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 ut + \(w_{x_1 } \) (u, ux uxx,…) = 0 (where w is an arbitrary local function of u and its xi 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.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 178, No. 3, pp. 346–362, March, 2014.
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Zenchuk, A.I. Solutions of multidimensional partial differential equations representable as a one-dimensional flow. Theor Math Phys 178, 299–313 (2014). https://doi.org/10.1007/s11232-014-0144-3
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DOI: https://doi.org/10.1007/s11232-014-0144-3