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Quasi-periodic solutions of nonlinear random Schrödinger equations. (English) Zbl 1148.35104

The authors seek time quasi-periodic solutions to the nonlinear random Schrödinger equation
\[ i\frac{\partial}{\partial t}=(\varepsilon\Delta +V)u+\delta| u| ^{2p}u\quad (p>0) \] on \(\mathbb Z^d\times[0,\infty)\), where \( 0<\varepsilon,\delta\ll1,\Delta\) is the discrete Laplacian:
\[ \Delta_{ij}=\begin{cases} 1, &\text{if }\| i-j\| _{l^1}=1,\\ 0, &\text{otherwise},\end{cases} \]
and \(V=\{v_j\}_{j\in \mathbb Z^d}\), the potential, is a family of time independent identically distributed random variables with common distribution \(g=\tilde{g}(v_j)dv_j,\tilde{g}\in L^{\infty}.\) For appropriate initial conditions \(u(0)\), the time quasi-periodic solutions to this equation are constructed.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
35B15 Almost and pseudo-almost periodic solutions to PDEs
35B25 Singular perturbations in context of PDEs
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References:

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