Necessary conditions and sufficient conditions for global existence for two-components nonlinear Schrödinger equations in the super critical case

In this paper, we consider two-components nonlinear Schrödinger equations in the super critical case. We establish a necessary condition and a sufficient condition of global existence of the solution for two-components nonlinear Schrödinger equations. These conditions are charge criterion of global existence in the super critical case, thereby extending the results in the critical case. Furthermore, we improve a blow-up condition.     

Sharp conditions of global existence and scattering for a focusing energy-critical Schrödinger equation

We consider a focusing energy-critical Schrödinger equation with subcritical perturbations and address question related to the sharp criterion of global existence and scattering. By analyzing the variational characteristics of this equation, we established two types of invariant flows. Then approximating this equation by the energy-critical nonlinear Schrödinger equations with the same initial data and combining the properties of the invariant flows, we obtain the sharp conditions of global existence for this equation. Moreover, when the solution is globally defined, we prove the scattering.     

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.     

Global existence for 2D nonlinear Schrödinger equations via high-low frequency decomposition method

We study global existence of solutions for the Cauchy problem of the nonlinear Schrödinger equation iut+Δu=|u|^2mu in the 2 dimension case, where m is a positive integer, m?2. Using the high-low frequency decomposition method, we prove that if then for any initial value φ∈Hs(R²), the Cauchy problem has a global solution in C(R,Hs(R²)), and it can be split into u(t)=eitΔφ+y(t), with y∈C(R,H¹(R²)) satisfying , where ? is an arbitrary sufficiently small positive number.     

Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential

We prove that a parametric nonlinear Schrödinger equation possesses a finite dimensional smooth global attractor in a suitable energy space.     

Instability of standing waves for a class of nonlinear Schrödinger equations

This paper discusses a class of nonlinear Schrödinger equations with different power nonlinearities. We first establish the existence of standing wave associated with the ground states by variational calculus. Then by the potential well argument and the concavity method, we get a sharp condition for blowup and global existence to the solutions of the Cauchy problem and answer such a problem: how small are the initial data, the global solutions exist? At last we prove the instability of standing wave by combing those results.     

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.     

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.     

On the Cauchy problem for the nonlinear Schrödinger equations with time-dependent linear loss/gain

This paper is devoted to the Cauchy problem for the nonlinear Schrödinger equation with time-dependent loss/gain which reads i_ut+Δu+λ|u|^αu+ia(t)u=0$i{u}_t+\mathrm{\Delta }u+\lambda {|u|}^{\alpha }u+ia(t)u=0$. This equation appears in the recent studies of Bose–Einstein condensates and optical systems. We obtain some global existence and blow-up results which depend on the size of the loss/gain coefficient. In particular, we prove the global existence for the energy critical nonlinearity. By scaling and compactness arguments, we also discuss asymptotic profiles and concentration properties of blow-up solutions.     

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).     

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.     

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.     

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,     

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.     

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.     

Wellposedness for the fourth order nonlinear Schrödinger equations

We study the local smoothing effects and wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in 1D

     

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.