On the energy exchange between resonant modes in nonlinear Schrödinger equations

Abstract

We consider the nonlinear Schrödinger equation $$ i\psi_t= -\psi_{xx}\pm 2\cos 2x \ |\psi|^2\psi,\quad x\in S^1,\ t\in \R$$ and we prove that the solution of this equation, with small initial datum $\psi_0=\e (\cos x+\sin x)$, will periodically exchange energy between the Fourier modes $e^{ix}$ and $e^{-ix}$. This beating effect is described up to time of order $\e^{-9/4}$ while the frequency is of order $\e^2$. We also discuss some generalizations

Multi-revolution composition methods for highly oscillatory differential equations

“…This effect is named "beating effect" in [15]. Another interesting part of the dynamics of this system concerns the modes 3 and −3 whose energies scale like ε 2 and may be regarded for this reason as a "finer" component of the dynamics.…”

“…In this section, we present some numerical experiments of the application of MRCMs to numerically integrate a problem considered in [7] and originally analyzed by B. Grébert and C. Villegas-Blas in [15]. It consists of a nonlinear Schrödinger equation in the with one-dimensional torus with a cubic nonlinearity |u| 2 u multiplied by an inciting term of the form 2 cos(2x),…”

“…The following nonlinear phenomenon is proved in [15]. [15] Consider the Fourier expansion u(t, x) = k∈Z ξ k (t)e ikx of the solution of (28).…”

Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

“…In comparison, using (19) and the change of variableŝ = ε −1 s, and the periodicity of t → e tA with period 1, the exact solution of (1) with f = 0 and (4) satisfies…”

“…The solutions of (10) and (11) can be expressed using the variation of constant formula (19), which yields, using e A = e −A = I,…”