Particular solutions to multidimensional PDEs with KdV-type nonlinearity

Abstract

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 $u_t-u{u}_{x_1}=0$ 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 $u_{x_1}u$).

13 pages

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Cited by in corpus (1)

• Particular solutions to multidimensional PDEs represented in the form of one-dimensional flow