Determinant form of modulation equations for the semiclassical focusing Nonlinear Schrödinger equation
نویسندگان
چکیده
We derive a determinant formula for the WKB exponential of singularly perturbed Zakharov-Shabat system that corresponds to the semiclassical (zero dispersion) limit of the focusing Nonlinear Schrödinger equation. The derivation is based on the RiemannHilbert Problem (RHP) representation of the WKB exponential. We also prove its independence of the branchpoints of the corresponding hyperelliptic surface assuming that the modulation equations are satisfied.
منابع مشابه
Interaction of Modulated Pulses in the Nonlinear Schrödinger Equation with Periodic Potential
We consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses. version: ...
متن کاملSemiclassical limit of the scattering transform for the focusing Nonlinear Schrödinger Equation
X iv :0 90 3. 26 48 v1 [ m at hph ] 1 5 M ar 2 00 9 Abstract. The semiclassical limit of the focusing Nonlinear (cubic) Schrödinger Equation (NLS) corresponds to the singularly perturbed Zakharov Shabat (ZS) system that defines the direct and inverse scattering transforms (IST). In this paper, we derive explicit expressions for the leading order terms of these transforms, which are called semic...
متن کاملSemiclassical soliton ensembles for the focusing nonlinear Schrödinger equation: recent developments
We give an overview of the analysis of the semiclassical (zerodispersion) limit of the focusing nonlinear Schrödinger equation via semiclassical soliton ensembles, and we describe some recent developments in this direction.
متن کاملUniversality in the profile of the semiclassical limit solutions to the focusing Nonlinear Schrödinger equation at the first breaking curve
We consider the semiclassical (zero-dispersion) limit of the one-dimensional focusing Nonlinear Schrödinger equation (NLS) with decaying potentials. If a potential is a simple rapidly oscillating wave (the period has the order of the semiclassical parameter ε) with modulated amplitude and phase, the space-time plane subdivides into regions of qualitatively different behavior, with the boundary ...
متن کاملMathematical and computational methods for semiclassical Schrödinger equations
We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or, at least, of the associated physical observables, is a formidable ma...
متن کامل