Bounded versus unbounded high-energy excitation at long times

Ascertain whether the long-time evolution of the nonlinear Schrödinger equation in the D-shape billiard above the chaos border leads to a bounded number of effectively populated linear eigenmodes (due to decreasing interaction matrix elements and effective nonlinearity) or to unbounded, subdiffusive-like excitation of arbitrarily high-energy modes; establish the parameter regimes and scaling behavior for each scenario.

Background

Drawing analogies with disordered lattice models (DANSE and Stark ladder) that exhibit subdiffusive unbounded spreading, the authors discuss competing mechanisms in the billiard: Rayleigh–Jeans condensation that traps most of the norm at low energy, versus possible coupling to higher modes whose interaction strengths may decay with energy.

Given these competing effects, the authors state that they cannot currently argue decisively for either a bounded or unbounded scenario, highlighting an unresolved issue in the long-time dynamics.