Approximate symmetries of the Boussinesq equation

Abstract

In this paper we show that all exact symmetries of the linear wave
equation are inherited by the Boussinesq type equation with a small parameter
as approximate symmetries in any order of precision. We find an approximate integral­
differential transformation of any order of precision, which transforms the
Boussinesq type equation into the linear wave equation. Using the
transformation and the symmetries, we get approximate solutions of the
Boussinesq type equation.

Mathematics Subject Classification (1991): 58D19, 35Q53.

Key words: The Boussinesq equation, the
linear wave equation; approximate symme­tries, Lie-Bäcklund transformations;
integral-differential transformations, approximately invariant solutions.

Quaestiones
Mathematicae 26(2003), 1-14.