For a three-mode reduced NLS model with stochastic forcing and damping at the boundary modes, the authors construct a unique invariant probability measure with polynomial convergence to the non-equilibrium steady state.
Modified scattering for the cubic Schr\"odinger equation on Diophantine waveguides
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abstract
We consider the cubic Schr\"odinger equation posed on a product space subject to a generic Diophantine condition. Our analysis shows that the small-amplitude solutions undergo modified scattering to an effective dynamics governed by interactions that induce no growth of high-order Sobolev norms. This is in sharp contrast with the infinite energy cascade scenario observed by Hani--Pausader--Tzvetkov--Visciglia in the absence of Diophantine conditions.
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Non-equilibrium steady state for a three-mode energy cascade model
For a three-mode reduced NLS model with stochastic forcing and damping at the boundary modes, the authors construct a unique invariant probability measure with polynomial convergence to the non-equilibrium steady state.