Multiple soliton solutions for the Whitham–Broer–Kaup model in the shallow water small-amplitude regime

Abstract

In this work we investigate the Whitham–Broer–Kaup model for dispersive long waves in the shallow water small-amplitude regime. We use the simplified form of Hirota's direct method to determine multiple soliton solutions for the integrable Whitham–Broer–Kaup model. We exhibit generalized multiple soliton solutions.

Simplified Homogeneous Balance Method and Its Applications to the Whitham-Broer-Kaup Model Equations

“…the expressions (14) become the 1-soliton solutions, 2-solliton solutions and 3-soliton solutions for the WBK model equations, respectively, these results coincide with those obtained by using the simplified form of Hirota's method in [6] one by one. In particular, when…”

“…Equations (1) and (2) become the variant Boussinesq equations. In the latest paper [6], the multiple soliton solutions of Equations (1) and (2) have been obtained by using the simplified form of Hirota's direct method.…”

“…In the present paper, we will apply a simplified homogeneous balance method to investigate the WBK model Equations (1) and (2), by this method a nonlinear transformation that from the solution for a linear equation to the solution for the WBK model equations is derived, and more type of solutions than those given in [6] are obtained via the nonlinear transformation successfully.…”

Optimal System and Group Invariant Solutions of the Whitham-Broer-Kaup System

“…In fact, these solitary waves must interact with other waves or spread on some nonconstant background. Here, the forms of solutions (29) and (36) directly embody this kind of interactions between the soliton and other nonlinear waves.…”

“…A large class of almost periodic solutions are found algebrogeometrically by Matveev and Yavor [27]. Some soliton excitations and periodic waves solutions without dispersion relation are obtained in [28,29]. The classical symmetries and corresponding one-dimensional optimal system of (1)-(2) are performed in [30].…”