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Axioms 2019, 8(2), 45; https://doi.org/10.3390/axioms8020045

Bäcklund Transformations for Nonlinear Differential Equations and Systems

Institute of Mathematics and Natural Sciences, North Caucasus Federal University, 1 Pushkin Street, 355009 Stavropol, Russia
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Received: 5 February 2019 / Revised: 4 April 2019 / Accepted: 7 April 2019 / Published: 11 April 2019
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
PDF [547 KB, uploaded 11 April 2019]

Abstract

In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases were considered. The BTs construction is based on the method proposed by Clairin. The solutions of the considered equations have been found using the BTs, with a unified algorithm. In addition, the work develops the Clairin’s method for the system of two third-order equations related to the integrable perturbation and complexification of the Korteweg-de Vries (KdV) equation. Among the constructed BTs an analog of the Miura transformations was found. The Miura transformations transfer the initial system to that of perturbed modified KdV (mKdV) equations. It could be shown on this way that, considering the system as a link between the real and imaginary parts of a complex function, it is possible to go to the complexified KdV (cKdV) and here the analog of the Miura transformations transforms it into the complexification of the mKdV.
Keywords: Bäcklund transformation; Clairin’s method; generalized Liouville equation; Miura transformation; Korteweg-de Vries equation Bäcklund transformation; Clairin’s method; generalized Liouville equation; Miura transformation; Korteweg-de Vries equation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Redkina, T.V.; Zakinyan, R.G.; Zakinyan, A.R.; Surneva, O.B.; Yanovskaya, O.S. Bäcklund Transformations for Nonlinear Differential Equations and Systems. Axioms 2019, 8, 45.

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