Generalization of the double reduction theory

Abstract

In a recent work [1,2] Sjöberg remarked that generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to find invariant solution for a non linear system of qth order partial differential equations with n independent and m dependent variables provided that the non linear system of partial differential equations admits a nontrivial conserved form which has at least one associated symmetry in every reduction. In order to give an application of the procedure we apply it to the nonlinear (2 + 1) wave equation for arbitrary function f (u) and g (u).

Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

“…Corollary 8 (the necessary and sufficient condition for reduced conserved form [9]). The conserved form = 0 of the PDE system (1) can be reduced under a similarity transformation of a symmetry to a reduced conserved form̃= 0 if and only if is associated with the conservation law , that is, [ , ]…”

“…Sjöberg [7,8] developed a double reduction formula for a nonvariational PDE of order with two independent and dependent variables to reduce it to an ODE of order ( − 1) provided that the PDE admits a nontrivial conserved vector associated with at least one symmetry. Recently, Bokhari et al [9] generalized the double reduction theory for the case of several independent variables. According to the generalized double reduction theory, a nonlinear system of th-order PDEs with independent and dependent variables can be reduced to a nonlinear system of ( − 1)th-order ODEs.…”

On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions

“…Theorem 3 (see [20]). Suppose that = 0 is a conservation law of the partial differential equation system (6).…”

Conservation laws and associated Lie point symmetries admitted by the transient heat conduction problem for heat transfer in straight fins

“…[35]). However, the advantage of the procedure undertaken below (the Kara and Mahomed theorem [26]) is that one may obtain physical solutions by double reduction using the conserved vector and the associated Lie point symmetry [36,37].…”