Dynamics of Shallow Water Waves with Various Boussinesq Equations

Abstract

Attempt has been made to construct the solitary waves and shock wave solutions or domain walls (in higher dimension) for various Boussinesq equations. The method of undetermined coefficients have been used to explore the exact analytical solitary waves and shock wave solutions in terms of bell-shaped sech p function and kinkshaped tanh p function for the considered equations. The Boussinesq equation in the (1 + 1)-dimensional, the (2 + 1)-dimensional and the (3 + 1)-dimensional equations are studied and the parametric constraint conditions and uniqueness in view of both solitary waves and shock wave solutions are determined. Such solutions can be valuable and desirable for explaining some nonlinear physical phenomena in nonlinear science described by the Boussinesq equations. The effect of the varying parameters on the development of solitary waves and shock wave solutions have been demonstrated by direct numerical simulation technique.

New Exact Solutions, Dynamical and Chaotic Behaviors for the Fourth-Order Nonlinear Generalized Boussinesq Water Wave Equation

“…where u = uðx, tÞ is a real function and a, b, and c are real nonzero arbitrary constants [9][10][11][12][13][14][15]. As far as the authors know, Lie symmetry method is used to analyze Equation (1) and some soliton wave solutions are obtained in Ref.…”

“…[12]. Some travelling wave solutions of Equation (1) were given by extended tanh method and rational method [13,14]; the solitary wave and shock wave solutions of Equation (1) were obtained by method of undetermined coefficients [15].…”

Solitary Wave Solutions for the Shallow Water Wave Equations and the Generalized Klein-Gordon Equation Using Exp(-ϕ(η) )-Expansion Method

“…The solitary wave phenomena are observed in various fields, such as in plasma physics, fluid dynamics, optical fibres, the Bose-Einstein condensates, biological systems, propagation of shallow water waves, etc. [1,2]. The shallow water waves describe the motion of water bodies that are seen in various places like sea beaches, lakes and rivers, and governed by Boussinesq equation [2][3][4].…”

Analytic rogue wave solutions for a generalized fourth‐order Boussinesq equation in fluid mechanics

“…In this work, we will investigate the generalized fourth‐order Boussinesq equation, which has been recently discussed in Kumar et al and Darvishi et al where u = u ( x , t ) is a physical quantity that stands for the energy field in fluid mechanics. t and x represent time and displacement, respectively.…”

“…Due to that three free parameters a , b , and c involved in Equation , the equation possesses greater tolerance and can be adapted to a wider range of circumstances . The derivative of variable can describe the rate of change for physical quantity.…”