Solutions of multidimensional partial differential equations representable as a~one-dimensional flow

A. I. Zenchuk

Teoretičeskaâ i matematičeskaâ fizika, Tome 178 (2014) no. 3, pp. 346-362

Résumé

We propose an algorithm for reducing an $(M{+}1)$-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow $u_t+w_{x_1}(u,u_x,u_{xx},\dots)=0$ (where $w$ is an arbitrary local function of $u$ and its $x_i$ derivatives, $i=1,\dots, M$) to a family of $M$-dimensional nonlinear PDEs $F(u,w)=0$, where $F$ is a general (or particular) solution of a certain second-order two-dimensional nonlinear PDE. In particular, the $M$-dimensional PDE might turn out to be an ordinary differential equation, which can be integrated in some cases to obtain explicit solutions of the original $(M{+}1)$-dimensional equation. Moreover, a spectral parameter can be introduced in the function $F$, which leads to a linear spectral equation associated with the original equation. We present simplest examples of nonlinear PDEs together with their explicit solutions.

Détail

Keywords: method of characteristics, integrability theory, boundary condition, particular solution, reduction to lower dimensions.