72 research outputs found
On the one-dimensional cubic nonlinear Schrodinger equation below L^2
In this paper, we review several recent results concerning well-posedness of
the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real
line R and on the circle T for solutions below the L^2-threshold. We point out
common results for NLS on R and the so-called "Wick ordered NLS" (WNLS) on T,
suggesting that WNLS may be an appropriate model for the study of solutions
below L^2(T). In particular, in contrast with a recent result of Molinet who
proved that the solution map for the periodic cubic NLS equation is not weakly
continuous from L^2(T) to the space of distributions, we show that this is not
the case for WNLS.Comment: 14 pages, additional reference
Focusing Singularity in a Derivative Nonlinear Schr\"odinger Equation
We present a numerical study of a derivative nonlinear Schr\"odinger equation
with a general power nonlinearity, . In the
-supercritical regime, , 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 Schr\"odinger equation with supercritical
power law nonlinearity.Comment: 24 pages, 17 figure
Stability of Solitary Waves for a Generalized Derivative Nonlinear Schr\"odinger Equation
We consider a derivative nonlinear Schr\"odinger equation with a general
nonlinearity. This equation has a two parameter family of solitary wave
solutions. We prove orbital stability/instability results that depend on the
strength of the nonlinearity and, in some instances, their velocity. We
illustrate these results with numerical simulations.Comment: 29 pages, 4 Figure
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