4,725 research outputs found
Modified scattering for the critical nonlinear Schr\"odinger equation
We consider the nonlinear Schr\"odinger equation in all dimensions , where and . We construct a class of initial values for which
the corresponding solution is global and decays as , like if and like if
. Moreover, we give an asymptotic expansion of those solutions
as . We construct solutions that do not vanish, so as to avoid
any issue related to the lack of regularity of the nonlinearity at . To
study the asymptotic behavior, we apply the pseudo-conformal transformation and
estimate the solutions by allowing a certain growth of the Sobolev norms which
depends on the order of regularity through a cascade of exponents
Continuous dependence for NLS in fractional order spaces
We consider the Cauchy problem for the nonlinear Schr\"odinger equation
in , in the -subcritical
and critical cases , where . Local existence of
solutions in is well known. However, even though the solution is
constructed by a fixed-point technique, continuous dependence in does not
follow from the contraction mapping argument. In this paper, assuming
furthermore , we show that the solution depends continuously on the
initial value in the sense that the local flow is continuous . If,
in addition, then the flow is Lipschitz. This completes
previously known results concerning the cases .Comment: Corrected typos. Simplified section 4. Results unchange
Standing waves of the complex Ginzburg-Landau equation
We prove the existence of nontrivial standing wave solutions of the complex
Ginzburg-Landau equation with periodic boundary conditions. Our result includes all
values of and for which , but
requires that be sufficiently small
Finite-time blowup for a complex Ginzburg-Landau equation with linear driving
In this paper, we consider the complex Ginzburg--Landau equation on , where
, and . By convexity
arguments we prove that, under certain conditions on ,
a class of solutions with negative initial energy blows up in finite time
On the instability for the cubic nonlinear Schrodinger equation
We study the flow map associated to the cubic Schrodinger equation in space
dimension at least three. We consider initial data of arbitrary size in ,
where , the critical index, and perturbations in H^\si, where
\si is independent of . We show an instability mechanism in some
Sobolev spaces of order smaller than . The analysis relies on two features
of super-critical geometric optics: creation of oscillation, and ghost effect.Comment: 4 page
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