A class of nonlinear wave equations containing the continuous Toda case

Abstract

We consider a nonlinear field equation which can be derived from a binomial lattice as a continuous limit. This equation, containing a perturbative friction-like term and a free parameter , reproduces the Toda case (in the absence of the friction-like term) and other equations of physical interest, by choosing particular values of . We apply the symmetry and the approximate-symmetry approach, and the prolongation technique. Our main aim is to check the limits of validity of different analytical methods in the study of nonlinear field equations. We show that the equation under investigation with the friction-like term is characterized by a finite-dimensional Lie algebra admitting a realization in terms of boson annhilation and creation operators. In the absence of the friction-like term, the equation is linearized and connected with Bessel-type equations. Examples of exact solutions are displayed, and the algebraic structure of the equation is discussed.