality reflects the conservation of the potential vorticity Ω (see Section 2) along the trajectories of the velocity vector fields of the original shallow water equations. Indeed, one can readily see that the leading term with respect to µ on the left-hand side in the second relation in (2.6) is (∂/∂t + (V, ∇))(curl3 uas /c2 ). This gives (d/dt0 )(curl3 uas /c2 ) ≈ 0, which is equivalent to (5.22) mod O(µ). Remark 9. In [14], we proved that the asymptotic solution (5.21) has exactly the same form as the initial function (5.14) if the Cauchy–Riemann conditions are satisfied along the trajectory X(t), i.e., ∂V2 ∂V1 ∂V2 ∂V1 (X(t)) = (X(t)) and (X(t)) = − (X(t)). ∂x1 ∂x2 ∂x2 ∂x1 This fact is based on the following formula for the Cauchy matrix Z(t): Z = g(t)Π(t), where g(t) is a smooth scalar function and Π(t) is the matrix of a rotation. 5.3. Fast Modes The modes corresponding to the Hamiltonians H ± propagate along the characteristics of the wave equation. The structure of the corresponding asymptotic solutions of the wave equation for localized initial data was studied in [11–13]. These functions are localized in neighborhoods of the wave fronts, i.e., the projections to the x-plane of the set in R4 obtained by shifting the circle x = 0, |p| = 1 along the trajectories of the Hamiltonian systems (4.12). As is well known, the wave fronts are not smooth in general; the singularities of these curves are the points at which the amplitude of the asymptotic solution increases; the order of the amplitude as µ → 0 is determined by the type of the corresponding singularity. 5.3.1. Reduction of two wave modes to the single mode corresponding to H + . Let us first show that the construction of two Lagrangian manifolds Λ± t can be reduced to the construction of Λ+ only, and hence we need only the solution to the problem t p˙ = −t Vx p − |p|∇c, x˙ = V + (p/|p|)c, p = n0 (ψ), x = n0 (ψ)α. (5.23) t=0

The manifold

Λ− t is Λ− t = −

t=0

p = P − (ψ, α, t), x = X − (ψ, α, t) | ψ ∈ [0, 2π), |α| < δ ,

(5.24)

−

where the pair P (ψ, α, t), X (ψ, α, t) is a solution to the Hamiltonian system p˙ = −t Vx p + |p|∇c, x˙ = V − (p/|p|)c, p t=0 = n0 (ψ), x t=0 = n0 (ψ)α.

(5.25)

Let us change the coordinates α, ψ on the manifold Λ− t by setting ψ = ψ ′ + π,

α = −α′ .

(5.26)

In this case, the functions P − (ψ ′ +π, −α′ , t), X − (ψ ′ +π, −α′ , t) form a solution (to the same system) satisfying the initial conditions p = n0 (ψ ′ + π) = −n0 (ψ ′ ), x = −n0 (ψ ′ + π)α′ = n0 (ψ ′ )α′ . (5.27) t=0

t=0

Setting P − (ψ ′ + π, −α′ , t) = −P (ψ ′ , α′ , t), X − (ψ ′ + π, −α′ , t) = X (ψ ′ , α′ , t), we see that the pair (P(ψ ′ , α′ , t), X (ψ ′ , α′ , t)) is a solution (to system (5.23)) corresponding to the sign “+” and with the initial data p t=0 = n0 (ψ ′ ), x t=0 = n0 (ψ ′ )α′ . (5.28) Thus, we can say that + ′ ′ + ′ ′ ′ ′ Λ− t = p = −P (ψ , α , t), x = X (ψ , α , t) | ψ ∈ [0, 2π), |α | < δ , where the functions (P + (ψ, α, t), x = X + (ψ, α, t)) define the manifold + + Λ+ t = p = P (ψ, α, t), x = X (ψ, α, t) | ψ ∈ [0, 2π), |α| < δ .

(5.29) (5.30)

Let us now find the vector ξ − and the amplitude A− in the coordinates ψ ′ , α′ . According to the definition of the vector ξ − , we have ξ − (p, x, t) Λ− = ξ − (−P + (ψ ′ , α′ , t), X + (ψ ′ , α′ , t), t) = −ξ + (P + (ψ ′ , α′ , t), X + (ψ ′ , α′ , t), t). (5.31) t


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Recall that, for the Fourier transform f˜(k) of any real-valued function f (y), we have f˜(−k) = f˜(k) (where the bar stands for complex conjugation). Thus, according to (4.17), (4.18), s |p|c(αn0 (ψ), 0) 0 0 1 η ˜ (ρn (ψ)) he u0 (ρn0 (ψ)), n0 (ψ)i − A− = − 0 2 c(αn (ψ), 0) c(x, t) Λt s 1 0 η˜0 (−ρn0 (ψ ′ )) |P + (ψ ′ , α′ , t)|c(α′ n0 (ψ ′ ), 0) = − he u (−ρn0 (ψ ′ )), n0 (ψ ′ )i + 2 c(α′ n0 (ψ ′ ), 0) c(X + (ψ ′ , α′ , t), t) s 0 1 0 η˜ (ρn0 (ψ ′ )) |P + (ψ ′ , α′ , t)|c(α′ n0 (ψ ′ ), 0) 0 ′ 0 ′ = − he u (ρn (ψ )), n (ψ )i + , 2 c(α′ n0 (ψ ′ ), 0) c(X + (ψ ′ , α′ , t), t) and

+

A− = −A . (5.32) Let us make an important simple remark concerning the definition of the Maslov canonical operator. Let Λ = {p = P, x = X } be a Lagrangian manifold with a distinguished point r ∗ = (P ∗ , X ∗ ) and let ϕ be a function on Λ. These objects enable one to construct the function Ψ(x) = KΛh (r ∗ ) ϕ, where KΛh (r ∗ ) is the Maslov canonical operator on Λ with the distinguished point r ∗ . Let us now replace Λ by the Lagrangian manifold Λ = {p = −P, x = X }, the point r ∗ by the point r ∗ = (−P ∗ , X ∗ ), and the function ϕ by the function ϕ on Λ, and construct the function Ψ′ (x) = KΛh (r ∗ ) ϕ. Then this function is complex conjugate to Ψ(x), Ψ′ (x) = Ψ(x). (5.33) p R ∞√ µ/ρ − − Let us apply this relation to the term µ/(2πi) 0 ρKΛ− A ξ dρ in the sum (4.20). The dist

tinguished point r = (P 0 , X 0 ) on Λ0 is the point with coordinates ψ ∗ , α = +0. Thus, after the passage to the coordinates ψ ′ = ψ + π, α′ = −α, the new coordinates of the distinguished point are ∗ ψ ′ = ψ ∗ + π, α′ = −0. With regard to this fact and to (5.31), (5.32), and (5.33), we can write µ/ρ

KΛ− A− ξ − t

=

µ/ρ KΛ+ t

A+ ξ +

exp

ni Z h

(ψ=ψ∗,α=+0)

(ψ=ψ∗ +π,α=−0)

pdx −

o iπ Ind γ (ψ = π, α = −0) → (ψ = 0, α = +0) . 2

According to the definition of the integral over Λ0 , Z (ψ=ψ∗ ,α=+0) Z (ψ=ψ∗ ,α=+δ) pdx = lim pdx = 2 lim δ = 0. (ψ=ψ∗ +π,α=−0)

δ→+0

(ψ=ψ∗ +π,α=−δ)

δ→+0

Since the path consisting of the points on Λ0 with negative α does not cross the singularity of Λ0 , we also have Ind{γ (ψ = ψ ∗ + π, α = −0) → (ψ = ψ ∗ , α = −0) } = 0. Thus, we must find the value (ψ=ψ∗,α=+0) 1 Ind{γ (ψ = ψ ∗ , α = −0) → (ψ = ψ ∗ , α = +0) } = lim [Arg det(C − iεB)] (ψ=ψ∗,α=−0) , π ε→+0 where B, C are given by ∂(−P1+ )/∂α ∂(−P1+ )/∂ψ 0 sin ψ B= = 0 − cos ψ , ∂(−P2+ )/∂α ∂(−P2+ )/∂ψ t=0 ∂X1+ /∂α ∂X1+ /∂ψ cos ψ −α sin ψ C= . = sin ψ α cos ψ ∂X2+ /∂α ∂X2+ /∂ψ t=0

Thus,

(ψ=0,α=δ) α=δ 1 1 lim [Arg det(C − iεB)] = lim [Arg det(α + iε)] = −1. π ε→+0 π ε→+0 (ψ=0,α=−δ) α=−δ


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Hence µ/ρ

µ/ρ

t

t

n iπ o

KΛ− A− ξ − = KΛ+ A+ ξ + exp and this gives

r

µ 2πi

Z

0

∞

√

µ/ρ ρKΛ− t

− −

A ξ dρ =

So we have proved the following assertion.

r

µ 2πi

Z

2

∞

0

,

√ µ/ρ + + ρKΛ+ A ξ dρ. t

Lemma 14. Considering the “wave” part (or the “fast” mode) of the solution, we can restrict ourselves to the Hamiltonian H + = hV, pi + |p|c and set Z ∞ Z ∞ np Xp √ µ/ρ ± ± √ µ/ρ + + o µ/(2πi) ρKΛ± A ξ dρ = 2 Re µ/(2πi) ρKΛ+ A ξ dρ . (5.34) 0

+,−

t

0

t

5.3.2. Focal points and the trajectories of the Hamiltonian H + . The Maslov index of nonsingular points and its relation to the Morse index. The trajectories (Pα+ , Xα+ ) possess all the properties described in Section 4. However, the term |p|c in the Hamiltonian implies some additional important properties. Lemma 15. 1. Any focal point on the trajectory is isolated with respect to time t. 2. The Maslov index of the nonsingular point r(ψ, α, t) ∈ Λ+ t with positive α is the Morse index of the trajectory (P + (ψ, α, t), X + (ψ, α, t)), t ∈ [0, t], i.e., the number of zeros of the Jacobian J+ or of |X + (ψ, α, t)| on this trajectory. For negative α, to obtain the Maslov index, one must add −1 to the Morse index of the corresponding trajectory. Remark 10. Let us stress the difference between the trajectories corresponding to the Hamiltonians H 0 and H + . In the first case, the (singular) set formed by the focal points moves along the trajectory, whereas in the other case the trajectories cross this set. Proof. Let the focal point occurring at t = tf have the coordinates ψf , αf . Consider the derivative dJ+ /dt(ψf , αf , tf ) of the Jacobian J+ = det(Xα+ , Xψ+ ). Since Xψ+ (ψf , αf , tf ) = 0, we obtain dJ+ /dt(ψf , αf , tf ) = det Xα+ , dXψ+ /dt (ψf , αf , tf ). It follows from the Hamiltonian system that dXψ+ dt

(ψf , αf , tf ) =

h ∂2H + ∂p∂x

Xψ+ + c

Pψ+

|P + |

+ cP +

1 i (ψf , αf , tf ). |P + | ψ

The first term on the right-hand side vanishes at the focal point because Xψ+ = 0 at the point. The same argument and the Lagrangian property gives hXα+ , Pψ+ i = hPα+ , Xψ+ i = 0 at the focal point. However, the vectors Pψ and Xα are nonzero, and thus P + = aPψ+ + bXα+ , where a = hP + , Pψ+ i/|Pψ+ |2 . Hence, |P + |2 |Pψ+ |2 − hP + , Pψ+ i2 dJ+ (ψf , αf , tf ) = c det(Xα+ , Pψ+ )(ψf , αf , tf ). + 2 + 3 dt |P | |Pψ |

(5.35)

Since hXα+ , P + i = 1, the first factor is positive and dJ+ /dt(ψf , αf , tf ) 6= 0. Using (4.6) and (5.35), we can find the Maslov index of nonsingular points on Λ+ t . Indeed, r(ψ,α,t) 1 1 r(ψ,α,t) lim Arg Jεσ = lim Arg Jεσ . π ε→+0 π ε→+0 r0 r(ψ,α,0)

Here Jεσ = det(C+ − iεB+ ) = det(Xα − iεPα , Xψ − iεPψ ) = u + iv. As was shown in Section 3, the jump of the Maslov index at the focal point is +1 if −vdu is positive and −1 if −vdu is negative. + We have v = −ε det(Xα , Pψ ) and du = dJ dt dt at the focal point. Thus −vdu > 0 by (5.35), and the jump of the Maslov index is always equal to +1. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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5.3.3. The fast mode asymptotics outside the focal points. The wave front given by 0 Γ+ t = {p = n (ψ), x = 0} is in singular position at t = 0 and becomes nonsingular very soon. It is clear that P + = n0 −(Vx (n0 α, 0))T n0 t−cx (n0 α, 0)t+O(t2 ),

X + = n0 α+V (n0 α, 0)t+n0 c(n0 α, 0)t+O(t2 ),

and J+ = tc(0, 0) + O(α) + O(t) for small t. Hence J+ > 0 if t is small and positive. This relation shows that there is a time interval t ∈ (δ, tcr ), δ > 0, on which the front γt+ = {p = P + (ψ, t), x = X + (ψ, t)}

in the configuration space is a smooth closed curve on R2x . In the case when tcr < ∞, the focal points can appear on the wave front Γ+ t , but usually they have a different structure with respect to time t, namely, they are isolated in the phase space as well. The boundary layer expansion enables one to crucially simplify the solution outside the focal points as well as in a neighborhood of the focal points, and the final formulas include only the wave front Γ+ t and related characteristics. Here we present formulas only for the solution outside the focal point, refer the reader to the papers [11–13] for the formulas for the solution in a neighborhood of the (isolated) focal points, and only mention now that, due to the additional integral against ρ in the general answer, these formulas are not based on Airy or Peirce functions as it takes place for oscillating solutions. We define a neighborhood of the wave front for a sufficiently small coordinate y, where |y| is the distance between the point x belonging to a neighborhood of the wave front and the wave front by itself. To this end, we take y > 0 for the external subset of the wave front and y < 0 for the internal subset. Then a point x of the neighborhood of the wave front is characterized by two coordinates, ψ(x, t) and y(x, t), where ψ(x, t) is defined by the condition that the vector y = x − X + (ψ, t) is orthogonal to the vector tangent to the wave front at the point X + (ψ, t), and thus hy, Xψ+ (ψ, t)i = 0. The phase is defined by the rule S(x, t) = hP + (ψ(x, t), t), x − X + (ψ(x, t), t)i.

(5.36)

The initial perturbations u0 , η 0 generate the function Z ∞ √ √ −iπ/4 U (y, ψ) = (e / 2π) c0 hn0 (ψ), u e0 (ρn0 (ψ))i + ηe0 (ρn0 (ψ)) ρ eizρ dρ,

c0 = c(0, 0),

0

where u e0 (k) and ηe0 (k) stand for the Fourier transforms (1.5) of the functions u0 (z) and η 0 (z), respectively. Let us now state the first proposition on the fast mode. Theorem 4. The relation η(x, t) = O(µ3/2 ) holds outside the above neighborhood of the front γt . For tcr > t > δ > 0, the relation r 1 c0 √ u(x, t) q = µ + (ψ, t), t) η(x, t) + c(X |Xψ (ψ, t)| h S(x, t) i 1 √ × Re U , ψ P + (ψ, t) c(X + (ψ,t),t) + o( µ) (5.37) µ ψ=ψ(t,x) c0 holds in some neighborhood (of the wave front γt ) independent of µ, η, u and outside some neighborhood of the focal points.

Note that, if c does not depend on t and the background velocity is V = 0, then it follows from the Hamiltonian system that |P + (ψ, t)| = c0 /c(X + (ψ)). Thus, the η-component q satisfies the q c0 so-called Green law the factor c(X + (ψ,t),t) with regard to the so-called divergency |Xψ+ (ψ, t)|

of the trajectories coming to the point x = X + (ψ, t). In this case, this formula was discussed in detail in [11–13]. Let us stress that the profile of the solution crucially depends on the form of the initial perturbation, and this is the main difference from the case in which the original equations include strong dispersion. In the last case, the influence of the initial perturbation is related to average characteristics of the initial perturbation (cf. [6, 9, 5]). RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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At the moments t > tcr , focal points occur on the wavefront. This gives the following effects: 1) the Maslov (or Morse) index of the point x = X + (ψ, t) can appear on the last formulas; 2) for some time t, several trajectories having different angles ψj can have close values X + (ψj , t). In this case, it is necessary to use the sum of contributions coming from different functions ψj (x, t), yj (x, t), Sj (x, t) with the index j, the Maslov index is mj = m(r(ψj (x, t), t)) and it coincides with the Morse index of the trajectory x = X + (ψj (x, t), τ ), τ ∈ (+0, t). Finally, the assertion generalizing Theorem 4 looks as follows. Theorem 5. In a neighborhood of the front and outside some neighborhood of the focal points, the wave field is the sum of the fields

u(x, t) η(x, t)

√ X q = µ

r

1

c0 + c(X (ψ, t), t)

|Xψ+ (ψ, t)| h S(x, t) i 1 √ −iπmj /2 + + c(X (ψ,t),t) × Re e U + o( µ). (5.38) , ψ P (ψ, t) µ ψ=ψj (t,x) c0 j

Moreover, η(x, t) = O(µ3/2 ) outside this neighborhood of the front γt .

The solution in a neighborhood of the focal points can also be simplified. Such a simplification was carried out in [11–13] in the case of V = 0, c = c(x). This requires an additional analysis of the wave front near the focal points, which is not discussed here. Proof of Theorems 4, 5. It suffices to prove the theorem in a strip of the front. Consider a nonsingular neighborhood of some nonsingular point X + (ψ, t) on the front γt+ . By definition, the Maslov index is the same at all points x in this neighborhood, and the contribution of this neighborhood into the canonical operator is (see (5.2)) r

µ 2π

s

|P + (ψ, α, t)| −i π −i π m(rt (ψ,α)) + e 4e 2 ξ |Xψ+ (ψ, α, t)|

Z

0

∞

√

ρ ρei µ α hξ0σ† , V˜ (ρn0 (ψ))idρ

α=α(x,t),ψ(x,t)

where ξ0+† (ψ, α) = ξ +† |p=n0 (ψ),x=αn0 (ψ) , ξ + = ξ + |p=P + (ψ,α,t),x=X + (ψ,α,t) , V (k) = ψ(x, t), α(x, t) is the solution to the system X + (ψ, α, t) = x.

, (5.39)

u ˜0 (k) , η˜0 (k)

and

(5.40)

By the properties of u0 (z), η 0 (z), the integral Z

0

∞

√

ρeiρy hξ0σ† , V˜ (ρn0 (ψ))idρ

defines a function of the argument α rapidly decaying as |y| → ∞ and α(x, t), and this function vanishes at the points x = X + (ψ, α, t)|α=0 ≡ X + (ψ, t). Thus, according to Subsection 3.6, we can replace α(x, t) at the factor 1/µ by its linear part of the Taylor expansion h∂α/∂x|x=X(X + (ψ,t)) , x − X + (ψ, t)i, where ψ(x, t) is the solution to the equation hx − X + (ψ, t), Xψ+ (ψ, t)i = 0, and by 0 if the factor 1/µ is absent. This replacement gives an O(µ) correction to the asymptotics. Since, for positive α, the Maslov index of X + (ψ, α, t) is the Morse index of the corresponding trajectory, we can use the last index in (5.37). The derivative is ∂α/∂x = P + , because α(x, t) is an action on Λ+ t . The use of these considerations, Lemma 12, and the explicit expressions for ξ + , ξ +† completes the proof. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Fig. 1. The center of the vortex and the front of the wave for the” depth “qdiverse t, . ‹ 2 2 2 2 D(x1 , x2 ) = 0.1(1 + 3 tanh (2x1 + x2 + 1)) cosh 4x1 + x2 5 ,

and the basic velocity V (x1 , x2 ) = 0.001((cos x2 + x1 ), x1 ). One can see the occurrence of focal points on the front in the third picture.

5.4. General Behavior of the Solution The formulas in Theorems 3–5 give a quantitative realization of the propagation of an initial localized perturbation on linear shallow water. The perturbation takes apart the vortical (slow) part and wave (fast) part. The vortical part moves along the trajectories x = X 0 (t) of the basic (background) velocity vector field V (x, t) and preserves its structure globally, keeping the localization near the point (the center of the vortex) x = X 0 (t) for any chosen t. In classical mechanics, this part corresponds to the motion of a focal point in the phase space. On the contrary, the wave part immediately changes its structure and very soon becomes localized in a neighborhood of the front, which is a “closed curve with a center” defined by the center the vortex x = X 0 (t). Thus, we observe the phenomena of “profile metamorphosis.” The points on the front of the wave run away from the point x = X 0 (t) with the velocity c(x, t), which is much greater than the velocity V (x, t) in real situations (see Fig. 1). For this reason, the wave mode is referred to as the fast mode. Its amplitude becomes very small very quickly, as long as the velocity c(x, t) or the depth of the basin becomes small. After this, the amplitude of the wave field grows up, and the linear theory stops working. We also note that, on the contrary, the amplitude of the vortical part decreases as c(x, t) (or the depth of the basin) becomes small. From this point of view, the vortical part of the wave describes the wave radiation. An important mathematical fact is that, after the vortical and wave parts disperse, one can study their asymptotics separately in the nonlinear theory as well. 6. CONCLUSION We present explicit formulas for asymptotic solutions of the Cauchy problem with localized initial data for linear hyperbolic systems. As in the parametrix theory (theory of propagation of singularities [19, 24, 35]), our formulas use wavefronts in the phase space, but these formulas preserve the dependence of the profile of the solution in the form of initial perturbation as well. Our formulas are illustrated by the example of linearized shallow water equation. In this case, we show how these formulas separate the vortices and the waves, and how this separation is reflected in the phase space related to classical mechanics. In particular, we show that the motion of the vortex corresponds to the motion of focal points (of singularities) in the phase space. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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APPENDIX. REDUCTION TO SCALAR CASES A.1. Commutation of the Maslov Canonical Operator with Scalar Differential and Pseudodifferential Operators 1 2 ˆ σ = Lσ p ˆ = −ih∇x , be a differential or a pseudodifferential operator in the ˆ , x, t, µ , p Let L sense of [26, 16] with the symbol Lσ (p, x, t, h) satisfying the corresponding conditions in [26, 16] and having the (asymptotic) expansion

Lσ = H σ (p, x, t) + hLσ1 (p, x, t) + h2 Lσ2 (p, x, t) + · · · .

(A.1)

t σ Let Λσt = gH σ be the family of the Lagrangian manifolds obtained from the manifold Λ0 with σ σ the help of the canonical transform generated by the Hamiltonian H . Let KΛσt A be the Maslov canonical operator on the family Λσt , t ∈ [0, T ], acting on a function Aσ defined on Λσ . The effective use of the Maslov canonical operator is based on the relation

−ih

d ∂ σ σ + Lσ KΛσt Aσ = −ihKΛσt + G | Aσ + O(h2 ), Λ t ∂t dtσ

Gσ = iLσ1 −

1 σ tr Hpx , 2

where d/dtσ stands for the derivative along the trajectories of the Hamiltonian system with the Hamiltonian H σ . Using this formula, we can reduce the solution of the Cauchy problem for the scalar equation ˆ σ ϕσ (x, t, h) ih∂ϕσ (x, t)/∂t = L (A.2) to the transport equation (4.7) on the manifolds Λσt . A.2. “Operator Separation of Variables” and the Peierls Substitution In the case of vector equations, the commutation formula is separated into several parts and is not as easy as the one above. The idea is to reduce the original vector equation to a family of scalar ones and then to use the commutation formula for each part separately. Recall (briefly and formally) this adiabatic procedure [8, 4] (ignoring the nonsmoothness of effective Hamiltonians for p = 0 by Subsection 4.5), which is close to the adiabatic approximation (the Born–Oppenheimer method) and looks convenient for systems of differential equations, especially if the solution has focal points. By this procedure, one can construct an asymptotic solution in two steps: 1) reduction to a scalar problem, 2) construction of an asymptotic solution of the scalar equation. The realization of the first step enables one to find the term Lσ1 independently of the presence of focal points in the asymptotic solution. Consider the vector equation ∂Ψ ih = (Hˆ0 − ihH1 )Ψ. (A.3) ∂t Decompose it into scalar ones. Following [8, 4], we seek a partial solution to (A.3) in the form Ψ=χ ˆσ ϕσ (x, t),

(A.4)

assuming that the scalar function ϕσ (x, t) satisfies the scalar equation ih

∂ϕσ (x, t) ˆ σ ϕσ (x, t). =L ∂t

(A.5)

ˆ σ are h-pseudodifferential operators, i.e., functions of operators p ˆ = −ih∇x and x, Here χ ˆσ and L and the subindex σ means the number of the corresponding partial solutions. The operators χ ˆσ are σ ˆ known as interwinding ones, the operators L are related to the well-known Peierls substitution. It is ˆ = −ih∇x and x do not commute, the construction of functions of of importance that the operators p operators is not unique, and it depends on the ordering of these operators. Two of these orderings (or “quantization”) are used rather often. These are the Feynman–Maslov and Weyl orderings RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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(see [24]). Our experience and technical reasons show that the Feynmann–Maslov “quantization” is more convenient in many practical calculations. In this case, we have 1 2 ˆ , x, h , χ ˆ σ = χσ p

1 2 ˆ σ = Lσ p ˆ , x, t, h , L

ˆ = −ih∇x p

(A.6)

with smooth symbols χ(p, x, h) and L(p, x, h) having the asymptotic expansions (A.1), χσ (p, x, t, h) = ξ σ (p, x, t) + hχσ1 (p, x, t) + h2 χσ2 (p, x, t) + · · · .

(A.7)

The indices over the operators mean the order of their action (see, e.g., [24]). The functions H σ (p, x, t) ≡ Lσ0 (p, x, t) are called effective Hamiltonians or modes. The choice of the function χσ (p, x, t, h) in the representation (A.4) is not unique. To make the choice, we add normalization conditions. If the operator of the original equation is selfadjoint, ′ ˆσ = δσσ′ , where δσ,σ′ is the Kronecker symbol [4]. then it is natural to impose the relation (χ ˆσ )∗ χ However, in some important examples (like the shallow water equation), the corresponding operator is not symmetric, and we can use a more reasonable choice from the physical point of view. Let some χ ˆσ be chosen. Substituting (A.4) into (A.3) and considering the Peierls substitution (A.2), we obtain the equaˆ σ − ih(∂ χˆσ /∂t) ϕσ = 0. Obviously, this holds for any ϕσ if ˆχ ˆσ L tion H ˆσ − χ σ

ˆ ˆχ ˆ σ − ih ∂ χ H ˆσ − χ ˆσ L = 0. ∂t

(A.8)

Let us now pass from the operator equation to that for the symbols of these operators. We obtain 1

1

2

2

H(p − ih∇x , x)χσ (p, x, t, h) = χσ (p − ih∇x , x, t, h)Lσ (p, x, t, h) + ihχσt (p, x, t, h).

(A.9)

We can represent (A.9) in the form 2 X ∂H0 ∂ H0 (p, x) − ih − ihH1 (x, t) χσ (p, x, t, h) ∂pj ∂xj j=1 1

2

= χσ (p − ih∇x , x, t, h)Lσ (p, x, t, h) + ihχσt (p, x, t, h).

(A.10)

Using the Taylor formula, we can write, at least formally, σ

χ (p −

1 2 ih∇x , x, t, h)

∞ X 1 ∂ |ν| χσ ∂ |ν| = χ (p, x, t, h) + (−i)|ν| h|ν| (p, x, t, h) . ν! ∂pν ∂xν σ

(A.11)

|ν|=1

Here ν = (ν1 , ν2 ) is a multiindex, and |ν| = ν1 + ν2 , ν! = ν1 !ν2 !, and ∂ |ν| /∂xν = ∂ |ν| /∂x1 ν1 ∂x2 ν2 as usual. Substituting this expansion and the asymptotic expansions (A.7) into (A.10) after equating the coefficients of the same powers of h to zero, we obtain the equation H0 ξ σ − ξ σ H σ = 0

(A.12)

and a chain of linear algebraic nonuniform equations for the coefficients Hχσk − χσk H σ = χσ0 Lσk + Fkσ ,

χσk (p, x, t), Lσk (p, x, t),

k = 1, 2, . . .

k > 1, (A.13)

Each Fkσ is a sum of products of derivatives ∂χσj /∂xj and (∂ ν χσj /∂pν )(∂ ν Lσm /∂xν ) with the indices j = 1, . . . , k − 1. We need F1σ

X 2 ∂H0 ∂ξ σ ∂H σ ∂ξ σ ∂ξ σ σ =i − + H1 ξ + . ∂pj ∂xj ∂xj ∂pj ∂t j=1


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Assume that, for each (p, x), H σ is a nondegenerate eigenvalue, ξ σ is the corresponding eigenvector, ξ σ† is the eigenvector of the conjugate matrix H0† , and hξ σ † , ξ σ i = 1. Consider (A.13) for χσk and take its inner product with the vectors ξ σ † . We obtain Lσk = −hξ σ † , Fkσ i and, in particular, (4.8). Formula (4.8) was used in Subsection 4.3 for (2.5). We refer for details and generalizations to [4]; see also [31]. We also note that the above scheme is local with respect to the variables (p, x); therefore, one can consider hyperbolic system (1.1) outside the point p = 0. Finally, note that, for the shallow water equation ξ σ† = W(x)ξ σ , where 1/2 0 0 0 1/2 0 W(x) = , one can readily see that the matrix H0 is selfadjoint in the space 2 0

0 1/(2c (x))

with the inner product (ζ ′ , Wζ ′′ ). It seems natural to chose the normalization conditions for the operators χ ˆσ in the form 2 1 1 2 ′ ′ ˆ , x, µ W(x)χσ p ˆ , x, µ = δσσ′ . (χ ˆσ )∗ W(x)χ ˆσ = δσσ′ = χσ ∗ p

This condition can be useful when constructing the corrections Lσ2 , Lσ3 , etc. ACKNOWLEDGMENTS

The authors are grateful to A. G. Kulikovskii and V. P. Maslov for useful discussions. This work is supported by the RFBR grant no. 08-01-00726 and the scientific agreement among the Department of Physics of the University of Rome “La Sapienza” and the Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow. REFERENCES 1. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1974) [in Russian]; Graduate Texts in Mathematics 60, Ed. by S. Axler, F. W. Gehring, and K. A. Ribet (Springer, Berlin, 1978). 2. V. I. Arnol’d, Singularities of Caustics and Wave Fronts (Kluwer, Dordrecht, 1990). 3. V. M. Babich, V. S. Buldyrev, and I. A. Molotkov, The Space–Time Ray Method, Linear and Nonlinear Waves (Leningrad. Univ., Leningrad, 1985) [in Russian]. 4. V. V. Belov, S. Yu. Dobrokhotov, and T. Ya. Tudorovskiy, “Operator Separation of Variables for Adiabatic Problems in Quantum and Wave Mechanics,” J. Engrg. Math. 55 (1–4), 183–237 (2006). 5. M. V. Berry,“Tsunami Asymptotics,” New J. of Phys. 7 (129), 1–18 (2005). 6. V. A. Borovikov and M. Ya. Kelbert, “The Field near the Wave Front in a Cauchy–Poisson Problem,” Izv. AN SSSR Ser. Mekh. Zhidk. Gaza (2), 173–174 (1984) [Fluid Dynamics 19 (2), 321–323 (1984)]. 7. L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Media II, Point Source and Bounded Beam (Springer, 1992). 8. S. Yu. Dobrokhotov, “Maslov’s Methods in the Linearized Theory of Gravitational Waves on a Liquid Surface,” Dokl. Akad. Nauk SSSR 269 (1), 76–80 (1983) [Sov. Phys. – Doklady 28, 229–231 (1983)]. 9. S. Yu. Dobrokhotov, P. N. Zhevandrov, and V. M. Kuzmina, “Asymptotic Behavior of the Solution of the Cauchy–Poisson Problem in a Layer of Nonconstant Thickness,” Mat. Zametki 53 (6), 141–145 (1993) [Math. Notes 53 (5–6), 657–660 (1993)]. 10. S. Yu. Dobrokhotov, P. N. Zhevandrov, V. P. Maslov, and A. I. Shafarevich, “Asymptotic Rapidly Decreasing Solutions of Linear Strictly Hyperbolic Systems with Variable Coefficients,” Mat. Zametki 49 (4), 31–46 (1991) [Math. Notes 49 (3–4), 355–365 (1991)]. 11. S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, and T. Tudorovski, “Asymptotic Theory of Tsunami Waves: Geometrical Aspects and the Generalized Maslov Representation,” in Publications of Kyoto Institute (RIMS), Kokyuroku 1457, Vol. 4, 118–153 (2006). 12. S. Dobrokhotov, S. Sekerzh-Zen’kovich, B. Tirozzi, and T. Tudorovski, “Description of Tsunami Propagation Based on the Maslov Canonical Operator,” Dokl. Akad. Nauk 409 (2) 1–5 (2006) [Dokl. Math. 74 (1), 592–596 (2006)]. 13. S. Dobrokhotov, S. Sekerzh-Zenkovich, B. Tirozzi, and B. Volkov, “Explicit Asymptotics for Tsunami Waves in Framework of the Piston Model,” Russ. J. Earth Sciences 8 (4), ES4003, 1–12 (2006). 14. S. Yu. Dobrokhotov, A. I. Shafarevich, and B. Tirozzi, “The Cauchy–Riemann Conditions and Localized Asymptotic Solutions of Linearized Equations of Shallow Water Theory,” Prikl. Mat. Mekh. 69 (5), 804–809 (2005) [J. Appl. Math. Mech. 69 (5), 720–725 (2005)]. RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS

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Vol. 15

No. 2

2008