Conservative Integrators for a Toy Model of Weak Turbulence
This work provides improved numerical tools for studying the long-time dynamics of weak turbulence in the 2D NLS, a problem of interest to mathematicians and physicists.
The authors developed and tested conservative numerical integrators for a toy model of weak turbulence, demonstrating that their schemes preserve invariants better than standard methods like Runge-Kutta, which is crucial for long-time simulations of energy transfer.
Weak turbulence is a phenomenon by which a system generically transfers energy from low to high wave numbers, while persisting for all finite time. It has been conjectured by Bourgain that the 2D defocusing nonlinear Schrödinger equation (NLS) on the torus has this dynamic, and several analytical and numerical studies have worked towards addressing this point. In the process of studying the conjecture, Colliander, Keel, Staffilani, Takaoka, and Tao introduced a "toy model" dynamical system as an approximation of NLS, which has been subsequently studied numerically. In this work, we formulate and examine several numerical schemes for integrating this model equation. The model has two invariants, and our schemes aim to conserve at least one of them. We prove convergence in some cases, and our numerical studies show that the schemes compare favorably to others, such as Trapezoidal Rule and fixed step fourth order Runge-Kutta. The preservation of the invariants is particularly important in the study of weak turbulence as the energy transfer tends to occur on long time scales.