In this paper the following Cauchy problem for the nonlinear Klein-Gordon equation is considered:
where , , , etc., is a small positive parameter, , and the nonlinear function with can be written in the form in a neighborhood of ; here , are linearly independent polynomials of defined by
is of the form , are homogeneous polynomials of degree 1, and is a smooth function of of degree 4, i.e. near . The main result of the paper is the following:
Theorem 1.1. For any integer there exists a positive constant such that for any the Cauchy problem (1), (2) has a unique classical solution . Moreover, the solution has a free profile in the sense that there exist and ( is the usual Sobolev space) such that
where is the solution to the Cauchy problem for the linear Klein-Gordon equation in with initial data and , .