Bilinearization and new soliton solutions of Whitham-Broer-Kaup equations with time-dependent coefficients

Abstract

In this paper, Whitham-Broer-Kaup (WBK) equations with time-dependent coefficients are exactly solved through Hirota's bilinear method. To be specific, the WBK equations are first reduced into a system of variable-coefficient Ablowitz-KaupNewell-Segur (AKNS) equations. With the help of the AKNS equations, bilinear forms of the WBK equations are then given. Based on a special case of the bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions and the uniform formulae of n-soliton solutions are finally obtained. It is graphically shown that the dynamical evolutions of the obtained one-, two-and three-soliton solutions possess time-varying amplitudes in the process of propagations.

Bilinearization and fractional soliton dynamics of fractional Kadomtsev-Petviashvili equation

“…Hirota's [3] bilinear method is a famous analytical method for constructing exact and explicit n-soliton solutions of non-linear PDE. Since put forward in 1970, Hirota's bilinear method has achieved considerable developments [4][5][6][7][8][9][10][11][12][13][14]. With the close attentions of fractional calculus and its applications [15][16][17][18][19][20][21][22][23][24], some of the natural questions are whether Hirota's bilinear method can be extended to non-linear PDE of fractional orders and what about the fractional soliton dynamics and integrability of fractional PDE.…”

A Novel Approach to a Time‐Dependent‐Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

“…In 2006, Zhang and Xia [30] generalized the F-expansion method by introducing a new and more general ansätz. The present paper is motivated by the desire to extend the generalized F-expansion method [30] to the new and more general tcdWBK system [31,32]:…”

Analytical insights into three models: Exact solutions and nonlinear vibrations

“…The second model to consider is the time-dependent-coefficient nonlinear Whitham-Broer-Kaup system 19,20 u…”

“…The present paper is motivated by the desire to show the analytical method 8 can be used for some special types of nonlinear vibration equations. For this purpose, we shall consider three models in this paper, they are the generalized nonlinear Schrödinger equation with distributed coefficients, 9,18 the time-dependent-coefficient nonlinear Whitham – Broer – Kaup system 19,20 and the fractional nonlinear vibration governing equation which can be thought of as a generalized form in the fractal space of the model for an embedded single-wall carbon nanotube. 16 Since carbon nanotube is discontinuous and to be more exactly the classical continuum mechanics becomes invalid, then the fractal calculus becomes optional (see those in literature 21–25 for examples).…”

“…The second model to consider is the time-dependent-coefficient nonlinear Whitham – Broer – Kaup system 19,20 where γ1, γ2 and γ3 representing different dispersion and dissipation forces are all differentiable functions of the time variable t, and c is an arbitrary constant. It is worth mentioning that equations (2) and (3) include some existing important models as special cases, such as the approximate equations for long water waves, 26 and the Boussinesq – Burgers equations.…”