Construction of Doubly Periodic Solutions via the Poincare-Lindstedt Method in the case of Massless Phi^4 Theory

TL;DR

This paper constructs doubly periodic solutions in (1+1)-dimensional massless Phi^4 theory using the Poincare-Lindstedt method, solving the principal resonance with elliptic functions and providing an asymptotic solution up to third order.

Contribution

It introduces a novel approach to solve the principal resonance in massless Phi^4 theory using elliptic functions and computer algebra, extending previous methods for periodic solutions.

Findings

Successfully constructed doubly periodic solutions with third-order accuracy.

Resolved the principal resonance problem using elliptic cosine functions.

Provided a REDUCE program for reproducing the solutions.

Abstract

Doubly periodic (periodic both in time and in space) solutions for the Lagrange-Euler equation of the (1+1)-dimensional scalar Phi^4 theory are considered. The nonlinear term is assumed to be small, and the Poincare-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 constructed (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.