The symmetry and some exact solutions of the nonlinear many-dimensional Liouville, d'Alembert and eikonal equations

Abstract

Multiparametrical exact solutions of the many-dimensional nonlinear d'Alembert, Liouville, sine-Gordon and eikonal equations arc obtained. The maximally extensive local invariance groups of the equations are determined and invariants of the extended Poincaré group are found.

Symmetry classification of quasi-linear PDE’s containing arbitrary functions

“…Except for this, a necessary condition in order that equation (18) may admit symmetries is that the function α has the form α(x, y) = x r β(y/x), where β is an arbitrary function.…”

“…Specifically, if the r.h.s. of equation (18) has the form α(x, y)u + β(x, y), the admitted symmetry is, not surprisingly, generated by X = c x + Ψ(x, y) ∂ ∂u where c is a constant and Ψ(x, y) is any solution of the PDE E[Ψ] = αΨ − c β .…”

The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries

“…One different but not unrelated approach to integrability of PDE's begins with the seminal contribution of Lie in classical symmetries of differential equations, which has been generalized thanks to the work of Bluman and Cole [6] and Olver and Rosenau [30,31], and Fushchich et al [15,16]. This new procedure deals with symmetries that leave invariant just a subset of all the possible solutions of the PDE under scrutiny [15,16].…”

“…This new procedure deals with symmetries that leave invariant just a subset of all the possible solutions of the PDE under scrutiny [15,16]. These symmetries, which do not form a group in the Lie sense, appear however to be extremely interesting for analyzing the integrability and properties of the PDE as well as its relationship with the PP.…”

“…It consists on finding symmetries not through the usual group-theoretical procedures but rather through the so-called Similarity Transformations. Usually one chooses an ansatz [15,16] that allows one to find a change of variables connected with the symmetries of the equation such that the PDE is reduced to an ODE which can be analyzed through the usual Painlevé tests for ordinary nonlinear differential equations. The question of how to choose an adequate set of ansätze leading to the right change of variables (the similarity transformations) has extensively been analyzed by Clarkson and Kruskal [8].…”