Complex plane representations and stationary states in cubic and quintic resonant systems

Abstract

Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives related to the physics of Bose-Einstein condensates and Anti-de Sitter spacetime) that admit special analytic solutions and an extra conserved quantity. Here, we develop and explore a complex plane representation for these systems modelled on the related cubic Szego and LLL equations. To demonstrate the power of this representation, we use it to give simple closed form expressions for families of stationary states bifurcating from all individual modes. The conservation laws, the complex plane representation and the stationary states admit furthermore a natural generalization from cubic to quintic nonlinearity. We demonstrate how two concrete quintic PDEs of mathematical physics fit into this framework, and thus directly benefit from the analytic structures we present: the quintic nonlinear Schroedinger equation in a one-dimensional harmonic trap, studied in relation to Bose-Einstein condensates, and the quintic conformally invariant wave equation on a two-sphere, which is of interest for AdS/CFT-correspondence.

arXiv commentv2: version accepted for publication

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• Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes
• Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
• Energy level splitting for weakly interacting bosons in a harmonic trap
• Turbulent cascades in a truncation of the cubic Szego equation and related systems
• Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials
• Breathing modes, quartic nonlinearities and effective resonant systems
• New islands of stability with double-trace deformations
• Time-periodicities in holographic CFTs
• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
• Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials