APPLICATION OF BACKLUND DIFFERENTIAL CONSTRAINT FOR CONSTRUCTING EXACT SOLUTIONS OF NONLINEAR HYPERBOLIC EQUATIONS

Abstract

Background. Finding exact solutions of nonlinear partial differential equations is one of the main problems of the nonlinear systems theory. A number of methods have been developed for integrable systems, but due to the complexity of various nonlinear equations, there is no single method and method for solving them. One of the effective methods is the use of Backlund differential constraints for constructing exact solutions of nonlinear equations. Backlund transformations make it possible to go to a simpler equation, and the use of differential constraints - to obtain a solution to one of the equations if the solution to the other is known. In addition, these transformations play an important role in integrable systems, since they reveal internal connections between various integrable properties. Recently, a lot of research has been done in this area. The aim of this work is to obtain solutions of nonlinear hyperbolic second-order partial differential equations using Bäcklund differential constraints. Materials and methods. Finding solutions to nonlinear differential equations using Bäcklund differential constraints is considered. The construction of Bäcklund transformations is based on the method proposed by Claren for second-order equations of the Monge-Ampere type. Results. For the nonlinear hyperbolic partial differential equations investigated in this work, exact solutions are obtained using Bäcklund differential constraints; the solution of one of the equations is proved if the solution of the other is known; various cases of obtaining solutions by this method are analyzed. Conclusions. The results are of interest for studying nonlinear partial differential equations. The found solutions can serve as a basis for further research of equations of this type, as well as for solving applied problems in various fields of natural science.

Keywords

• nonlinear partial differential equations
• hyperbolic equations
• bäcklund transformations
• claren's method
• differential constraints
• liouville's equation