N-soliton solutions, Bäcklund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg–de Vries equation

Abstract

In this paper, a generalized variable-coefficient fifth-order Korteweg–de Vries equation is investigated. Based on the Hirota bilinear method and symbolic computation, the N-soliton solutions, Bäcklund transformation and Lax pair are presented. Furthermore, the characteristic-line method is applied to discuss the solitonic propagation and collision under the effects of the variable coefficients, from which the following conclusions can be derived: (i) solitonic amplitude decreases as the positive coefficient of the line-damping term increases; (ii) coefficients of the dispersive and dissipative terms determine the solitonic direction and speed by changing the sign and absolute value of the solitonic velocity; (iii) the appearances of the characteristic lines depend on the forms of the variable coefficients.

Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

“…In this paper, with dependent-variable transformations (8) and (27), we have converted (1) to the Manakov system [i.e., (9)], and (7) to a linear equation [i.e., (30)] together with Expression (29). Moreover, we have derived some new solutions for (1) and (7) based on Transformations (27) and (8), seen as Solutions (18), (24), and (36), respectively.…”

Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary waves

“…Following the similar procedures in [54][55][56][57][58], the characteristic line and velocity v for each soliton of (1) can be respectively derived as…”

Soliton-like solutions to the generalized Burgers-Huxley equation with variable coefficients

“…Due to the inhomogeneities of media and nonuniformities of boundaries, the variable-coefficient nonlinear evolution equations can be used to describe the real physical backgrounds [1]. Variable coefficients are able to represent, e.g., the fluid density, viscosity, velocity, inhomogeneities in optic fibers and external force on plasmas with cosmic dust [2].…”

“…One of the important exact solutions is the quotedblleft soli-ton solutionquotedblright , because the soliton approach is universal in different fields of modern physics. In order to obtain the exact solutions, a number of ansatz methods have been developed, such as, the subsidiary ordinary differential equation method (sub-ODE method for short) [3,4], solitary wave ansatz method [5,6], sine-cosine method [7,8], Hirota bilinear method [1,9], F-expansion method [10], tanh method [11][12][13][14], and so on. Without these modern methods of integrability, many such equations would not have been solved, thus leaving many scientific questions unanswered [5].…”