New conditional symmetries and exact solutions of the nonlinear wave equation

Abstract

We have constructed the new class of conditional symmetries of the nonlinear wave equation in 1+3 dimensions. These symmetries enable us to obtain broad families of new exact solutions of the nonlinear wave equation containing up to the four arbitrary functions.

Ansatzes and exact solutions for nonlinear Schrodinger equations

“…In the case of Q-conditional invariance a solution of the system (26) gives an ansatz which will reduce Eq. (25). Very often investigation of reduction conditions or Q-conditional invariance gives more ansatzes than the classical Lie method.…”

“…Q-conditional invariance for the equation (1) with special cases of the function F was studied in [23], and for (18) -in [24,25].…”

General Results on Conditional Symmetry for the Two-Dimensional Nonlinear Wave Equation

“…the classical texts [11,12,13]) of the same equation with a certain additional condition. Conditional symmetries of the multidimensional nonlinear wave equations are specifically discussed in [15,19,20].…”

A Precise Definition of Reduction of Partial Differential Equations

“…Let us emphasize that all the functions ω(x) defined by formulae (35), (36), (38), (39), (41) give rise to conditionally-invariant Ansätze for the field u(x) of the form (25). Using these one can construct broad families of new (non-Lie) exact solutions even for such a well studied model as the nonlinear wave equation (see, also [46]). Consider, for example, the confomally-invariant nonlinear wave equation…”