BACKLUND TRANSFORMATIONS FOR LIOUVILLE’S EQUATIONS WITH EXPONENTIAL NONLINEARITY

Abstract

Background. The study of Backlund 's transformations is one of the current topics in the theory of differential equations in partial derivatives. Such transformations are used to find solutions to nonlinear differential equations, including solitonic equations. At the same time, they represent an example of the differential-geometric structure generated by differential equations. Backlund transformations made it possible to obtain not only pairs of equations, but also the solution of one if the solution of the other was known. Transformation data played an important role in integrable systems, as it revealed internal relationships between different integrable properties, such as the definition of symmetry, the presence of a Hamiltonian structure. In recent years, many studies have been carried out in this area. The aim of the work is to obtain new Backlund transformations and auto-transformations for generalized Liouville’s equations with indicative-degree nonlinearity having a multiplier dependent on the first derivatives. Materials and methods. This paper examines the construction of Backlund transformations for nonlinear equations in second-order partial derivatives of salt-ton type with logarithmic nonlinearity and hyperbolic lie-neon part. The construction of transformations is based on the method suggested by Clairen for second-order equations of the Monge-Ampere type. Results. For the equations examined in the article, new equations have been found by means of the Backlund transformations, which make it possible to find solutions to the original nonlinear equations, as well as to identify internal connections between different integrable equations. Conclusions. The results are of interest in the study of nonlinear differential equations in partial derivatives, particularly soliton equations. The new equations obtained by differential linkages can be used for further research of equations of this type, as well as in solving many applications in physics and engineering.

Keywords

• nonlinear equations in partial derivatives
• hyperbolic equations
• backlund transformations
• the clairan method
• differential bonds
• the liouville’s equation