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arXiv2025

Lie symmetries and travelling wave solutions of the nonlinear waves in the inhomogeneous Fisher-Kolmogorov equation

In this work we consider a Fisher-Kolmogorov equation depending on two exponential functions of the spatial variables. We study this equation from the point of view of symmetry reductions in partial differential equations. Through two-dimensional abelian subalgebras, the equation is reduced to ordinary differential equations. New solutions have been derived and interpreted.

math.AP↗

arXiv2017

Conservation laws, symmetries, and line soliton solutions of generalized KP and Boussinesq equations with p-power nonlinearities in two dimensions

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 $p\neq 0$, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers $p\neq0$. 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.

math-ph↗