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TMF, 2014, Volume 178, Number 3, Pages 346–362 (Mi tmf8597)  

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

A. I. Zenchuk

Institute of Chemical Physics, RAS, Chernogolovka, Moscow Oblast, Russia

Abstract: We propose an algorithm for reducing an $(M{+}1)$-dimensional nonlinear partial differential equation (PDE) representable in the form of a one-dimensional flow $u_t+w_{x_1}(u,u_x,u_{xx},…)=0$ (where $w$ is an arbitrary local function of $u$ and its $x_i$ derivatives, $i=1,…, M$) 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.

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

DOI: https://doi.org/10.4213/tmf8597

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English version:
Theoretical and Mathematical Physics, 2014, 178:3, 299–313

Bibliographic databases:

PACS: 02.30.Jr 02.30.Ik 47.35.-i
MSC: 35Q35 35Q53
Received: 19.09.2013

Citation: A. I. Zenchuk, “Solutions of multidimensional partial differential equations representable as a one-dimensional flow”, TMF, 178:3 (2014), 346–362; Theoret. and Math. Phys., 178:3 (2014), 299–313

Citation in format AMSBIB
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