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.     

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.     

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.     

Semiclassical stationary states of Nonlinear Schrödinger equations with an external magnetic field

In this paper we obtain multiple solutions of the nonlinear Schrödinger equation with an external magnetic field

     

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.     

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.     

Existence and concentration of ground states of coupled nonlinear Schrödinger equations

In this paper we consider the existence and concentration of ground states of coupled nonlinear Schrödinger equations with trap potentials. When the interaction between two states is repulsive, we prove the existence of ground states. Then concentration phenomenon of these ground states is studied as the small perturbed parameter (Planck constant) approaches zero. Roughly speaking, we prove that components of the ground states concentrate at the unique global minimum points of their potentials. Moreover, we prove the existence of ground states when the interaction is attractive.     

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.     

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:

     

Multiple solutions for quasilinear Schrödinger equations involving critical exponent

By using Lions’ second concentration-compactness principle and concentration-compactness principle at infinity to prove that the (PS) condition holds locally and by minimax methods and the Krasnoselski genus theory, we establish the multiplicity of solutions for a class of quasilinear Schrödinger equations arising from physics.     

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.     

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.     

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.     

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)$H^1(\mathbb{R})$.     

NODAL BOUND STATES WITH CLUSTERED SPIKES FOR NONLINEAR SCHRDINGER EQUATIONS

We consider the following nonlinear Schr¨odinger equations -ε2△u + u = Q(x)|u|p-2u in RN, u ∈ H1(RN),where ε is a small positive parameter, N ≥ 2, 2 p ∞ for N = 2 and 2 p 2N/N-2 for N ≥ 3. We prove that this problem has sign-changing(nodal) semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q(x).Moreover, the number of this type of solutions will go to infinity as ε→ 0+.     

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.     

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.     

Infinitely many small solutions for a modified nonlinear Schrödinger equations

In the present paper, we study the Modified Nonlinear Schrödinger Equations (MNSE). Without any growth condition on the nonlinear term, we obtain the existence of infinitely many small solutions for MNSE by a dual approach.     

Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields

In this paper, we study the nonlinear Schrödinger equation with electromagnetic fields