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 Korteweg–de Vries and Boussinesq equations with $p$-power nonlinearities, arise in many physical applications and are interesting from the analytic standpoint because of their critical behavior. We study analogous nonlinear $p$-power generalizations of the integrable Kadomtsev–Petviashvili and Boussinesq equations in two dimensions. For all $p\ne0$, we present a Hamiltonian formulation of these two generalized equations. We derive all Lie symmetries including those that exist for special powers $p\ne0$. We use Noether's theorem to obtain conservation laws arising from the variational Lie symmetries. Finally, we obtain explicit line soliton solutions for all powers $p>0$ and discuss some of their properties.