A class of nonlinear wave equations containing the continuous Toda case

TL;DR

This paper studies a nonlinear wave equation derived from a lattice model, exploring its symmetries, algebraic structure, and solutions, and examining the validity of analytical methods across different parameter regimes.

Contribution

It introduces a generalized nonlinear wave equation with a friction term, analyzes its symmetry properties, algebraic structure, and solutions, and assesses the applicability of analytical techniques.

Findings

The equation admits a finite-dimensional Lie algebra with a boson operator realization.

In the absence of friction, the equation reduces to a linear Bessel-type equation.

Exact solutions are constructed and the algebraic structure is characterized.

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 $\gamma$, reproduces the Toda case (in absence of the friction-like term) and other equations of physical interest, by choosing particular values of $\gamma$. We apply the symmetry and the approximate symmetry approach, and the prolongation technique. Our main purpose 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 absence of the friction-like term, the equation is linearized and connected with equations of the Bessel type. Examples of exact solutions are displayed, and the algebraic structure of the equation is discussed.