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A New Approach to Direct Discretization of Wave Kinetic Equations with Application to a Nonlinear Schrodinger System in 2D

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arxiv 2509.03432 v1 pith:O7N4JNTD submitted 2025-09-03 math.NA cs.NA

A New Approach to Direct Discretization of Wave Kinetic Equations with Application to a Nonlinear Schrodinger System in 2D

J. W. Banks , J. Shatah This is my paper
classification math.NA cs.NA
keywords waveapproachnonlinearapproximationcollisiondescribedirectensemble
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Wave Kinetic Equations (WKEs) are often used to describe the evolution of ensemble averaged wave amplitudes for nonlinear wave systems. In the present manuscript we describe a new approach to direct numerical simulation of solutions to WKEs. This new method relies on a piecewise polynomial approximation of the resonant manifold, followed by numerical quadrature of the collision integral. The approach is general in nature, and is discussed in detail here for a particular nonlinear Schrodinger model in 2 spatial dimensions. Detailed convergence studies demonstrate 2nd-order accuracy for model collision integrals, and self-convergence studies for the WKE show near 2nd-order rates. Furthermore, comparison of the WKE approximation to ensemble averages of the NLS illustrate the efficacy of the method and the validity of the WKE, for both isotropic and an-isotropic solutions.

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Cited by 3 Pith papers

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  1. Entropy Structures and Long-Time Relaxation for 3-Wave Kinetic Equations

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    For a broad class of 3-wave kinetic equations, a new one-sided entropy mechanism yields global L^1_loc weak solutions and forces local relaxation to zero as t tends to infinity.

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    New entropy structures from one-sided balance conditions on interaction weights yield global weak L1_loc solutions to 3-wave kinetic equations and prove their local relaxation to zero equilibrium.

  3. Global time-analytic strong solutions for a class of 3-wave kinetic equations

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    For radial nonnegative data, a regularized 3-wave kinetic equation has a unique global strong solution that stays nonnegative and is time-analytic.