Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials

We establish the existence and multiplicity of semiclassical bound states of the following nonlinear Schrödinger equation:

     

Instability of bound states for abstract nonlinear Schrödinger equations

We study the instability of bound states for abstract nonlinear Schrödinger equations. We prove a new instability result for a borderline case between stability and instability. We also reprove some known results in a unified way.     

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.     

Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions for the nonlinear Schrödinger equation with E being a critical frequency in the sense that . We show that if the zero set of W−E has several isolated connected components Z_i(i=1,…,m) such that the interior of Z_i is not empty and ∂Z_i is smooth, then for ?>0 small there exists, for any integer k,1?k?m, a standing wave solution which is trapped in a neighborhood of , where is any given subset of . Moreover the amplitude of the standing wave is of the level . This extends the result of Byeon and Wang (Arch. Rational Mech. Anal. 165 (2002) 295) and is in striking contrast with the non-critical frequency case , which has been studied extensively in the past 20 years.     

Asymptotics of odd solutions of quadratic nonlinear Schrödinger equations

We consider the Cauchy problem for a quadratic nonlinear Schrödinger equation in the case of odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Asymptotics of odd solutions for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for a cubic nonlinear Schrödinger equation in the case of an odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Soliton solutions for quasilinear Schrödinger equations, II

For a class of quasilinear Schrödinger equations, we establish the existence of ground states of soliton-type solutions by a variational method.     

Some dispersive estimates for Schrödinger equations with repulsive potentials

We prove the local smoothing effect for Schrödinger equations with repulsive potentials for n?3. The estimates are global in time and are proved using a variation of Morawetz multipliers. As a consequence we give sharp constants to measure the attractive part of the potential and its rate of decay, which turns out to be different whether dimension 3 or higher are considered. Also a notion of zero resonance arises in a natural way. Our smoothing estimate allows us to use Sobolev inequalities and treat nonradial perturbations.     

On the existence of signed and sign-changing solutions for a class of superlinear Schrödinger equations

This paper deals with a semilinear Schrödinger equation whose nonlinear term involves a positive parameter λ and a real function f(u) which satisfies a superlinear growth condition just in a neighborhood of zero. By proving an a priori estimate (for a suitable class of solutions) we are able to avoid further restrictions on the behavior of f(u) at infinity in order to prove, for λ sufficiently large, the existence of one-sign and sign-changing solutions. Minimax methods are employed to establish this result.     

Nonlinear Schrödinger equation with a point defect

We study the nonlinear Schrödinger equation with a delta-function impurity in one space dimension. Local well-posedness is verified for the Cauchy problem in H¹(R)$H^1(\mathbb{R})$. In case of attractive delta-function, orbital stability and instability of the ground state is proved in H¹(R)$H^1(\mathbb{R})$.     

Soliton dynamics for a general class of Schrödinger equations

The soliton dynamics for a general class of nonlinear focusing Schrödinger problems in presence of non-constant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation.     

Nonlinear Schrödinger equations with coupled Hartree-type terms and rotation

We consider the Cauchy problem for a class of magnetic Schrödinger system with local and nonlocal nonlinearities. The problem stems from a typical model describing the mean-field dynamics of rotating many-body bosons in a confining trap. We present sufficient conditions which yield global well-posedness or finite time blowup solutions to the system.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Standing waves for nonlinear Schrödinger equations with singular potentials

We study semiclassical states of nonlinear Schrödinger equations with anisotropic type potentials which may exhibit a combination of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials.     

On a class of Schrödinger systems with discontinuous nonlinearities

Using a non-smooth critical point theory for locally Lipschitz functionals, we investigate a class of stationary Schrödinger systems with subcritical discontinuous nonlinearities and lower bounded potentials that blow up at infinity. The existence of nontrivial solution is obtained.     

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.     

Nodal type bound states of Schrödinger equations via invariant set and minimax methods

For nonlinear Schrödinger equations in the entire space we present new results on invariant sets of the gradient flows of the corresponding variational functionals. The structure of the invariant sets will be built into minimax procedures to construct nodal type bound state solutions of nonlinear Schrödinger type equations.