The Nonlinear Schroedinger equation: solitons dynamics
نویسندگان
چکیده
In this paper we investigate the dynamics of solitons occurring in the nonlinear Schroedinger equation when a parameter h → 0. We prove that under suitable assumptions, the the soliton approximately follows the dynamics of a point particle, namely, the motion of its barycenter q h (t) satisfies the equation ¨ q h (t) + ∇V (q h (t)) = H h (t) where sup t∈R |H h (t)| → 0 as h → 0.
منابع مشابه
Hylomorphic solitons
This paper is devoted to the study of solitary waves and solitons whose existence is related to the ratio energy/charge. These solitary waves are called hylomorphic. This class includes the Q-balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation (NKG), as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schroedinger equat...
متن کاملGeometric Characterization of Solitons
The aim of this work is to describe a geometric definition of localized solutions of NLS. In the linear case we have the RAGE Theorem, which relates localized solutions to the pure point spectrum of the Hamiltonian: Localized solutions of the linear Schroedinger equation are linear combinations of L eigenfunctions of the Hamiltonian. In particular, they are almost periodic functions of time. Fo...
متن کاملDerivation of Gross-pitaevskii Version of Nonlinear Schroedinger Equation from Scale Invariance
It is known that in the usual 3-D space, the Schroedinger equation can be derived from scale-invariance. In view of the fact that, according to modern physics, the actual dimension of proper space may be di erent from 3, it is desirable to analyze what happens in other spatial dimensions D. It turns out that while for D ≥ 3 we still get only the Schroedinger's equation, for D = 2, we also get t...
متن کاملOn the limits of validity of nonparaxial propagation equations in Kerr media.
Recent generalizations of the standard nonlinear Schroedinger equation (NLSE), aimed at describing nonparaxial propagation in Kerr media are examined. An analysis of their limitations, based on available exact results for transverse electric (TE) and transverse magnetic (TM) (1+1)-D spatial solitons, is presented. Numerical stability analysis reveals that nonparaxial TM soltions are unstable to...
متن کاملSpatial Optical Solitons in Waveguide Arrays
We overview theoretical and experimental results on spatial optical solitons excited in arrays of nonlinear waveguides. First, we briefly summarize the basic properties of the discrete nonlinear Schrödinger (NLS) equation frequently employed to study spatially localized modes in arrays, the so-called discrete solitons. Then, we introduce an improved analytical model that describes a periodic st...
متن کامل