Perturbation theory for the nonlinear Schrödinger equation with a random potential

doi:10.1088/0951-7715/22/12/004

Perturbation theory for the nonlinear Schrödinger equation with a random potential

Shmuel Fishman et al 2009 Nonlinearity 22 2861-2887 doi: 10.1088/0951-7715/22/12/004

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Shmuel Fishman¹, Yevgeny Krivolapov¹ and Avy Soffer²
¹ Physics Department, Technion - Israel Institute of Technology, Haifa 32000, Israel
² Mathematics Department, Rutgers University, New-Brunswick, NJ 08903, USA
E-mail: fishman@physics.technion.ac.il, evgkr@tx.technion.ac.il and soffer@math.rutgers.edu

Recommended by E S Titi

Abstract. A perturbation theory for the nonlinear Schrödinger equation in 1D on a lattice was developed. The small parameter is the strength of the nonlinearity. For this purpose secular terms were removed and a probabilistic bound on small denominators was developed. It was shown that the number of terms grows exponentially with the order. The results of the perturbation theory are compared with numerical calculations. An estimate on the remainder is obtained and it is demonstrated that the series is asymptotic.

Mathematics Subject Classification: 72.15.Rn, 73.20.Fz, 11.10.Lm, 31.15.xp

Print publication: Issue 12 (December 2009)
Received 26 January 2009, in final form 7 September 2009
Published 30 October 2009

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Perturbation theory for the nonlinear Schrödinger equation with a random potential

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