Exact Solutions of Generalized Boussinesq‐Burgers Equations and (2+1)‐Dimensional Davey‐Stewartson Equations

Abstract

We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and (2+1)-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the (2+1)-dimensional Davey-Stewartson equations.

Lie group analysis of nonlinear inviscid flows with a free surface under gravity

“…Integrating (2.7) for water (ρ = 1), we obtain For convenience, we rewrite the above system of governing equations and auxiliary conditions in a simplified form by redefining the variables as follows [20]:…”

“…Mekheimer et al studied an electrically conducting Jeffrey fluid [18] and a hydro-magnetic Maxwell fluid in a porous medium [19]. Mhlanga and Khalique studied generalized Boussinesq-Burgers equations [20]. Parmar and Timol applied a group theoretic approach to natural heat convection and mass transfer for an inclined surface [21].…”

Global dynamics of the Boussinesq‐Burgers system with large initial data

“…Only a few results on its variants were studied (e.g. see and reference therein) in the whole space . When ϵ = μ > 0, the existence of traveling waves of the system was established in in the whole space corresponding to the bore‐like initial data When u + = u − , w + = w − and ϵ = μ = 0, the existence of weak solutions and classical solutions of problem ‐ has been established in and , respectively.…”

“…Only a few results on its variants were studied (e.g. see [9,10] and reference therein) in the whole space R. When " D > 0, the existence of traveling waves of the system (1.2) was established in [11] in the whole space R corresponding to the bore-like initial dataWhen u C D u , w C D w and " D D 0, the existence of weak solutions and classical solutions of problem (1.2)-(1.3) has been established in [11] and [12], respectively. For the bounded interval, inspiblack by an idea from [13], Ding and Wang [14] established the This section is devoted to proving Theorem 1.1.…”

Global existence and asymptotic behavior of the Boussinesq–Burgers system

“…Compared to the KdV-Burgers equation, the Boussinesq-Burgers system (1.1) is not so widely studied. There are a few results on its variants (e.g., see [8,18] and references therein) for the whole interval R. As we know, the only result of the Boussinesq-Burgers system (1.1) is the existence of traveling wave solutions obtained in [21] in the whole interval R with bore-like data, where ε = μ. The goal of this paper is to study the initial-boundary value problem of the Boussinesq-Burgers system in a bounded interval.…”