Solvable cubic resonant systems

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.

arXiv commentv2: 23 pages, 1 figure, minor corrections, published version

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• Complex plane representations and stationary states in cubic and quintic resonant systems
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• Turbulent cascades in a truncation of the cubic Szego equation and related systems
• Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials
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• Time-periodicities in holographic CFTs
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• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
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