Exact peakon solutions of a generalized CH-KP equation (Q2909864)

Consider the partial differential equation NEWLINE\[NEWLINE(u_t+ 2ku_x- (u^2)_{xxt}- auu_x)_x+ u_{yy}= 0\tag{\(\ast\)}NEWLINE\]NEWLINE for \(a< 0\). The authors look for travelling wave solutions of \((\ast)\) of the form NEWLINE\[NEWLINEu(x,y,t)= \widetilde U(|x+y- ct|),\tag{\(\ast\ast\)}NEWLINE\]NEWLINE where \(c\) is the wave speed. Substituting \((\ast\ast)\) into \((\ast)\), the authors derive after integration a planar system of two autonomous ordinary differential equations. Studying the corresponding phase portrait, they derive conditions under which an explicit solution can be obtained.