Complex structure and solutions of classical nonlinear equation with the interaction $u^4_4$

TL;DR

This paper studies the complex structure of solutions to a nonlinear wave equation with a quartic interaction in four-dimensional Minkowski space, demonstrating the complex analyticity of the dynamics, wave, and scattering operators on finite energy initial data.

Contribution

It introduces a complex structure for the nonlinear wave equation and proves the complex analyticity of the associated operators on the space of initial data.

Findings

Operators are complex analytic on the space of initial data.

The nonlinear wave and scattering operators are well-defined and complex analytic.

The complex structure facilitates analysis of solutions to the nonlinear equation.

Abstract

We consider the (real) nonlinear wave equation $$\Box u + m^2 u + \lambda u^3 = 0, \quad m > 0, \quad \lambda > 0, $$ 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 $R(\varphi, \pi) = \varphi + i\mu^{-1}\pi$ 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 $RU(t)R^{-1},$ $RWR^{-1},$ and $RSR^{-1}$ are defined correctly and are complex analytic on the complex Hilbert space $H^1({\R}^3,\C).$