Exact and Approximate Symmetries and Approximate Conservation Laws of Differential Equations with a Small parameter

Abstract

The frameworks of Baikov-Gazizov-Ibragimov (BGI) and Fushchich-Shtelen (FS) approximate symmetries have proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. While for algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation, for second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order BGI approximate symmetries of the perturbed ODE, and can be systematically computed. We present a relation between BGI and FS approximate point symmetries for perturbed ODEs. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq ODE reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs, including Benjamin-Bona-Mahony ODE reduction are provided. The frameworks of BGI and FS approximate symmetries are used to study symmetry properties of partial differential equations with a small parameter. In general, we show that unlike in the ODE case, unstable point symmetries of an unperturbed PDE do not necessarily yield local approximate symmetries for the perturbed equation. We classify stable point symmetries of a one-dimensional wave model in terms of BGI and FS frameworks. We find a connection between BGI and FS approximate local symmetries for a PDE family. We classify approximate point symmetries for a family of one-dimensional wave equations with a small nonlinear term, and construct a physical approximate solution for a family that includes a one dimensional wave equation describing the wave motion in a hyperelastic material with a single family of fibers. For this model, we find wave breaking times numerically and using the approximate solution. A complete classification of exact and approximate point symmetries of the two-dimensional wave equation with a general small nonlinearity is presented. We investigate approximate conservation laws of systems of perturbed PDEs. We apply the direct mul tiplier method to obtain new approximate conservation laws for perturbed PDEs including nonlinear heat and wave equations. We show that the direct method generalizes the Noether’s theorem for construction of approximate conservation laws by proving that an approximate multiplier corresponds to an approximate local symmetry of an approximately variational problem. We present two formulas relating to construct ad ditional approximate conservation laws for a system of perturbed PDEs. We illustrate these formulas using perturbed wave equation and nonlinear telegraph system. An application for using approximate conservation laws to construct potential systems and approximate potential symmetries is provided