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 $p\ne 0$, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers $p\ne 0$. 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.

Corrected final version

References in corpus (1)

• A nonlinear generalization of the Camassa-Holm equation with peakon solutions

Cited by in corpus (5)

• Symmetry multi-reduction method for partial differential equations with conservation laws
• Modulation theory for soliton resonance and Mach reflection
• A formula for symmetry recursion operators from non-variational symmetries of partial differential equations
• Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation
• Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs