Construction of Doubly Periodic Solutions via the Poincaré–Lindstedt Method in the case of Massless φ Theory

Doubly periodic (periodic both in time and in space) solutions for the Lagrange–Euler equation of the (1 + 1)–dimensional scalar φ theory are considered. The nonlinear term is assumed to be small, and the Poincaré– Lindstedt method is used to find asymptotic solutions in the standing wave form. The principal resonance problem, which arises for zero mass, is solved if the leading-order term is taken in the form of a Jacobi elliptic function. It have been proved that the choice of elliptic cosine with fixed value of module k (k ≈ 0.451075598811) as the leading-order term puts the principal resonance to zero and allows us to construct (with accuracy to third order of small parameter) the asymptotic solution in the standing wave form. To obtain this leading-order term the computer algebra system REDUCE have been used. We have appended the REDUCE program to this paper.