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.

Highlights

  • Nonlinear partial differential equations are widely used to describe the so-called “fine processes”, such as propagation of nonlinear waves in dispersive media [1]

  • 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 Korteweg-de Vries (KdV) and here the analog of the Miura transformations transforms it into the complexification of the modified KdV (mKdV)

  • We were able to show that when considering the system as a relation between the real and imaginary parts of a complex function, we can pass to the complexified KdV (cKdV), and the analog of the

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Summary

Introduction

Nonlinear partial differential equations are widely used to describe the so-called “fine processes”, such as propagation of nonlinear waves in dispersive media [1]. In the classical works [2,6] the Bäcklund transformations (BTs) were considered for the couple of differential second order partial differential equations and presented in form of a system of relations and containing independent variables, functions of the said equations, and their first-order derivatives. In this way, we were able to show that when considering the system as a relation between the real and imaginary parts of a complex function, we can pass to the cKdV, and the analog of the. We were able to show that when considering the system as a relation between the real and imaginary parts of a complex function, we can pass to the cKdV, and the analog of the Miura transformations transforms it into the complexification of mKdV

Bäcklund Transformations for Special Cases of Liouville Equations
Bäcklund Transformations for Three-Dimensional Liouville Equation
Bäcklund Transformations for System of Two Third-Order Equations
Analog of Miura Transformations
Conclusions

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