New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

Abstract

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota's bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz-Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived 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 shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

Inverse scattering transform for new mixed spectral Ablowitz-Kaup-Newell-Segur equations

“…Since the steps of IST method solving the initial value problem (IVP) of non-linear PDE are very similar to those of Fourier transform used to deal with the IVP of linear equations, the IST method is often referred to as non-linear Fourier analysis [2]. With the development of the IST method, more and more analytical methods have been developed for solving nonlinear PDE, such as Hirota's bilinear method [3][4][5][6][7], Painleve truncation expansion [8][9][10], homogeneous balance method [11,12], auxiliary equation methods [13][14][15], variational iteration method (VIM) [16], homotopy perturbation method (HPM) [17], exp-function (EXP) method [18][19][20], and so on. Thanks to the powerful VIM, HPM, and EXP method proposed by Ji-Huan He, a large number of exact and approximate solutions have been obtained in different fields.…”

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.…”