Nonlinear superposition of solutions of linear d'Alembert equation

Abstract:
The invariance of the linear one-dimensional d'Alembert equation under the Legendre transformation allows us to obtain nonlinear nonlocal superposition formulae for solutions of this equation. We have used the classical Lie symmetry of the multidimensional d'Alembert equations to obtain their reduction to the one-dimensional case and than to construct the corresponding superposition principles. The examples of constructing of new solution of the d'Alembert equation from two its known solutions are considered.