Approximate double-periodic solutions in $(1+1)$-dimensional $\varphi ^4$-theory

Double-periodic solutions of the Euler–Lagrange equation for the $(1+1)$-dimensional scalar $\varphi^4$-theory are considered. The nonlinear term is assumed to be small, and the Poincarй method is used to seek asymptotic solutions in the standing-wave form. The principal resonance problem, which arises for zero mass, is resolved if the leading-order term is taken in the form of a Jacobi elliptic function.