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.     

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.     

Analytic smoothing effect for global solutions to nonlinear Schrödinger equations

We prove the global existence of analytic solutions to the Cauchy problem for the cubic Schrödinger equation in space dimension n?3 for sufficiently small data with exponential decay at infinity. Minimal regularity assumption regarding scaling invariance is imposed on the Cauchy data.     

Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields

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

     

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.     

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.     

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.     

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.     

Ground state solutions for the nonlinear Schrödinger-Maxwell equations

In this paper we study the nonlinear Schrödinger-Maxwell equations

     

On the standing wave for a class of nonlinear Schrödinger equations

This paper is concerned with the standing wave for a class of nonlinear Schrödinger equations

iφt+Δφ−²|x|φ+μ|φ|^p−1φ+γ|φ|^q−1φ=0,     

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.     

On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations

The existence and concentration behavior of a nodal solution are established for the equation

     

A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations

In an abstract setting we prove a nonlinear superposition principle for zeros of equivariant vector fields that are asymptotically additive in a well-defined sense. This result is used to obtain multibump solutions for two basic types of periodic stationary Schrödinger equations with superlinear nonlinearity. The nonlinear term may be of convolution type. If the superquadratic term in the energy functional is convex, our results also apply in certain cases if 0 is in a gap of the spectrum of the Schrödinger operator.     

Blow-up solutions to the nonlinear Schrödinger equation with oscillating nonlinearities

In this article we investigate the possibility of finite time blow-up in H¹(R²) for solutions to critical and supercritical nonlinear Schrödinger equations with an oscillating nonlinearity. We prove that despite the oscillations some solutions blow up in finite time. Conversely, we observe that for a given initial data oscillations can extend the local existence time of the corresponding solution.     

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.     

Exact explicit traveling wave solutions for two nonlinear Schrödinger type equations

The traveling wave solutions of the generalized nonlinear derivative Schrödinger equation and the high-order dispersive nonlinear Schrödinger equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions and periodic wave solutions) are given.     

Global existence for some radial, low regularity nonlinear Schrödinger equations

We prove the nonlinear Schrödinger equation has a local solution for any energy - subcritical nonlinearity when u₀ is the characteristic function of a ball in Rn. Additionally, we establish the existence of a global solution for n?3 when and α?2. Finally, we establish the existence of a global solution when the initial function is radial, the nonlinear Schrödinger equation has an energy subcritical nonlinearity, and the initial function lies in Hρ+?(Rn)∩H1/2+?(Rn)∩H1/2+?,1(Rn).