Determination of approximate nonlinear self-adjointenss and approximate conservation law

Abstract

Approximate nonlinear self-adjointness is an effective method to construct approximate conservation law of perturbed partial differential equations (PDEs). In this paper, we study the relations between approximate nonlinear self-adjointness of perturbed PDEs and nonlinear self-adjointness of the corresponding unperturbed PDEs, and consequently provide a simple approach to discriminate approximate nonlinear self-adjointness of perturbed PDEs. Moreover, a succinct approximate conservation law formula by virtue of the known conservation law of the unperturbed PDEs is given in an explicit form. As an application, we classify a class of perturbed wave equations to be approximate nonlinear self-adjointness and construct the general approximate conservation laws formulae. The specific examples demonstrate that approximate nonlinear self-adjointness can generate new approximate conservation laws.

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References in corpus (1)

• New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order

Cited by in corpus (3)

• Nonlinear Boundary Conditions for Energy and Entropy Stable Initial Boundary Value Problems in Computational Fluid Dynamics
• Nonlinear and Linearised Primal and Dual Initial Boundary Value Problems: When are they Bounded? How are they Connected?
• Approximate conditions admitted by classes of the Lagrangian $\mathcal{L}=\frac{1}{2}\left(-u^{\prime 2}+u^2\right)+{\epsilon }^i{G}_i(u,u^{\prime },u^{\prime \prime })$