Solvable cubic resonant systems

Weakly nonlinear analysis of resonant PDEs in recent literature has generateda number of resonant systems for slow evolution of the normal mode amplitudesthat 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 analysisprovides an additional conserved quantity for all of these systems, which hasbeen previously known for the resonant system of the two-dimensional Gross-Pitaevskii equation, but not for any other cases.