Complex structure and solutions of classical nonlinear equation with the interaction $u_4^4$

We consider the (real) nonlinear wave equation $$\mathrm{\square }u+m^{2}u+\lambda u^3=0,m>0,\lambda >0,$$ on four--dimensional Minkowski space. We introduce the complex structure and show that the (nonlinear) operator of dynamics, the wave and scattering operators define complex analytic maps on the space of initial Cauchy data with finite energy. In other words, let $R(\phi ,\pi )=\phi +i{\mu }^{-1}\pi$ be the map of initial data on the positive frequency part of the solution of the free Klein--Gordon equation with these initial data. The operators $RU(t)R^{-1},$ $RW{R}^{-1},$ and $RS{R}^{-1}$ are defined correctly and are complex analytic on the complex Hilbert space $H^1({\mathbb{R}}^3,\text{C})\mathrm{.}$