Exact Solutions and Conservation Laws of the Generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt System via Lie Symmetry and Riccati Equation Methods

Introduction

Nonlinear partial differential equations (NPDEs) are fundamental in simulating diverse phenomena in physics, biology, and engineering. This encompasses their applications in areas such as optics, communications, and chemical processes [1, 2–3]. On the basis of the research on the NPDEs in shallow water wave dynamics, we conduct a study on the extended Generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt (GWBKBK) system [4, 5–6]. The system is expressed as follows:

1

where u(x, t) represents the horizontal wave velocity, v(x, t) corresponds to the wave displacement from its equilibrium position, and the coefficients and are constants.

System (1) has seen some special cases presented:

For , system (1) reduces into a Whitham-Broer-Kaup system, which is capable of simulating dispersive long waves in shallow waters [7].

When and , it becomes a long-wave system that simulates shallow water waves with diffusion effects [8].

Setting and , transforms the system into a dispersiveless long-wave system, useful for modeling long waves in shallow environments [9].

For and , the system represents a Kaup-Boussinesq model for shallow water wave phenomena [10].

When , and , it models dispersive long waves in shallow water environments [11].

Setting , and , the system becomes a (1+1)-dimensional Broer-Kaup system that models long waves in shallow water [12].

For , and , it represents a Kaup-Boussinesq system modeling shallow water wave dynamics [13].

When and , it reduces to a variant of the Boussinesq system, useful for modeling water waves [14].

Setting , and , it becomes a (1+1)-dimensional Broer-Kaup-Kupershmidt system, also relevant for modeling long waves in shallow water [15].

The derivation of exact solutions for NPDEs is essential for understanding complex physical and mathematical phenomena. Significant efforts have been made to derive exact solutions for various NPDEs in recent years. Various methods have been proposed to solve the problem of finding exact solutions, including: Bäcklund transformation [16], Hirota’s bilinear method [17], Darboux transformation [18], Lie symmetry analysis [19], exp-function method [20], G’/G expansion method [21], F-expansion method [22] and Jacobi elliptic function expansion method [23].

However, to the best of our knowledge, Lie symmetry analysis has not been previously applied to the GWBKBK system (1) in the literature. In this work, we employ Lie symmetry to reduce the...