Soliton Solution of Good Boussinesq Equation Lu Trong Khiem Nguyen1 Lehrstuhl f¨ur Mechanik-Materialtheorie, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany.
Abstract Using Hirota’s direct bilinear method, we develop the soliton solution of the good Boussinesq equation. A mathematical justification of the solution formula is presented. Besides, the obtained result is compared to the existing solution obtained by the Wronskian formulation and a detailed discussion on their structures is provided. Keywords: Hirota’s bilinear method, Soliton equation, Boussinesq equation PACS: 02.30.Ik, 02.30.Jr
1. Introduction
The soliton theory plays an important role in modeling different processes of wave propagation. The Korteweg-de Vries (KdV) equation describes the shallow-water waves propagating in one direction [1], whereas the Boussinesq (BSQ) equation allows the description of waves propagating in two directions [2, 3]. Besides, the BSQ equation finds its application to other physical phenomena, for instance, magnetosound waves in plasma [4], magnetoelastic waves in antiferromagnetics [5] and electromagnetic waves in nonlinear dielectrics [6]. Additionally, one of the rich fields of physics in which the soliton usually appears is the Frenkel-Kontorova model [7] . The KdV equation was first numerically investigated by Zabusky and Kruskal to discover the particle-like behavior of the solutions [8]. Hereby, such behavior explains the suffix -on in the soliton solution. The name soliton equation is consequently coined for a class of equations which is integrable for the soliton solutions. These equations have attracted attention of many mathematicians as well as physicists since the early of 1970s due to its integrability for exact solutions by using the Hirota’s bilinear method or, alternatively, the inverse scattering transform. The discovery of an infinite number of conservation laws of the KdV equation by Miura helped in developing the inverse scattering transform [9]. Around the same time, Hirota developed independently a direct method to deal with such kind of equation and successfully solved the KdV equation [10]. This method was subsequently applied to the modified KdV equation [11], the sine-Gordon equation [12], and the BSQ equation [13]. The method proves its effectiveness due to easy construction of closed forms of the solutions. The BSQ equation with constant coefficients can be expressed in the following form utt + αu xx + β(u2 ) xx + γu xxxx = 0.
(1)
If γ is positive, we classify it as the good BSQ equation since it is well-posed. On the other hand, if γ is negative, it is realized as the bad BSQ equation. Recently, the more general form of the BSQ equation utt + αu xx + β(uk ) xx + γu xxxx = 0, which is called the Boussinesq equation with power law, has also been intensively studied for exact solutions. The
1
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variant equation utt + αu xx + β(u2m ) xx + γu xxxx = 0,
m∈N
was solved for 1-soliton solution by using the Ansatz method and the G0 /G expansion method in two papers [18, 19], respectively. Then, these results were extended further to obtain more different analytical solutions by the mapping method and the singular soliton solutions are specifically constructed by the Weierstrass elliptic function method [20]. In addition to the BSQ equation with the nonlinearity of even power, the cubic Boussinesq equation utt − u xx − 2(u3 ) xx + u xxxx = 0 was examined for the solitary wave with the aid of the tanh method [21]. In addition, the phenomenon of shockwaves associated with this Boussinesq-type equation was examined in [22, 23]. Two-dimension versions of the Boussinesq equation have been treated in the recent work [24] by the Ansatz method, which turns out equivalent to the tanh method. Recently, the Wronskian technique has been developed to obtain many other types of solutions of soliton equations such as rational and positon solutions [14, 15, 16]. Later, Chun-Xia Liu et. al. employed and upgraded the Wronskian formulation to obtain further the so-called negaton and complexiton solutions of the BSQ equation [17]. Despite this method is powerful, it, in some cases, fails to extract the continuous soliton solution. The bad variant of equation (1) utt − u xx − 6(u2 ) xx − u xxxx = 0 was investigated by Hirota [13] for multi-collision of solitons. In the rest of the paper, we will consider equation 1 utt − u xx + (u2 ) xx + u xxxx = 0, 3
(2)
which was treated in [17]. The paper is organized as follows. In Section 2, the bilinear form other than that in [17] is derived and the closed form of N-soliton solution is subsequently stated. Its mathematial verification is postponed to the Section 3. The last section adds some remarks on the algebraic structures of the established solutions and those obtained by the Wronskian formulation in [17]. Their difference is, thereby, indicated. A short conclusion follows. 2. Exact N-soliton solutions
We look for the solution of equation (2) in the form of bi-logarithm transformation u(x, t) = 2∂2x log f (x, t) + a = 2
f f xx − f x2 + a. f2
Substituting this into equation (2), integrating the obtained equation with respect to x twice and choosing the zero constants of integration, we obtain 1 (2 log f )tt − (2 log f ) xx + [(2 log f ) xx ]2 + 2a(2 log f ) xx + (2 log f ) xxxxx = 0. 3 We adopt here the definition of the bilinear differential operator defined by Hirota !n !m ∂ ∂ ∂ ∂ n m D x Dt f · g = − − f (x, t)g(ξ, τ)|ξ→x,τ→t , ∂x ∂ξ ∂t ∂τ
2
(3)
where the arrow indicates that the variables ξ and τ should be replaced with x and t after the differentiation, respectively. Each term in the above equation can be rewritten in terms of these operators as follows 2( f ftt − ft2 ) D2t f · f = , f2 f2 2( f f xx − f x2 ) D2x f · f = , = f2 f2 2( f f − f 2 ) 2 2) 2( f f xxxx − 4 f x f xxx + 3 f xx xx x = − 3 2 f f2 D2 f · f 2 D4 f · f = x 2 −3 x 2 . f f
(2 log f )tt = (2 log f ) xx (2 log f ) xxxx
Substituting these expressions into equation (3), simplifying and multiplying it by the common denominator f 2 , it can be realized in the following bilinear form 1 [D2t − αD2x + D x ] f · f = 0, 3
α = 1 − 2a.
(4)
or in the more explicit form 1 2 f ftt − ft2 − α( f f xx − f x2 ) + ( f f xxxx − 4 f x f xxx + 3 f xx ) = 0. 3 Therefore, solving equation (2) becomes solving the above equation which seems much more complicated but is indeed bilinear in the Hirota’s sense. In this context, we call a solution f = f (x, t) of equation (4) a fundamental function. N-soliton solution . By investigating the pattern of the low-order solution using the perturbation method, we propose here the solution f (x, t) of the bilinear form (4) for N-soliton f f xx − f x2 + a, f2 N X X γ(i1 , . . . , in ) exp(ηi1 + · · · + ηin ), f (x, t) = 1 +
u(x, t) = 2
n=1 C Nn
where
ηi = ki x + ωi t + δi ,
ω2i = ki2 (α − ki2 /3), (n) Y
γ(i1 , . . . , in ) =
m
