N-soliton solutions for the integrable bidirectional sixth-order Sawada–Kotera equation

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Abstract

In this work, the integrable bidirectional sixth-order Sawada–Kotera equation is examined. The equation considered is a KdV6 equation that was derived from the fifth order Sawada–Kotera equation. Multiple soliton solutions and multiple singular soliton solutions are formally derived for this equation. The Cole–Hopf transformation method combined with the Hirota’s bilinear method are used to determine the two sets of solutions, where each set has a distinct structure.

Introduction

In [1], [2], [3], [4], [5], [6], [7], [8], the Painlevé analysis and other methods were used to test the complete integrability of the sixth-order nonlinear equation (KdV6), given byuxxxxxx+auxuxxxx+buxxuxxx+cux2uxx+dutt+euxxxt+fuxuxt+gutuxx=0,where a, b, c, d, e, f, and g are arbitrary parameters. In [2], [4], it was indicated that there are four distinct cases of relations between the parameters for Eq. (1) to pass the Painlevé test. The four KdV6 models were given by5x-1vtt+5vxxt-15vvt-15vxx-1vt-45v2vx+15vxvxx+15vvxxx-vxxxxx=0,5x-1vtt+5vxxt-15vvt-15vxx-1vt-45v2vx+752vxvxx+15vvxxx-vxxxxx=0,utt-uxxxt-2uxxxxxx+18uxuxxxx+36uxxuxxx-36ux2uxx=0,and the fourth model is given byx3+8uxx+4uxxut+uxxx+6ux2=0.

The bidirectional models (2), (3) were derived by Dye and Parker in [1] by using the Lax pair of Sawada–Kotera and Kaup–Kupershmidt equations, respectively. The third model is equivalent to the Drinfeld–Sokolov–Satsuma–Hirota system of coupled KdV equations [2], of which a fourth-order recursion operator was found in [4]. This model was examined in [7]. However, the fourth KdV6 model that was developed in [2] turned to be completely integrable as shown in [2], [5], [6], [8]. The fourth form of the KdV6 Eq. (5) can be rewritten asuxxxxxx+20uxuxxxx+40uxxuxxx+120ux2uxx+uxxxt+8uxuxt+4utuxx=0,where a=20,b=40,c=120,d=0,e=1=112(f+g),f=8,g=4,f=2g.

In [2], a Lax pair, Bäcklund self-transformation, travelling wave solutions and third order generalized symmetries were found for the case (5). Moreover, it was found that the KdV6 Eq. (5) is associated with the same spectral problem as of the potential KdV equation.

In the literature, three powerful methods, namely, the inverse scattering method, the Bäcklund transformation method [1], [2], [3], [4], [5], [6], [7], [8], and the Hirota bilinear method [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20] were thoroughly used to derive the multiple-soliton solutions of these equations. The Hirota’s bilinear method is rather heuristic and possesses significant features that make it practical for the determination of multiple soliton solutions [21], [22], [23], [24], [25], [26], [27], [28] for a wide class of nonlinear evolution equations in a direct method. Moreover, the tanh method [29], [30], [31], [32] was used to determine single soliton solution. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.

Hirota and Ito in [9] examined the phenomenon of two solitons near resonant state, two solitons at the resonant state, and two solitons become after colliding with each other, where it was proved in [9] that two solitons become singular after colliding with each other, in the sense that regular solitons with sech2 profiles are transmitted into singular solitons with cosech2 profiles through the interaction. The resonant phenomenon will be examined here. This requires the examination of the phase shift to decide if resonance exists or do not exist.

In this work, the bidirectional Sawada–Kotera (KdV6) (2) will be studied using the Cole–Hopf transformation method combined with the Hirota’s bilinear method. Our goal is to derive multiple regular soliton solutions and multiple singular soliton solutions for this model. Using the potential v=ux, the bidirectional Sawada–Kotera model can be written as a KdV6 model in the form5utt+5uxxxt-15uxuxt-15uxxut-45(ux)2uxx+15uxxuxxx+15uxuxxxx-uxxxxxx=0.

The Hirota’s bilinear method is well known in the literature, therefore, our focus will be on applying this method.

Section snippets

Analysis of the method

As stated before, we will examine the KdV6 Eq. (7) to determine multiple solitons solutions and multiple singular soliton solutions.

Discussion

The bidirectional Sawada–Kotera equation is examined for complete integrability. In this work, we proved the integrability by the determination of N-soliton solutions for finite N. Moreover, the multiple singular soliton solutions for this equation were also determined. The resonance relation does not exist for this equation.

References (32)

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