Approximate Standing Wave Solutions inMassless ' 4

Doubly periodic solutions for the Lagrange{Euler equation of the (1+1)-dimensional scalar ' 4 theory are studied. Provided that nonlin-ear term is small, the Poincare{Lindstedt asymptotic method is used to nd asymptotic solutions in the standing wave form. It is proved that using the Jacobi elliptic function cn as a zero approximation one can solve the problem of the main resonance, appearing in the case of zero mass, and construct a uniform expansion both to the rst and to the second order. 1 Two kinds of doubly periodic solutions Our investigation is dedicated to the construction of doubly periodic classical elds in the (1+1)-dimensional ' 4 theory. We study the model of an isolated real scalar eld '(x; t), described by the Lagrangian density: