Construction of doubly periodic solutions 
via the Poincare-Lindstedt method 
in the case of massless $\varphi^4$ theory

Sergey Yu. Vernov Institute for Nuclear Physics of Moscow State University, Vorobievy Gory, Moscow, 119899, Russia                                                            Abstract

Doubly periodic solutions for the Lagrange-Euler equation of the (1+1)-dimensional scalar $\varphi^4$ theory are studied. Provided that nonlinear term is small, the Poincare-Lindstedt method is used to find asymptotic solutions in the standing wave form. If the mass of the scalar field is zero, then the standard zero approximation:

$$\varphi_0(x,t)=A\sin(x)\sin(t)$$

doesn't allow to construct a uniform expansion even to the first order. Such expansion can be found if and only if a zero approximation contains infinite number of harmonics:

$$\varphi_0(x,t)=\sum_{n=1}^{\infty}a_n\sin(nx)\sin(nt)$$   with the coefficients a_n obeying some infinite system of algebraic nonlinear equations. An approximate solution of this system have been found by the computer algebra system REDUCE. It is proved that using the Jacobi elliptic function cn as a zero approximation, one can find an exact solution of this system and construct a uniform expansion, containing only doubly periodic functions in the standing wave form, both to the first and to the second order.


1998-07-29