This article is cited in 12 papers

The geometric series method for constructing exact solutions to nonlinear evolution equations

A. V. Bochkarev, A. I. Zemlyanukhin

Saratov Technical University, Saratov, Russia

Abstract: It is proved that, for the majority of integrable evolution equations, the perturbation series constructed based on the exponential solution of the linearized problem is geometric or becomes geometric as a result of changing the variable in the equation or after a transformation of the series. Using this property, a method for constructing exact solutions to a wide class of nonintegrable equations is proposed; this method is based on the requirement for the perturbation series to be geometric and on the imposition of constraints on the values of the coefficients and parameters of the equation under which the sum of the series is the solution to be found. The effectiveness of using the diagonal Padé approximants the minimal order of which is determined by the order of the pole of the solution to the equation is demonstrated.

Key words: geometric series, perturbation method, evolution equation, exact solution, Padé approximant.

UDC: 517.957:517.537.3

Received: 03.02.2016
Revised: 10.01.2017

DOI: 10.7868/S0044466917070079

 English version: Computational Mathematics and Mathematical Physics, 2017, 57:7, 1111–1123

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