Integration of the $\varphi^4$ model in elliptic Jacobi functions and investigation of them by the phase plane method

Solutions are obtained for the $\varphi^4$ model for different relationships between the signs of the constants in the Hamiltonian in the form of Jacobi elliptic functions. Such essentially nonlinear solutions, investigated on the phase plane, go over into kinks or solitons in the limiting ease for the parameter E on the separatrices S. For the lowest state $E_{\min}=U(\varphi_0)$ (vacuum), the solutions are transformed into a vacuum condensate (harmonic oscillations). Expansion of the solutions near the vacuum corresponds to the result of perturbation theory.