Conservation Laws, Symmetries, and Line Soliton Solutions of Generalized KP and Boussinesq Equations with p-Power Nonlinearities in Two Dimensions

Abstract

Nonlinear generalizations of integrable equations in one dimension, such as the KdV and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting in analysis due to critical behaviour. This paper studies analogous nonlinear p-power generalizations of the integrable KP equation and the Boussinesq equation in two dimensions. Several results are obtained. First, for all p = 0, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers p = 0. Third, Noether's theorem is applied to obtain the conservation laws arising from the Lie symmetries that are variational. Finally, explicit line soliton solutions are derived for all powers p > 0, and some of their properties are discussed.

Hamiltonian Structure, Symmetries and Conservation Laws for a Generalized (2 + 1)-Dimensional Double Dispersion Equation

“…x [11]. However, we remark that this maximal algebra is also admitted by Equation (10) with a = 0 and d = 0 for any f (v x ).…”

“…x , the generalized (2 + 1)-dimensional Boussinesq with p-power nonlinearity is obtained. The Hamiltonian structure for this equation was obtained in [11]. In addition, if β = 0 and p = 1, this Hamiltonian formulation is one of the Hamiltonian structures [15] of the ordinary Boussinesq Equation 1.…”

“…The 2D generalized double dispersion (2D gDD) family of Equations (8) unifies the previous Equations (1)- (4) and (7). Some (2 + 1)-dimensional equations in this family were considered in recent literature and were shown to admit interesting exact solutions such as line solitons and lump solutions [11][12][13].…”

“…xx + βu yy , p = 0 (7) was considered in [11], where the point symmetries, conservation laws, and line soliton solutions were derived. In this work, we consider a (2 + 1)-dimensional generalization of the double dispersion Equation 4, given by:…”

Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

“…Recently, double reduction of PDEs was performed by using the interrelation between symmetries and conservation laws [16][17][18]. Lately, non-linear p−power generalisations of the KP and Boussinesq equations were studied and line soliton solutions were constructed for p > 0 [19].…”