Discrete bound states in a toy model for weak turbulence and implications for the invariant measure

Key Points

• The aim is to analyze frequency cascades in a weak turbulence model described by the cubic defocusing nonlinear Schrödinger equation.
• Derived a Hamiltonian dynamical system to study the nonlinear Schrödinger equation.
• Classified energy minimizers for fixed mass.
• Characterized the invariant measure near the energy minimizers.
• Identified discrete bound states in the model system.
• Established a connection between energy minimizers and the invariant measure.
• Demonstrated implications for understanding weak turbulence dynamics.