Stabilization for a Fourth Order Nonlinear Schrödinger Equation in Domains with Moving Boundaries

In the present paper we study the well-posedness using the Galerkin method and the stabilization considering multiplier techniques for a fourth-order nonlinear Schrödinger equation in domains with moving boundaries. We consider two situations for the stabilization: the conservative case and the dissipative case.     

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if

     

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces Hs, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the Hs norm of the solution, in terms of the Hs norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in Hs.     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

Global existence of small classical solutions to nonlinear Schrödinger equations

We study the global Cauchy problem for nonlinear Schrödinger equations with cubic interactions of derivative type in space dimension n?3$n?3$. The global existence of small classical solutions is proved in the case where every real part of the first derivatives of the interaction with respect to first derivatives of wavefunction is derived by a potential function of quadratic interaction. The proof depends on the energy estimate involving the quadratic potential and on the endpoint Strichartz estimates.     

Quasi-periodic solutions of nonlinear Schrödinger equations of higher dimension

It is shown that there are plenty of quasi-periodic solutions of nonlinear Schrödinger equations of higher spatial dimension, where the dimension of the frequency vectors of the quasi-periodic solutions are equal to that of the space.     

On the Finiteness of the Discrete Spectrum of a Four-Particle Lattice Schrödinger Operator

A Hamiltonian describing four bosons that move on a lattice and interact by means of pair zero-range attractive potentials is considered. A stronger version of the Hunziker–Van Vinter–Zhislin theorem on the essential spectrum is established. It is proved that the set of eigenvalues lying to the left of the essential spectrum is finite for any interaction energy of two bosons and is empty if this energy is sufficiently small.     

On Some Discrete Variational Problems

Making use of a discrete version of the P.-L. Lions concentration-compactness principle, we establish some results on the existence of nontrivial solutions for nonlinear stationary discrete equations of the Schrödinger type.     

Uniform energy decay of Schrodinger equations with variable coefficients

This paper establishes the exponential decay of the solutionsof the Schrödinger equation with variable coefficientsin the principal part subject to dissipative feedback actingin the Dirichlet boundary condition. The approach adopted usesa lifting argument in the topology of the solution along withRiemannian geometry method.     

A counter example to Strichartz estimates for the inhomogeneous Schrödinger equation

We disprove Strichartz estimates for the solution of the inhomogeneous Schrödinger equation in a certain range of the Lebesgue exponents values.     

Markov链的瞬时态与Martin边界的个数

在本文中,我们证明了含有限瞬时态的Markov链的瞬时态的个数不超过它的Martin积极流出边界的个数。     

Improved inhomogeneous Strichartz estimates for the Schrödinger equation

We study inhomogeneous Strichartz estimates for the Schrödinger equation for dimension n?3. Using a frequency localization, we obtain some improved range of Strichartz estimates for the solution of inhomogeneous Schrödinger equation except dimension n=3.     

The growth in time of higher Sobolev norms of solutions to Schrödinger equations on compact Riemannian manifolds

In this paper, we shall estimate the growing speed for higher Sobolev norms of the solutions to Schrödinger equations on Riemannian manifolds (d?2), under some bilinear Strichartz estimate assumptions.     

On the Blowing up of Solutions for one System of Schrödinger Equations

We consider one system of nonlinear Schrödinger equations and find the conditions when its solution blows up.     

Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear Schrödinger equations −Δu+V(x)u=f(x,u), x∈RN, where the primitive of the nonlinearity f is of superquadratic growth near infinity in u and the potential V is allowed to be sign-changing.     

Poisson geometry of the filament equation

Summary The Hasimoto transformation (relating vortex filament flow to the nonlinear Schrödinger equation) is interpreted in the context of Poisson geometry with the aid of a compact formula for its differential. A useful relationship is derived between Killing fields for soliton solutions of the filament flow and the sequence of commuting Hamiltonian flows.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.