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

2013/09/20 by A. I. Zenchuk, Zenchuk, A. I.
Engineering · Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Fluid Dynamics and Turbulent Flows #Mathematical and Theoretical Analysis #advanced mathematical theories #nlin.SI

paper · pdf · doi:10.48550/arxiv.1309.5171

16 pages

arxiv created 2013/09/20 · openalex publication_date 2013/09/20 · arxiv updated 2013/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We represent an algorithm reducing the (M+1)-dimensional nonlinear partial differential equation (PDE) representable in the form of one-dimensional flow u_t + w_x1(u,u_x,u_xx,…)=0, (where w is an arbitrary local function of u and its x_i-derivatives, i=1,…,M) to the family of M-dimensional nonlinear PDEs F(u,w)=0, where F is general (or particular) solution of a certain second order two-dimensional nonlinear PDE. Particularly, the M-dimensional PDE might be an ODE which, in some cases, may be integrated yielding the explicite solutions to the original (M+1)-dimensional PDE. Moreover, the spectral parameter may be introduced into the function F which yields a linear spectral equation associated with the original PDE. Simplest examples of nonlinear PDEs with explicite solutions are given.

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