Symmetries for a class of explicitly space- and time-dependent (1+1)-dimensional wave equations. (English) Zbl 0948.35081

Euler, M.; Euler, N.

Shkil, Mykola (ed.) et al., Symmetry in nonlinear mathematical physics. Proceedings of the second international conference, Kyiv, Ukraine, July 7-13, 1997. Memorial Prof. W. Fushchych conference. Vol. 1. Kyiv: Institute of Mathematics of the National Academy of Sciences of Ukraine. 70-78 (1997).

In this paper the nonlinear wave equation \(\partial^2 u/\partial x_0^2-\partial^2 u/\partial x_1^2+f(x_0,x_1,u)=0\), where \(f\) is an arbitrary smooth function of its arguments, is considered from the symmetry standpoint. The form of the most general Lie point symmetry generator of this equation is obtained. The classes of functions \(f\), for which the equation in question admits a one-parameter Lie point symmetry group, are constructed. Then, the authors investigate the possible form of generators of conformal transformations, assuming the usual form of generators of Lorentz and scaling transformations, and study the wave equations invariant under such operators. The symmetry groups of obtained equations are used for the construction of ansätze and reductions of these equations to ordinary differential equations. \(Q\)-conditional (nonclassical) symmetries of the wave equation are also considered. Namely, the determining equations for the coefficients of a \(Q\)-conditional symmetry operator are found and their compatibility is investigated.
For the entire collection see [Zbl 0882.00038].

Reviewer: R.O.Popovych (Kyïv)

MSC:

35L70	Second-order nonlinear hyperbolic equations
58J70	Invariance and symmetry properties for PDEs on manifolds
35C05	Solutions to PDEs in closed form
35A30	Geometric theory, characteristics, transformations in context of PDEs

Keywords:

Lie symmetry; group classification; invariance; reduction; invariance algebra; Lorentz algebra; conditional symmetry; nonclassical symmetry