Abstract: A virial relation is an integral identity involving the solution of a differential equation. The identity can be derived from the equation itself or, if the equation can be formulated as a variational problem, from the infinitesimal variations in the action functional. A well-known example of a viral relation relates the time-average kinetic and potential energies of an n-particle system under the influence of central forces (usually referred to as the virial theorem). In mathematics virial relations have been used extensively, for example, in deriving necessary conditions for the existence of solutions of differential equations.
In this article we present a systematic approach to deriving virial relations for almost periodic solutions of nonlinear wave equations (NLWs) of the form;
Our objective is twofold. First, we want to illuminate a method that has been known and used in various guises for many years in mathematics and physics and is well-suited for problems arising in nonlinear differential equations. Second, we apply this method to derive necessary conditions for the existence of almost periodic solutions of NLW.
The main results of this article are the following. We first derive a class of integral identies that must be satisfies by almost periodic solutions of NLW. Then, by choosing a particular subclass of these we are able to prove the following theorem.
Theorem: Suppose u is an almost periodic solution of NLW. Let F(z) be an antiderivative of f(z). If F(z) - cf(z) is less than or equal to 0 for some c in [N-2/2N, 1/2] (N is the spatial dimension), then u is independent of time.