Some generalizations of the nonlocal transformations approach

Abstract

Some generalizations of a method of nonlocal transformations are proposed: a con­nection of given equations via prolonged nonlocal transformations and finding of an adjoint solution to the solutions of initial equation are considered. A concept of nonlocal transformation with additional variables is introduced, developed and used for searching symmetries of differential equations. A problem of inversion of the nonlocal transforma­tion with additional variables is investigated and in some cases solved. Several examples are presented. Derived technique is applied for construction of the algorithms and for­mulae of generation of solutions. The formulae derived are used for construction of exact solutions of some nonlinear equations.

Keywords

• nonlocal transformations
• Lie classical symmetry
• nonlocal symmetries
• additional variables
• formulae for generation of solutions
• nonlinear superposition principle
• Backlund transformations