Focusing Singularity in a Derivative Nonlinear Schrödinger Equation
نویسندگان
چکیده
We present a numerical study of a derivative nonlinear Schrödinger equation with a general power nonlinearity, |ψ| ψx. In the L-supercritical regime, σ > 1, our simulations indicate that there is a finite time singularity. We obtain a precise description of the local structure of the solution in terms of blowup rate and asymptotic profile, in a form similar to that of the nonlinear Schrödinger equation with supercritical power law nonlinearity.
منابع مشابه
Nonlinear Schrödinger Equation with a White-noise Potential: Phase-space Approach to Spread and Singularity
We propose a phase-space formulation for the nonlinear Schrödinger equation with a white-noise potential in order to shed light on two problems: the rate of dispersion and the singularity formation. Our main tools are the energy laws and the variance identity. The method is completely elementary. For the problem of dispersion, we show that in the absence of dissipation the ensemble-average disp...
متن کاملSelf-focusing in the complex Ginzburg1Landau limit of the critical nonlinear Schrödinger equation
We analyze self-focusing and singularity formation in the complex Ginzburg1Landau equation (CGL) in the regime where it is close to the critical nonlinear Schrödinger equation. Using modulation theory [Fibich and Papanicolaou, Phys. Lett. A 239 (1998) 167], we derive a reduced system of ordinary differential equations that describes self-focusing in CGL. Analysis of the reduced system shows tha...
متن کاملFocusing Quantum Many-body Dynamics: the Rigorous Derivation of the 1d Focusing Cubic Nonlinear Schrödinger Equation
We consider the dynamics of N bosons in one dimension. We assume that the pair interaction is attractive and given by Nβ−1V (N ·) where ∫ V 6 0. We develop new techniques in treating the N−body Hamiltonian so that we overcome the diffi culties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity...
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 کاملSmoothing for the Fractional Schrödinger Equation on the Torus and the Real Line
In this paper we study the cubic fractional nonlinear Schrödinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a ful...
متن کامل