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We consider the derivative NLS equation with general quadratic nonlinearities. In [2] the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension . Here we prove a similar result for large initial data in all dimensions .
Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also...
Sample path large deviations
for the laws of the solutions of stochastic nonlinear
Schrödinger equations when the noise converges to zero are
presented. The noise is a complex additive Gaussian noise. It is
white in time and colored in space. The solutions may be global or
blow-up in finite time, the two cases are distinguished. The
results are stated in trajectory spaces endowed with topologies
analogue to projective limit topologies. In this setting, the
support of the law of the solution is...
We give the definitions of exact and approximate controllability for
linear and nonlinear Schrödinger equations, review fundamental criteria
for controllability and revisit a classical “No-go” result
for evolution equations due to Ball, Marsden and Slemrod.
In Section 2 we prove corresponding results on non-controllability
for the linear Schrödinger equation and distributed additive control,
and we show that the Hartree equation of quantum chemistry with bilinear
control is not controllable...
This paper gives a rigorous derivation
of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611]
to characterize the energy of vortex filaments
in a rotationally forced Bose-Einstein condensate. This
functional is derived as a Γ-limit
of scaled versions of the Gross-Pitaevsky
functional for the wave function of such a condensate. In most situations,
the vortex filament energy functional is either unbounded below
or has only trivial minimizers, but
we establish the existence...
We study the local behaviour of solutions of the following type of equation,-Δu - V(x)u + g(u) = 0 when V is singular at some points and g is a non-decreasing function. Emphasis is put on the case when V(x) = c|x|-2 and g has a power-like growth.
In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the critical and super-critical...
The nonlinear heat equation with a fractional Laplacian , is considered in a unit ball . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.
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