Conservative Integrators for a Toy Model of Weak Turbulence

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 Schr\"odinger 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 two numerical schemes for integrating this model equation. It has two invariants, and our two schemes each preserve one of the two quantities. We prove convergence in one case, and our numerical studies show that both schemes compare favorably to the trapezoidal method and fixed step 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.