Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

Abstract

Abstract. We have constructed new formulae for generation of solutions for the nonlinear heat equation and for the Burgers equation that are based on linearizing nonlocal transformations and on nonlocal symmetries of linear equations. Found nonlocal symmetries and formulae of nonlocal nonlinear superposition of solutions of these equations were used then for construction of chains of exact solutions. Linearization by means of the Legendre transformations of a second-order PDE with three independent variables allowed to obtain nonlocal superposition formulae for solutions of this equation, and to generate new solutions from group invariant solutions of a linear equation.

New Nonlocal Symmetries of Diffusion-Convection Equations and Their Connection with Generalized Hodograph Transformation

“…To prove the statement formulated above we will first rewrite Equation (27), using equality Equation (20) in the form…”

“…The characteristic Equation (27) determines nonlocal transformation with additional variables, which connects Equations (4) and (7) taking into account Equation (21) and condition Equation (20). Additional independent variable v(x, t) is determined by Equation (7) and additional dependent variable b(v, t) is an arbitrary solution of the Equation (21).…”

“…where u(x; t) and b(v; t) satisfy Equations (21) and (20). Thus, the appropriate "lacking" operator for Equation (4) should be written in the form, depending on b(v, t)…”

“…As Equation (4) is connected with the potential system Equations (15) and (16), one can construct the projection of the corresponding transformation Equation (44) onto the space of variables (x, t, u(x, t)) using substitution Equation (20) …”

“…This approach is based on the nonlocal transformations technic [16][17][18][19][20][21], which we regularly use for study of symmetries of the given nonlinear equations in [1] and in the current paper.…”