Solvable cubic resonant systems

TL;DR

This paper constructs a broad class of infinite-dimensional Hamiltonian resonant systems with cubic nonlinearities, revealing their integrability and energy recurrence properties, extending known results from specific physical equations.

Contribution

It introduces a large class of solvable cubic resonant systems, adding a conserved quantity and generalizing properties previously observed in specific physical models.

Findings

Existence of a conserved quantity for all systems

Systems exhibit periodic energy returns

Extension of known properties to a broader class

Abstract

Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties. Despite being infinite-dimensional Hamiltonian systems with cubic nonlinearities in the equations of motion, these resonant systems admit special analytic solutions, which furthermore display periodic perfect energy returns to the initial configurations. Here, we construct a very large class of resonant systems that shares these properties that have so far been seen in specific examples emerging from a few standard equations of mathematical physics (the Gross-Pitaevskii equation, nonlinear wave equations in Anti-de Sitter spacetime). Our analysis provides an additional conserved quantity for all of these systems, which has been previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.