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Register a new userEnergy transfer for solutions to the nonlinear Schrodinger equation on irrational tori
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We analyze the energy transfer for solutions to the defocusing cubic nonlinear Schrodinger (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.
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We consider a damped/driven nonlinear Schrodinger equation in an $n$-cube $K^{n}subsetmathbb{R}^n$, $n$ is arbitrary, under Dirichlet boundary conditions [ u_t- uDelta u+i|u|^2u=sqrt{ u}eta(t,x),quad xin K^{n},quad u|_{partial K^{n}}=0, quad u>0, ] where $eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ | u(t)|_m^2 le C u^{-m}, $ uniformly in $tge0$ and $ u>0$. In this work we prove that for small $ u>0$ and any initial data, with large probability the Sobolev norms $|u(t,cdot)|_m$ of the solutions with $m>2$ become large at least to the order of $ u^{-kappa_{n,m}}$ with $kappa_{n,m}>0$, on time intervals of order $mathcal{O}(frac{1}{ u})$.
An explicit lifespan estimate is presented for the derivative Schrodinger equations with periodic boundary condition.
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrodinger equations (NLS) that we gave in cite{CM15APDE}. We consider a NLS with a Schrodinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the refined profile, a quasi--periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in cite{CM15APDE}, giving us also a better understanding of the Fermi Golden Rule.
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