N-soliton solutions for the integrable bidirectional sixth-order Sawada–Kotera equation
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 bywhere 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 byand the fourth model is given by
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 aswhere .
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 , the bidirectional Sawada–Kotera model can be written as a KdV6 model in the form
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)
- et al.
Integrable KdV systems: recursion operators of degree four
Phys. Lett. A
(1999) KdV6: an integrable system
Phys. Lett. A
(2008)The integrable KdV6 equations: multiple soliton solutions and multiple singular soliton solutions
Appl. Math. Comput.
(2008)The Cole–Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld–Sokolov–Satsuma–Hirota equation
Appl. Math. Comput.
(2009)- et al.
A note on the integrable KdV6 equation: multiple soliton solutions and multiple singular soliton solutions
Appl. Math. Comput.
(2009) - et al.
Symbolic methods to construct exact solutions of nonlinear partial differential equations
Math. Comput. Simul.
(1997) Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method
Appl. Math. Comput.
(2007)Multiple-front solutions for the Burgers equation and the coupled Burgers equations
Appl. Math. Comput.
(2007)New solitons and kink solutions for the Gardner equation
Commun. Nonlinear Sci. Numer. Simul.
(2007)Multiple-soliton solutions for the Boussinesq equation
Appl. Math. Comput.
(2007)
The Hirota’s direct method and the tanh–coth method for multiple-soliton solutions of the Sawada–Kotera–Ito seventh-order equation
Appl. Math. Comput.
Multiple-front solutions for the Burgers–Kadomtsev–Petvisahvili equation
Appl. Math. Comput.
Multiple-soliton solutions for the Lax–Kadomtsev-Petvisahvili (Lax–KP) equation
Appl. Math. Comput.
The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves
Appl. Math. Comput.
Multiple-soliton solutions of two extended model equations for shallow water waves
Appl. Math. Comput.
The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations
J. Comput. Appl. Math.
Cited by (21)
A new method of generating the lump molecules and localized interaction solutions to the (2+1)-dimensional SK equation
2023, Physics Letters, Section A: General, Atomic and Solid State PhysicsA new set and new relations of multiple soliton solutions of (2 + 1)-dimensional Sawada–Kotera equation
2021, Communications in Nonlinear Science and Numerical SimulationLie symmetry analysis and different types of solutions to a generalized bidirectional sixth-order Sawada–Kotera equation
2017, Chinese Journal of PhysicsCitation Excerpt :In the preceding section, the explicit power series solutions of Eq. (1.1) are derived. In this section,we will construct another form of solutions for Eq. (1.1) with the use of Bell polynomials [33–35] and symbolic computation method [36–52]. Based on above detailed analysis, the following theorem is easily established:
A hierarchy of new nonlinear evolution equations and generalized bi-Hamiltonian structures
2015, Applied Mathematics and ComputationCitation Excerpt :The development of the inverse transform has shown that certain nonlinear evolution equations possess a number of remarkable properties, including the existence of solitons, an infinite set of conservation laws [4] and Hamiltonian formalism [5,6]. Furthermore, the key feature of an integrable nonlinear evolution equation is the fact that it can be expressed as a compatibility pair of two linear spectral problems, i.e., a Lax pair, which plays an important role in the inverse scattering transformation, Darboux transformation [7–9] and so on [10–13]. However, it is difficult to associate the nonlinear evolution equations with corresponding spectral problems.
Single soliton solutions of the coupled nonlinear Klein-Gordon equations with power law nonlinearity
2014, Applied Mathematics and ComputationMultiple soliton solutions and fusion interaction phenomena for the (2+1)-dimensional modified dispersive water-wave system
2013, Applied Mathematics and ComputationCitation Excerpt :With the aid of these solutions, people may give a good insight into the physical aspects of the problems, so searching for explicit solutions of NPDEs plays an important role in solving the nonlinear problem. To find new explicit solutions, some effective methods have been proposed, such as the inverse scattering method [1], Darboux transformation (DT) [2,3], Bäcklund transformation [4], the mapping method and extended mapping method [5,6], the Hirota bilinear method [7–10] etc. Among these methods, the bilinear method is an effective direct method to construct exact solutions of NPDEs.