A Generalized KdV-Family with Variable Coefficients in (2 + 1

IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.10 OCTOBER 2005

2548

PAPER

Special Section on Nonlinear Theory and its Applications

A Generalized KdV-Family with Variable Coefficients in (2 + 1) Dimensions Tadashi KOBAYASHI†a) and Kouichi TODA††b) , Nonmembers

SUMMARY Variable-coefficient generalizations of integrable nonlinear equations are one of exciting subjects in the study of mechanical, physical, mathematical and engineering sciences. In this paper, we present a KdV family (namely, KdV, modified KdV, Calogero-Degasperis-Fokas and Harry-Dym equations) with variable coefficients in (2 + 1) dimensions. key words: WTC method, integrable equations, variable coefficients, Lax pairs

1.

Introduction

Modern theories of nonlinear science have been highly and widely developed over the last half century. Particularly, integrable nonlinear systems have attracted great interests of a number of mathematicians and physicists. One of the reason for such attentions is algebraic solvability of the integrable systems. In addition to their theoretical importance, they have remarkable applications that many physical systems are thought of as perturbations of integrable systems such as hydrodynamics, nonlinear optics, plasma and certain field theories and so on [1]. The notion of the integrable nonlinear systems [2] is frequently not defined precisely but rather is characterized generally by various interrelated common features including the space-localized solutions (or solitons), Lax pairs, Bäcklund transformations and Painlevé properties [3], [4]. Finding new integrable systems is an important, but difficult problem because of their ambiguous definition and undeveloped mathematical background. Moreover solitons and the nonlinear evolution equations are a major subject in mechanical and engineering sciences as well as mathematical and physical ones. Among them, the celebrated Korteweg-de Vries(KdV) and Kadomtsev-Petviashvili(KP) equations possess certain impressive properties. For instance, a real ocean is inhomogeneous and the dynamics of nonlinear waves is very influenced by refraction, geometric divergence and so on. The Manuscript received January 21, 2005. Manuscript revised April 29, 2005. Final manuscript received June 6, 2005. † The author is with the High-Functional Design G, LSI IP Development Div., ROHM CO.,LTD., Kyoto-shi, 615-8585 Japan. †† The author is with Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10 D-37073 Gottingen, Germany (Temporality), and with the Department of Mathematical Physics, Toyama Prefectural University, Toyama-ken, 939-0398 Japan (Permanency). a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1093/ietfec/e88–a.10.2548

problem of evolution of transverse perturbations of a wave front is of theoretical and practical interest. The physical phenomena in which many integrable nonlinear equations with constant coefficients arise tend to be very highly idealized. Therefore, equations with variable coefficients may provide various models for real physical phenomena, for example, in the propagation of small-amplitude surface waves, which runs on straits or large channels of slowly varying depth and width. On the hand, there has been much interest in the study of generalizations with variable coefficients of completely integrable nonlinear equations [5]–[18]. For discovery of new integrable nonlinear systems, many researchers have mainly investigated autonomous and lower dimensional nonlinear systems. Thus many autonomous (1 + 1)-dimensional integrable nonlinear systems have been found. On the other hand, there are few research studies to find nonautonomous integrable nonlinear systems, since they are essentially complicated. And their results are still in its early stages. Analysis of higherdimensional systems is also an active topic in integrable systems. Since then, the study of integrable nonlinear equations in higher dimensions has attracted much more attention. So our goal in this paper is to construct integrable a (2 + 1) dimensional variable-coefficients version of the KdV family, namely, KdV, modified KdV, Calogero-DegasperisFokas and Harry-Dym equations. It is widely-known that the Painlevé test in the sense of the Weiss-Tabor-Carnevale(WTC) method [3], [4] is a powerful tool for investigating integrable equations with variable coefficients. We have discussed the following higherdimensional KdV-type equation with variable coefficients for u = u(x, z, t) [19], [20]: ut + a(x, z, t)u + b(x, z, t)u x + c(x, z, t)uz +d(x, z, t)uuz + e(x, z, t)u x ∂−1 x uz + f (x, z, t)u xxz + g(x, z, t) = 0,

(1)

where d(x, z, t) + e(x, z, t) 0, f (x, z, t) 0 and subscripts with respect to independent variables denote partial ∂u ∂2 u , u xz = etc, and derivatives, for example, u x = ∂x ∂x∂z x

∂−1 x u :=

u(s)ds. Here a(x, z, t), b(x, z, t), · · ·, g(x, z, t) are

functions of two spatial variables x, z and one temporal variable t. Equation (1) includes one of the widely-known integrable higher-dimensional KdV equation: 1 1 ut + uuz + u x ∂−1 (2) x uz + u xxz = 0, 2 4

c 2005 The Institute of Electronics, Information and Communication Engineers Copyright

KOBAYASHI and TODA: A GENERALIZED KdV-FAMILY WITH VARIABLE COEFFICIENTS IN (2 + 1) DIMENSIONS

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which is called the Calogero-Bogoyavlenskii-Schiff (CBS) equation [21]–[25]. We have presented two sets of coefficients of Eq. (1) to pass the Painlevé test in the sense of the WTC method. This paper is organized as follows. In Sect. 2, we will review the WTC method and the process of the test for Eq. (1) in brief. In Sects. 3 and 4 we will consider a KdV family, namely, KdV, modified KdV, Calogero-DegasperisFokas and Harry-Dym equations with variable coefficients in (2 + 1) dimensions corresponding the special but interesting case of a integrable higher-dimensional KdV equation with variable coefficients given in Sect. 2. Section 5 will be devoted to conclusions. 2.

Painlevé Test of Eq. (1)

In this section we would like to review the WTC method and the process of the test for Eq. (1) in brief. Experience has shown that, when a system possesses the Painlevé property, one is integrable. Weiss et al. [3] said that a partial differential equation (PDE) has the Painlevé property when the solutions of the PDE are single-valued about the movable, singularity manifold. They have proposed a technique which determine whether or not a given PDE is integrable, that we call the WTC method: When the singularity manifold is determined by φ(z1 , · · · , zn ) = 0,

(3)

and u = u(z1 , · · · , zn ) is a solution of PDE given, then we assume that α

u=φ

∞

u jφ j,

U0 =

(7)

is given. Then, substituting expansion (6) with α = −1 into Eq. (5), the recursion relations for the U j are presented as follows: ( j − 1)( j − 4)( j − 6)( j + 1) f (x, z, t)φ3x φz U j = F(U j−1 , · · · , U0 , φt , φ x , φz , · · ·),

(8)

where the explicit dependence on x, z, t of the right-hand side comes from that of the coefficients. It is found that the resonances occur at j = −1, 1, 4, 6. Then one can check the arbitrariness of resonance functions u1 , u4 and u6 though we have omitted it in this paper because of the limitation of pages. We have succeeded in finding of two higherdimensional KdV equations with variable coefficients. One of them is given as d (t) f (t) 4 − ut + α(z, t) − β(t) + cz (z, t) + u 3 d(t) f (t) 2 + x α(z, t) − β(t)b + cz (z, t) u x + c(z, t)uz 3 d(t) u x ∂−1 + d(t)uuz + x uz + f (t)u xxz + g(z, t) 2 = 0, (9) where α(z, t), β(t) and g(t) are arbitrary functions, and denotes a derivative with respect to variable t. In next section we will give modified equations of the special but interesting Eq. (9). 3.

(4)

12 f (x, z, t) φx , d(x, z, t) + e(x, z, t)

Calogero-Bogoyavlenskii-Schiff and Modified Equations with Variable Coefficients

j=0

where φ = φ(z1 , · · · , zn ), u j = u j (z1 , · · · , zn ), u0 0 ( j = 0, 1, 2, · · ·) are analytic functions of z j in a neighborhood of the manifold (3), and α is a negative integer (so-called the leading order). Substitution of expansion (4) into the PDE determines the value of α and defines the recursion relations for u j . When expansion (4) is correct, the PDE possesses the Painlevé property and is conjectured to be integrable. We briefly show the Painlevé test for Eq. (1). For that, a non-local term of Eq. (1) should eliminate. When defining u = U x , we have a potential version of Eq. (1): U xt + a(x, z, t)U x + b(x, z, t)U xx + c(x, z, t)U xz +d(x, z, t)U x U xz + e(x, z, t)U xx Uz + f (x, z, t)U xxxz + g(x, z, t) = 0.

(5)

We are now looking for a solution of Eq. (1) in the Laurent series expansion: U = φα

∞

U jφ j,

(6)

j=0

where U j are analytic functions in a neighborhood of φ = 0. In this case, leading order is −1 and

Setting the following condition of variable coefficients: 3 G (t) , c(z, t) = 0, 4 G(t) 1 f (t) = , g(z, t) = 0 4

α(z, t) − β(t) = d(t) = 1,

(10)

Eq. (9) reads 1 1 G (t) ut + uuz + u x ∂−1 u x uz + u xxz − 2 4 G(t) xG (t) − u x = 0. 2G(t)

(11)

Here G(t) is an arbitrary function of the temporal variable t. We would like to call Eq. (11) a generalized CalogeroBogoyavlenskii-Schiff(GCBS) equation. Because Eq. (11) is a higher-dimensional integrable version of the general KdV(GKdV) equation [9], [17]: xG (t) 3 1 G (t) u− u x = 0. ut + uu x + u xxx − 2 4 G(t) 2G(t)

(12)

Actually Eq. (11) can be reduced to Eq. (12) by the dimensinal-reduction z = x. And using Lax-pair Gener-


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wx 1 1 + σaw + σbG(t) ∂ x ∂2x + G(t) w w 1 ≡ LGCDF , (21) G(t)

ating Technique (LGT) [26]–[28]† , we obtain its Lax pair given by 1 2 1 LGKdV − λ, ∂x + u − λ ≡ g(t) g(t) 1 −1 xG (t) ∂x T = ∂z LGKdV + ∂ x uz − 2 G(t) 1 G (t) − uz − + ∂t , 4 G(t) L =

L =

(13)

T = ∂z LGCDF + T + ∂t ,

which a and b are arbitrary constants, σ2 = −1, and T is a very complicated form with many terms. Here λ also satisfies the non-isospectral condition (16). Then the Lax Eq. (15) gives a higher-dimensional GCDF equation:

(14)

∂ with ∂ x ≡ (similarly in what follows for ∂t , etc). The ∂x Lax equation for Lax pairs is defined as [L, T ] ≡ LT − T L = 0,

1 1 w x w xz 1 w xx wz − wt + w xxz − 4 2 w 4 w 2 w 1 1 w2x wz 1 x − w x ∂−1 + a2 w2 wz + x 2 2 2 w 8 w z 4 1 1 2 2 + abG(t)wz + a w x ∂−1 x (w )z 2 8 wz 1 1 + abG(t)w x + b2G(t)2 2 4 4 w 1 2 G (t) 2 −1 1 w + b G(t) w x ∂ x − 2 8 w z 2G(t) xG (t) − w x = 0. 2G(t)

(15)

which gives corresponding integrable equations. Here λ = λ(z, t) is the spectral parameter and satisfies the nonisospectral condition [29]–[31]: λt = λλz .

(16)

We present higher-dimensional versions of the modified GKdV(MGKdV) [10] and Calogero-DegasperisFokas(CDF) [32], [33] equations from the Lax pair (13), (14) for the GCBS Eq. (11) as follows; (i) Modified GCBS equation Using LGT, we obtain a following Lax pair: 1 2 ∂ x + v∂ x − λ G(t) 1 LGmKdV − λ, ≡ (17) G(t) 1 2 T = ∂z LGmKdV + ∂−1 x vz ∂ x 2 1 −1 1 −1 2 1 xG (t) + v∂ x vz − ∂ x v z − vz − ∂x 2 8 4 2G(t) + ∂t . (18) L =

Here λ satisfies the non-isospectral condition (16). Then the Lax Eq. (15) gives a higher-dimensional MGKdV equation, or modified GCBS equation: 1 1 1 2 vt − v2 vz − v x ∂−1 x v z + v xxz 4 8 4 xG (t) G (t) v− v x = 0. − 2G(t) 2G(t)

(19)

One can easily check that by the dimensinal-reduction z = x, Eq. (19) can be reduced to the MGKdV equation [10]: xG (t) 3 1 G (t) vt − v2 v x + v xxx − v− v x = 0. (20) 8 4 2G(t) 2G(t) (ii) Higher-dimensional generalized CDF equation with variable coefficients Using LGT, we obtain a following Lax pair:

(22)

4.

(23)

Hodograph Transformation and Higher-Dimensional Generalized Harry-Dym Equation

It is well-known that the hodograph transformation [34] links the modified KdV equation and the Harry-Dym equation [35]. We extend this transformation as follows: 1 1 ∂X = √ u∂ x . √ G(t) G(t)

(24)

Acting itself twice, the following operator is obtained, 1 2 1 2 2 ∂X = u ∂ x + uu x ∂ x G(t) G(t) 1 2 2 1 = u ∂x + √ u x ∂X . G(t) G(t) √ Setting G(t)u x = −v, the following relation: 1 2 1 2 2 u ∂x , ∂X + v∂X = G(t) G(t)

(25)

(26)

is obtained. The right hand side seems to be an operator L given a generalized Harry-Dym(GHD) equation for u = u(x, t) because the left hand side is one of the MGKdV Eq. (20) for v = v(X, t). Here using LGT for the operator L (26) the Lax pair for u = u(x, z, t) is given by L = †

1 1 2 2 u ∂x − λ ≡ LGHD − λ, G(t) G(t)

See also Appendix.

(27)


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2 T = u∂z LGHD + u3 1 + ∂−1 (u /u ) ∂3x z x 3 3 +(u2 − u3 )∂2x ∂z + u2 u x − 2u2 uz + uuz 2 2 3 2 −1 + u u x ∂ x (uz /u2 ) ∂2x + ∂t , 2

5.

(28)

which λ satisfies the non-isospectral condition (16). Then the Lax Eq. (15) gives higher-dimensional GHD equation: 1 1 1 1 ut + u3 u xxx − uu x u xz + uu xx uz + u2 u xxz 4 4 4 4 (t) 1 3 G 2 u = 0, + u u xxx ∂−1 x (uz /u ) − 4 2G(t)

(29)

respectively. Setting z = x, Eq. (29) is reduced to the (1 + 1)dimensional GHD equation: 1 G (t) u = 0. ut + u3 u xxx − 4 2G(t)

(30)

However by the transformations of variables: u(x, z, t) = G(t)1/2 U(ξ, η, τ), ξ = x, 3/2 η = G(t)1/2 z, τ = ∂−1 , t G(t)

(31)

Eq. (29) reads a higher-dimensional HD equation for U = U(ξ, η, τ) with constant coefficients [36]: 1 1 1 Uτ + U 3 Uξξξ − UUξ Uξη + UUξξ U η 4 4 4 1 1 2 + U 2 Uξξη + U 3 Uξξξ ∂−1 ξ (U η /U ) = 0. 4 4

1 1 2 2 u ∂ x + ∂y ≡ LGHD + ∂y , G(t) G(t)

Modern theories of nonlinear science have been highly and widely developed. In mechanical, physical, mathematical and engineering sciences, nonlinear systems play an important role, especially those with variable coefficients for certain realistic situations. In this paper, we have obtained higher-dimensional generalized KdV (11), modified generalized KdV (19), Calogero-Degasperis-Fokas (23) and Harry-Dym (29), (35) equations with variable coefficients. Equation (11) is a higher-dimensional version of (1 + 1) dimensional KdV equations with variable coefficients appeared in [5], [7], [8], [11]–[14]. In [19] we have reported some exact solutions of higher-dimensional equations with variable coefficients given in this paper. And we are searching variablecoefficients versions of the Sine-Gordon and Nonlinear Schrödinger equations in (2 + 1) dimensions by the WTC method. We will study relations of integrable nonlinear equations with variable coefficients derived in this paper to realistic situations in mechanical, physical, mathematical and engineering sciences. Though our program is going well now, there are still many things worth studying to be seen. Acknowledgments

(32)

We have given only the extension of the operator T to higher-dimensions with the additional spatial coordinate z. Of course the extension of the operator L can be done. Let us here mention the extension of the operator L for GHD Eq. (30). When we extend the operator L (27) such that L=

Conclusions

(33)

then an operator is given by 1 2 −1 2 T = u∂ x LGHD + u 3G(t)∂ x uy /u + u x ∂2x + ∂t 2 . (34) Here u = u(x, y, t) with the additional spatial coordinate y. We are able to construct the following equation from the Lax Eq. (15) :

2 −1 2 u ∂ x uy /u 3 1 3 G (t) y u+ = 0, (35) ut + u u xxx − 4 2G(t) 4 u respectively. Ignoring y-dependence, Eq. (35) is reduced to the (1 + 1)-dimensional GHD one (30). We have not found such a transformation as (31) for Eq. (35) yet.

Symbolic calculations to derive the Lax pairs appeared in this paper were partially carried out at the Yukawa Institute Computer Facility, Kyoto University, Japan. The authors would like to thank Dr. A. Nakamula, Dr. T. Tsuchida and anonymous referees for useful comments. This work was in part supported by the Sasagawa Scientific Research Grant (#13-089K) from The Japan Science Society, the First-Bank of Toyama Scholarship Foundation and Grant-in-Aid for Scientific Research (#15740242) from the Ministry of Education, Culture, Sports, Science and Technology. References [1] G.L. Lamb Jr., Elements of Soliton Theory, Wiley, New York, 1980. [2] V.E. Zakharov, ed., What Is Integrability?, Springer-Verlag, 1991. [3] J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” J. Math. Phys., vol.24, pp.522–526, March 1983. [4] R. Conte, ed., The Painlevé Property One Century Later, SpringerVerlag, 2000. [5] T. Brugarino and P. Pantano, “The integration of Burgers and Korteweg-de Vries equations with nonuniformities,” Phys. Lett. A, vol.80, pp.223–224, Dec. 1980. [6] F. Calogero and A. Degasperis, “Exact solution via the spectral transform of a generalization with linearly x-dependent coefficients of the modified Korteweg-de Vries equation,” Lett. Nuovo Cimento, vol.22, pp.270–279, March 1978. [7] W. Oevel and W.-H. Steeb, “Painlevé analysis for a time-dependent Kadomtsev-Petviashvili equation,” Phys. Lett. A, vol.103, pp.239– 242, July 1984.


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[8] W.-H. Steeb and J.A. Louw, “Parametrically driven Sine-Gordon equation and Painlevé test,” Phys. Lett. A, vol.113, pp.61–62, Dec. 1985. [9] T. Chou, “Symmetries and a hierarchy of the general KdV equation,” J. Phys. A, vol.20, pp.359–366, Feb. 1987. [10] T. Chou, “Symmetries and a hierarchy of the general modified KdV equation,” J. Phys. A, vol.20, pp.367–374, Feb. 1987. [11] N. Joshi, “Painlevé property of general variable-coefficient versions of the Korteweg-de Vries and non-linear Schrödinger equations,” Phys. Lett. A, vol.125, pp.456–460, Dec. 1987. [12] L. Hlavatý, “Painlevé analysis of nonautonomous evolution equations,” Phys. Lett. A, vol.128, pp.335–338, April 1988. [13] T. Brugarino, “Painlevé property, auto-Bäcklund transformation, Lax pairs, and reduction to the standard form for the Kortewegde Vries equation with nonuniformities,” J. Math. Phys., vol.30, pp.1013–1015, May 1989. [14] T. Brugarino and A.M. Greco, “Painlevé analysis and reducibility to the canonical form for the generalized Kadomtsev-Petviashvili equation,” J. Math. Phys., vol.32, pp.69–72, Jan. 1991. [15] Y.-T. Gao and B. Tian, “On a variable-coefficient modified KP equation and a generalized variable-coefficient KP equation with computerized symbolic computation,” Int. J. Mod. Phys. C, vol.12, pp.819– 833, July 2001. [16] Y.-T. Gao and B. Tian, “Variable-coefficient balancing-act algorithm extended to a variable-coefficient MKP model for the rotating fluids,” Int. J. Mod. Phys. C, vol.12, pp.1383–1389, Nov. 2001. [17] M. Wang, Y. Wang, and Y. Zhou, “An auto-Bäcklund transformation and exact solutions to a generalized KdV equation with variable coefficients and their applications,” Phys. Lett. A, vol.303, pp.45–61, Oct. 2002. [18] R.C. Cascaval, “Variable coefficient KdV equations and waves in elastic tubes,” Preprint, 2002. [19] T. Kobayashi and K. Toda, “Extensions of nonautonomous soliton equations to higher dimensions,” Preprint, 2005. [20] T. Kobayashi and K. Toda, “Extensions of nonautonomous nonlinear integrable systems to higher dimensions,” Proc. Inte’l Symp. on Nonlinear Theory and Its Applications, vol.1, pp.279–282, Dec. 2004. [21] F. Calogero, “A method to generate solvable nonlinear evolution equations,” Lett. Nuovo Cimento, vol.14, pp.443–447, March 1975. [22] O.I. Bogoyavlenskii, “Overturning solitons in new two-dimensional integrable equations,” Math. USSR-Izv., vol.34, pp.245–259, Feb. 1990. [23] J. Schiff, “Integrability of Chern-Simons-Higgs vortex equations and a reduction of the selfdual Yang-Mills equations to threedimensions,” NATO ASI Ser. B, vol.278, pp.393–405, Plenum, New York, 1992. [24] S.-J. Yu, K. Toda, N. Sasa, and T. Fukuyama, “N soliton solutions to the Bogoyavlenskii-Schiff equation and a quest for the soliton solution in (3 + 1) dimensions,” J. Phys. A, vol.31, pp.3337–3347, April 1998. [25] K. Toda, S.-J. Yu, and T. Fukuyama, “The Bogoyavlenskii-Schiff hierarchy and integrable equations in (2+1) dimensions,” Rep. Math. Phys., vol.44, pp.247–254, Aug. 1999. [26] S.-J. Yu and K. Toda, “Lax pairs, Painlevé properties and exact solutions of the Calogero Korteweg-de Vries equation and a new (2 + 1)dimensional equation,” J. Nonlinear Math. Phys., vol.7, pp.1–13, Feb. 2000. [27] K. Toda, “Extensions of soliton equations to non-commutative (2+1) dimensions,” JHEP Proc. Workshop on Integrable Theories, Solitons and Duality, PRHEP-unesp2002/038, March 2003. [28] M. Hamanaka and K. Toda, “Towards noncommutative integrable systems,” Phys. Lett. A, vol.316, pp.77–83, Sept. 2003. [29] P.A. Clarkson, P.R. Gordoa, and A. Pickering, “Multicomponent equations associated to non-isospectral scattering problems,” Inv. Prob., vol.13, pp.1463–1476, Dec. 1997. [30] P.R. Gordoa and A. Pickering, “Nonisospectral scattering problems:

[31] [32] [33]

[34] [35] [36]

A key to integrable hierarchies,” J. Math. Phys., vol.40, pp.5749– 5786, Nov. 1999. P.G. Estévez, “A nonisospectral problem in (2+1) dimensions derived from KP,” Inv. Prob., vol.17, pp.1043–1052, Aug. 2001. A.S. Fokas, “A symmetry approach to exactly solvable evolution equations,” J. Math. Phys., vol.21, pp.1318–1325, June 1980. F. Calogero and A. Degasperis, “Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform,” J. Math. Phys., vol.22, pp.23–31, Jan. 1981. N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, D. Reidel, Dordrecht, 1985. M. Błaszak, Multi-Hamiltonian Theory of Dynamical Systems, Springer-Verlag, 1998. Yu. S, K. Toda, and T. Fukuyama, “A quest for the integrable equation in (3+1) dimensions,” Theor. Math. Phys., vol.122, pp.256–259, Feb. 2000.

Appendix:

Note on LGT and Non-isospectral Problems

• LGT: We briefly introduce LGT, taking the ordinary KdV equation as an example. Let us take an operator L: L = ∂2x + u ≡ LKdV .

(A· 1)

Here ∂ x is a differential operator whose action on an arbitrary function F(x) is given by ∂ x (F) = F x + F∂ x . The form of operator T is given by + ∂t , T = ∂ x (L) + T KdV

(A· 2)

is an unknown operator and ∂ x (L) = ∂3x + where T KdV u∂ x + u x . If operators T is suitably chosen, the Lax Eq. (15) for the operators (A· 1) and (A· 2) gives the KdV equation and the exact form of the operator T KdV in parallel. The Lax Eq. (15) reads ] = 0. [L, ∂ x (L) + ∂t ] + [L, T KdV

(A· 3)

The first term in the left-hand side of Eq. (A· 3) gives [L, ∂ x (L) + ∂t ] = −u x ∂2x − (ut + uu x ).

(A· 4)

that So we can adopt the form of the operator T KdV involves, at least, a first order differential operator: = U∂ x + V, T KdV

(A· 5)

where U and V are functions of u, u x , u xxx , ..., and possibly also of x and t. Then the second term in the left-hand side of Eq. (A· 4) gives ] = 2U x ∂2x + (2 V x + U xx )∂ x [L, T KdV +V xx − Uu x .

(A· 6)

Substituting (A· 4) and (A· 6) into Eq. (A· 3), we obtain ux u (A· 7) U = + C1 , V = − + C2 , 2 4 where C1 and C2 are integration constants. Note here that two integral constants can be taken zero without


2553

loss of generality. Then the (ordinary) KdV equation: 3 1 (A· 8) ut + uu x + u xxx = 0, 2 4 is derived. • Non-isospectral problems: Let a non-isospectral Lax pair with a wave function ψ = ψ(x, z, t) be defined as Lψ = λ(z, t)ψ, z + ∆, −ψt = Tψ ≡ OP∂

(A· 9) (A· 10)

and ∆ are differential operators without ∂z . where OP Then the Lax Eq. (15) reads the following formulation: [L, T + ∂t ]ψ = (λt − λz OP)ψ.

(A· 11)

For example, when L = LKdV , the operator T is given by z + ∆ + ∂t T = T + ∂t = OP∂ = ∂z LKdV + T + ∂t .

(A· 12)

Because there is no terms of ∂z in the operator T , we impose = LKdV = ∂2x + u = λ. OP

(A· 13)

Equation (A· 11) is reduced to the following equation: [L, T + ∂t ]ψ = (λt − λλz )ψ.

(A· 14)

So if λt = λλz , the commutation relation gives the CBS Eq. (2).

Tadashi Kobayashi was born in 1979. He received the Master degree in Infomatics from Kyoto University in 2004. His research area is Soliton theory.

Kouichi Toda was born in 1971. He received the MSc and Ph.D. degrees in Mathematical Physics from Ritsumeikan University in 1998 and 2001, respectively. His research area is Integrable Systems and Mathematical Physics.