On the construction of conservation laws: A mixed approach

Abstract

A new approach, combining the Ibragimov method and the one by Anco and Bluman, with the aim of algorithmically computing local conservation laws of partial differential equations, is discussed. Some examples of the application of the procedure are given. The method, of course, is able to recover all the conservation laws found by using the direct method; at the same time we can characterize which symmetry, if any, is responsible for the existence of a given conservation law. Some new local conservation laws for the Short Pulse equation and for the Fornberg-Whitham equation are also determined.

On Construction of a Super Hierarchy of the Wadati-Konno-Ichikawa Equation

“…where pðM j Þ = 0, pðΓ j Þ = 1. Substituting (31) into (30) and comparing the coefficients of the same powers of λ, we obtain…”

“…In Section 4, we derive infinite conservation laws of the super WKI equation by resorting to the spectral parameter expansions. For the applied and analytic aspects on conservation laws, one can refer to [27][28][29][30]. We can refer to the two most recent results on the mixed method for the calculation of conservation laws studied in [29,30].…”

Approximate Conservation Laws of Nonvariational Differential Equations

“…For unperturbed differential equations without a variational principle, the available methods for obtaining conservation laws are, but not limited to: solving the divergence expression by integration, the standard multiplier approach [21,22], the multiplier approach on the solutions of the equation [23], the partial Noether approach [24], a direct construction method [25] and very significantly, the newly proposed mixed method [26]. The latter can characterize which symmetry, if any, is accountable for a given conservation law.…”