Soliton dynamics in the discrete nonlinear Schrödinger equation

Boris Malomed, Michael I. Weinstein

Research output: Contribution to journalArticlepeer-review


Using a variational technique, based on an effective Lagrangian, we analyze static and dynamical properties of solitons in the one-dimensional discrete nonlinear Schrödinger equation with a homogeneous power nonlinearity of degree 2σ+ 1. We obtain the following results. (i) For σ < 2 there is no threshold for the excitation of a soliton; solitons of arbitrary positive energies, W = Σ|un|2, exist. (ii) Range of multistability: there is a critical value of σ, σcr ≈ 1.32, such that for σcr < σ < 2, there exist three soliton-like states in a certain finite intermediate range of energies, two stable and one unstable (while there is no multistable regime in the continuum NLS equation). For energies below and above this range, there is a unique soliton state which is stable, (iii) For σ ≥ 2, there exists an energy threshold for formation of the soliton. For all σ ≥ 2 there exist two soliton states, one narrow and one broad. The narrow soliton is stable, while the broad one is not. (iv) We find an energy criterion for the excitation of solitons by initial configurations which are narrowly concentrated in few lattice sites.

Original languageEnglish
Pages (from-to)91-96
Number of pages6
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Issue number1-3
StatePublished - 2 Sep 1996


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