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We consider the long-term weakly nonlinear evolution governed by thetwo-dimensional nonlinear Schr\"{o}dinger (NLS) equation with an isotropicharmonic oscillator potential. The dynamics in this regime is dominated byresonant interactions between quartets of linear normal modes, accuratelycaptured by the corresponding resonant Hamiltonian system. In the framework ofthis system, we identify Fermi-Pasta-Ulam-like recurrence phenomena, wherebythe normal-mode spectrum passes in close proximity of the initialconfiguration, and two-mode states with time-independent mode amplitude spectrathat translate into long-lived breathers of the original NLS equation. Wecomment on possible implications of these findings for nonlinear optics andmatter-wave dynamics in Bose-Einstein condensates.

Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap