Applications of Nonlocally Related PDE Systems

Abstract

In Chapter 3, it was shown how one can systematically construct a set (tree) of PDE systems nonlocally related to a given PDE system. In particular, local conservation laws of a PDE system lead to augmented nonlocally related (potential) systems that explicitly include nonlocal (potential) variables. Moreover, further nonlocally related PDE systems (nonlocally related subsystems) arise when one or more dependent variables (including dependent variable(s) arising after a point transformation that involves an interchange of dependent and independent variable(s)) are excluded from a PDE system or its potential systems, through differential relations. In Section 3.5, an algorithm for the construction of an extended tree of nonlocally related systems was outlined. In particular, n local conservation laws of a given PDE system lead to a tree of up to 2ⁿ-1 nonlocally related potential systems. A tree is further extended by considering subsystems of both the given PDE system and its nonlocally related potential systems as well as by considering potential systems arising from conservation laws (whose multipliers have an essential dependence on potential variables) of its nonlocally related potential systems.