432 research outputs found
Finite-time blowup for a complex Ginzburg-Landau equation
We prove that negative energy solutions of the complex Ginzburg-Landau
equation blow up in finite time,
where \alpha >0 and \pi /2<\theta <\pi /2. For a fixed initial value , we
obtain estimates of the blow-up time as . It turns out that stays bounded (respectively, goes to
infinity) as in the case where the solution of the
limiting nonlinear Schr\"odinger equation blows up in finite time
(respectively, is global).Comment: 22 page
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
Flat blow-up solutions for the complex Ginzburg Landau equation
In this paper, we consider the complex Ginzburg Landau equation
The study aims to investigate the finite time blowup phenomenon. In
particular, for fixed , the existence of finite time
blowup solutions for an arbitrary large is still unknown.
Especially, Popp, Stiller, Kuznetsov, and Kramer formally conjectured in 1998
that there is no blowup (collapse) in such a case. In this work, considered as
a breakthrough, we give a counter example to this conjecture. We show the
existence of blowup solutions in one dimension with arbitrarily given
and . The novelty is based on two main contributions: an
investigation of a new blowup scaling (flat blowup regime) and a suitable
modulation
Blow-up of solutions for weakly coupled systems of complex Ginzburg-Landau equations
Blow-up phenomena ofvweakly coupled systems of several evolution equations,
especially complex Ginzburg-Landau equationsvis shown by a straightforward ODE
approach not so-called test-function method, which gives the natural blow-up
rate. The difficulty of the proof is that, unlike the single case, terms which
come from the fact that the Laplacian cannot be absorbed into the weakly
coupled nonlinearities. A similar ODE approach is applied to heat systems by
Mochizuki to obtain the lower estimate of lifespan.Comment: 17page
An explicit unconditionally stable numerical method for solving damped nonlinear Schr\"{o}dinger equations with a focusing nonlinearity
This paper introduces an extension of the time-splitting sine-spectral (TSSP)
method for solving damped focusing nonlinear Schr\"{o}dinger equations (NLS).
The method is explicit, unconditionally stable and time transversal invariant.
Moreover, it preserves the exact decay rate for the normalization of the wave
function if linear damping terms are added to the NLS. Extensive numerical
tests are presented for cubic focusing nonlinear Schr\"{o}dinger equations in
2d with a linear, cubic or a quintic damping term. Our numerical results show
that quintic or cubic damping always arrests blowup, while linear damping can
arrest blowup only when the damping parameter \dt is larger than a threshold
value \dt_{\rm th}. We note that our method can also be applied to solve the
3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics
of a collapsing and exploding Bose-Einstein condensate (BEC).Comment: SIAM Journal on Numerical Analysis, to appea
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