Conservation laws and exact solutions of a generalized Kadomtsev–Petviashvili (KP)‐like equation

摘要

Rosenau and Hyman introduced compactons as a solution of the K(m,n)$$ Kleft(m,nright) $$ equation which generalizes the celebrated Korteweg–De Vries (KdV) equation. The inclusion of the generalized Kfm,gn$$ Kleft({f}amp;amp;#x0005E;m,{g}amp;amp;#x0005E;nright) $$ equation as a central part of the Kadomtsev–Petviashvili (KP) equation results in a generalized KP‐like equation KPfm,gn$$ KPleft({f}amp;amp;#x0005E;m,{g}amp;amp;#x0005E;nright) $$. In this article, we present the general form of conservation laws for the nonlinear KPfm,gn$$ KPleft({f}amp;amp;#x0005E;m,{g}amp;amp;#x0005E;nright) $$ equation, in terms of unknown functions f$$ f $$ and g$$ g $$, by employing the multipliers approach. For suitable choices of m,n,f$$ m,n,f $$, and g$$ g $$, the derived conservation laws are utilized to obtain conservation laws for several variants of the KP equation, including the logarithmic KP‐like equation, the generalized Gardner KP equation, and the KP equation with p‐power non‐linearity. The double reduction theory is employed to construct reductions and exact solutions of various KP‐like generalized equations, including the KPum,un$$ KPleft({u}amp;amp;#x0005E;m,{u}amp;amp;#x0005E;nright) $$ equation for different values of m$$ m $$ and n$$ n $$. The structure of these solutions is analyzed by some graphs that illustrate soliton and compacton solutions for different parameter values.