A Hybridizable Discontinuous Galerkin Method for the non--local Camassa--Holm--Kadomtsev--Petviashvili equation

This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment of $x$ and $y$ derivatives. The non-local operator ${\mathrm{\partial }}_x^{-1}{u}_y$ is localized through an auxiliary variable $v$ satisfying $v_x=u_y$, allowing efficient element-by-element computations. We prove energy stability of the semi-discrete scheme and derive $\mathcal{O}(h^{k+1\mathrm{/}2})$ convergence in space. Numerical experiments validate the theoretical results and demonstrate the method’s capability to accurately resolve smooth solutions and peaked solitary waves (peakons).