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Perturbation theory for the nonlinear Schrödinger equation with a random potential. (English) Zbl 1204.37074

The author consider the problem of dynamical localization of waves in a nonlinear Schrödinger equation with a random potential term on a lattice. Regarding this equation, in this paper a perturbation theory in power of the density (denoted in the paper by \(\beta\) (beta)) was developed and bounds on the various terms were obtained. The work is only partialy rigorous. In some parts it relies on conjectures that the author test numerically. A similar result for a nonlinear equation of a different structure was obtained in [G. Benettin, J. Fröhlich and A. Giorgilli, Commun. Math. Phys. 119, No. 1, 95–108 (1988; Zbl 0825.58011)]. The present paper has the potential to develop into a method for the solution of some type of nonlinear differential equations. The constructed solution is a series in the eigenfunctions of the linear problem. The authors do not apply the standard perturbation theory for the coefficients. Removing the secular terms requires renormalization of the original linear Hamiltonian by shifting the energies. The entropy problem is solved by bounding an appropiate recursive relation. A general probabilistic bound on the terms of the perturbation theory is derived and the quality of the perturbation theory is tested. The remainder terms are controlled by a boostrap argument. In the final the open problems are listed.
Reviewer: M. Marin (Brasov)

MSC:

37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q55 NLS equations (nonlinear Schrödinger equations)
35F20 Nonlinear first-order PDEs
35Q41 Time-dependent Schrödinger equations and Dirac equations
35J10 Schrödinger operator, Schrödinger equation
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0825.58011
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