A certain class of solutions of the nonlinear wave equation

Publisher Website
Google Scholar
Abstract
In this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation ∇2u=Φ(u,(∇u‖∇u)). A special class of solutions is discussed for which (∇u‖∇u) is constant on each level of the function u. It is proved that levels of such solutions form in the space of independent variable’s hypersurfaces with all principal curvatures constant. The general form of such hypersurfaces is given. Then it is proved that via the method of characteristics it is possible to construct (in principle) all the solutions of the discussed class. They may be obtained by integration of an ODE of second order using a special class of the polynomial functions. Some new solutions are given for equations⧠v=4Av3+3Bv2+2cv+D, ⧠v=μ exp v, ⧠v=sin v, ⧠v=cosh v, and ⧠v=sinh v.

This publication has 3 references indexed in Scilit: