Abstract
The concept of an approximate multiplier (integrating factor) is introduced. Such multipliers are shown to give rise to approximate local conservation laws for differential equations that admit a small perturbation. We develop an explicit, algorithmic and efficient method to construct both the approximate multipliers and their corresponding approximate fluxes. Our method is applicable to equations with any number of independent and dependent variables, linear or nonlinear, is adaptable to deal with any order of perturbation and does not require the existence of a variational principle. Several important perturbed equations are presented to exemplify the method, such as the approximate KdV equation. Finally, a second treatment of approximate multipliers is discussed.
Highlights
The last three decades have seen the development of approximate Lie symmetries [1,2,3] and their subsequent applications to differential equations admitting a perturbation term
We seek to formulate a direct and schematic method—one without the involvement of an action principle to find the conservation laws of a given system of perturbed differential equations. To achieve this we prove that nontrivial approximate conserved quantities on the solution space of a given differential equation can be constructed from approximate multipliers, where the approximate multiplier depends on a predetermined set of variables and derivatives
The paper is structured as follows: we propose the relevant theory on approximate multipliers and some important properties
Summary
The last three decades have seen the development of approximate Lie symmetries [1,2,3] and their subsequent applications to differential equations admitting a perturbation term. 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. In the Appendix A, we discuss an alternate formulation of multipliers for perturbative equations
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