First integrals for non linear hyperbolic equations

TL;DR

This paper develops a method to relate initial and final data of nonlinear wave equations using convergent asymptotic series, inspired by quantum field theory algebraic structures.

Contribution

It introduces a novel algebraic framework and analytic approach for constructing convergent series solutions to nonlinear hyperbolic equations.

Findings

Constructed formal asymptotic series using generating functions.

Established an analytic setting ensuring convergence of series.

Linked Cauchy data at different times through nonlinear representation formulas.

Abstract

Given a solution of a nonlinear wave equation on the flat space-time (with a real analytic nonlinearity), we relate its Cauchy data at two different times by nonlinear representation formulas in terms of asymptotic series. We first show how to construct formally these series by mean of generating functions based on an algebraic framework inspired by the construction of Fock spaces in quantum field theory. Then we build an analytic setting in which all these constructions really make sense and give rise to convergent series.