Particular solutions to multidimensional PDEs with KdV-type nonlinearity

We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) ut+∂x2nux1−ux1u=0 (here n is any integer) reducing it to the ordinary differential equation (ODE). In a simplest case, n=1, the ODE is solvable in terms of elementary functions. Next choice, n=2, yields the cnoidal waves for the special case of Zakharov–Kuznetsov equation. The proposed method is based on the deformation of the characteristic of the equation ut−uux1=0 and might also be useful in study of the higher-dimensional PDEs with arbitrary linear part and KdV-type nonlinearity (i.e. the nonlinear term is ux1u).