1304.6864 (A. I. Zenchuk)

Particular solutions to multidimensional PDEs with KdV-type nonlinearity    [PDF]

A. I. Zenchuk

We consider a class of particular solutions to the (2+1)-dimensional nonlinear partial differential equation (PDE) $u_t +\partial_{x_2}^n u_{x_1} - u_{x_1} u =0$ (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 $u_t-uu_{x_1}=0$ and might also be useful in study the higher dimensional PDEs with arbitrary linear part and KdV-type nonlinearity (i.e. the nonlinear term is $u_{x_1} u$).

View original: http://arxiv.org/abs/1304.6864