On approximate symmetry and approximate solutions of the nonlinear wave equation with a small parameter

Abstract

The concept of approximate symmetry is introduced. We describe all nonlinearities F (u) with which the non-linear wave equation 2u + λu 3 + εF (u) = 0 with a small parameter ε is approximately scale and conformally invariant. Some approximate solutions of wave equations in question are obtained using the approximate symmetry.

Asymptotic symmetries of difference equations on a lattice

“…In such a way we have shown that also in the discrete case the FKPP equation possesses an asymptotic scaling symmetry which gives the correct asymptotic behaviour of the system. Long time ago Ibragimov and Fushchich [1,18] introduced the notion of approximate symmetry (for differential equations); this is based on some small parameter ε appearing in the equation, and is a symmetry "up to terms of order ε". One could wonder what is the relation between approximate symmetries in this sense and our approach to the front solutions of the FKPP equation; note however that no explicit small parameter ε is present in the FKPP equation.…”

“…Another interesting notion in this field is that of approximate symmetries [1,18]. Here one considers transformations which leave invariant the lower order approximations to the solution of a system depending on a small parameter.…”

Approximate Waiting-Time for a Thin Liquid Drop Spreading under Gravity

“…Another approach is that of finding approximate conditional symmetries admitted by the model equation as presented by Mahomed and Qu [13]. In this paper we use the approach adopted by Fushchich [10], M. Euler [7] and N. Euler [8,9]. In this paper the application of this approach is novel.…”

Approximate Functional Variable Separation for the Quasi-Linear Diffusion Equations withWeak Source

“…For decades, quite a few methods for tackling perturbed nonlinear evolution equations have been developed, such as the approximate Lie group theory [17], the approximate symmetry method [18], the approximate conditional symmetry method [19], the approximate potential symmetry method [20], the Lie group technique [21], the approximate generalized conditional symmetry approach [22], the approximate symmetry reduction for Cauchy problems of the perturbed PDEs [23], and so on.…”