Energy transfer for solutions to the nonlinear Schrödinger equation on irrational tori

Abstract

We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrödinger (NLS) initial value problem on 2D irrational tori. Moreover we complement the analytic study with numerical experimentation. As a biproduct of our investigation we also prove that the quasi-resonant part of the NLS initial value problem we consider, in both the focusing and defocusing case, is globally well-posed for initial data of finite mass.

Breather Solutions to a Two-dimensional Nonlinear Schrödinger Equation with Non-local Derivatives

“…We solve (1) on a 2D periodic domain, starting from an initial field ψ 0 ≡ ψ(x, t = 0), via a pseudospectral method [24][25][26] with 128 × 128 modes (with higher resolution results available in supplemental material [27]). Our numerical method treats the linear term via an integrating factor, and the nonlinear term via an explicit 4th order Runge-Kutta scheme.…”

Large deviations principle for the cubic NLS equation

“…We speculate though that a possible extension to a 2D setting may be possible for irrational tori. There in fact, it has been proved [42] that the irrationality of the torus completely decouples the resonant set into two 1D resonant sets, one for each coordinate.…”

“…With this we mean the expected behaviour of certain initial data with frequency localised support that evolve into solutions whose support lives mostly in higher frequencies. This phenomenon is connected to the question of asymptotic growth of Sobolev norms of solutions to the Schrödinger equation whose study was started by Bourgain [10] and later continued in [56,55,58,26,25,42,4,13,16,39] among others. In addition to such theoretical results, there are many interesting simulations and experimental results about transfer of energy to high frequencies, see for example [50,17,41,42].…”

Almost Global Existence for Some Hamiltonian PDEs with Small Cauchy Data on General Tori

“…A different point of view is to give upper bounds on the possible growth of the Sobolev norm in terms of the time t. This problem, as already remarked, has been tackled widely for linear equations. However we mention [18,44,45] and the recent result [41] dealing with nonlinear equations. A dual point of view is to study possible instability of solutions, namely to show that even solutions evolving from small initial data could show a large growth of the Sobolev norm by waiting for sufficiently long time.…”