Bäcklund Transformations for Liouville Equations with Exponential Nonlinearity

Abstract

This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with logarithmic nonlinearity and hyperbolic linear parts. The construction of transformations is based on the method proposed by Clairin for second-order equations of the Monge–Ampere type. For the equations studied in the article, using the Bäcklund transformations, new equations are found, which make it possible to find solutions to the original nonlinear equations and reveal the internal connections between various integrable equations.

Keywords

• nonlinear equations in partial derivatives
• hyperbolic equations
• Bäcklund transformations
• Clairin’s method
• differential relationships
• the Liouville equation