Long time dynamics of the 3D radial NLS with the combined terms

In this paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation(NLS) with the combined terms iu_t+△u=-|u|~4u+|4|~(p-1)u,1+4/3p5 in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with subcritical threshold energy.This extends algebraic perturbation in a previous work of Miao,Xu and Zhao[Comm.Math.Phys.,318,767-808(2013)]to all mass supercritical,energy subcritical perturbation.     

Dynamics of some coupled nonlinear Schrödinger systems in

In this paper, we show the scattering and blow up result of the solution for some coupled nonlinear Schrödinger system with static energy less than that of the ground state in , where . We first use the Nehari manifold approach and the Schwarz symmetrization technique to construct the ground state and obtain the threshold energy of scattering solution, then use Payne–Sattinger's potential well argument and Kenig–Merle's compactness‐rigidity argument to show the aforementioned dichotomy result. As we know, it is the first attempt to show the scattering result for the coupled nonlinear Schrödinger system. Copyright © 2013 John Wiley & Sons, Ltd.     

Multifrequency NLS Scaling for a Model Equation of Gravity‐Capillary Waves

This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrödinger (NLS) equation as well as other related systems starting from a model coming from the gravity‐capillary water wave system in the long‐wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrödinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence. © 2012 Wiley Periodicals, Inc.     

On the Cauchy problem for the nonlinear Schrodinger equations including fractional dissipation with variable coefficient

This paper is devoted to the Cauchy problem for the nonlinear Schrodinger equation with time‐dependent fractional damping term. We prove the local existence result, and we study the global existence and blow‐up solutions.     

Correspondence between the NLS equation for optical fibers and a class of integrable NLS equations

The propagation of the optical field complex envelope in a single‐mode fiber is governed by a one‐dimensional cubic nonlinear Schrödinger equation with a loss term. We present a result about L²‐closeness of the solutions of the aforementioned equation and of a one‐dimensional nonlinear Schrödinger equation that is Painlevé integrable. Copyright © 2014 John Wiley & Sons, Ltd.     

Remark on energy‐critical NLS in 5D

We reprove global well‐posedness and scattering for the defocusing energy‐critical nonlinear Schrödinger equation in five dimensions. Inspired by the recent work of Killip and Visan, we adapt the Dodson's strategy ‘long‐time Strichartz estimate’ used in the work on mass‐critical nonlinear Schrödinger equation sets. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

The rigorous derivation of the T2 focusing cubic NLS from 3D

We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the x-direction and a well of infinite-depth in the z-direction. Physical experiments for these systems are scarce. We find that, to fulfill the empirical requirement for observing NLS dynamics in experiments, namely, that the kinetic energy dominates the potential energy, it is necessary to impose an extra restriction on the system parameters. This restriction gives rise to an unusual coupling constant.     

Nodal bound state of nonlinear problems involving the fractional Laplacian

This paper is concerned with the following nonlinear fractional Schrödinger equation where ε>0 is a small parameter, V(x) is a positive function, 0<s<1, and . Under some suitable conditions, we prove that for any positive integer k, one can construct a nonradial sign‐changing (nodal) solutions with exactly k maximum points and k minimum points near the local minimum point of V(x).     

Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Local exact controllability of the one‐dimensional NLS (subject to zero‐boundary conditions) with distributed control is shown to hold in a H¹‐neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. Copyright © 2007 John Wiley & Sons, Ltd.     

Derivation of Korteweg‐de Vries flow equations from fourth order nonlinear Schrödinger equation

We perform a multiple scale analysis on the fourth order nonlinear Schrödinger equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg‐de Vries flow equations in the bi‐Hamiltonian form with the corresponding Hamiltonian functions. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

The attractor of the dissipative coupled fractional Schrödinger equations

In the present paper, we consider the dissipative coupled fractional Schrödinger equations. The global well‐posedness by the contraction mapping principle is obtained. We study the long time behavior of solutions for the Cauchy problem. We prove the existence of global attractor. Copyright © 2013 John Wiley & Sons, Ltd.     

Local well‐posedness of critical nonlinear Schrödinger equation on Zoll manifolds of odd‐growth

In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well‐poesdness in the critical space . This extends the recent results in the literature to the Zoll manifolds of dimension d≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution of perturbed Schrödinger system with critical nonlinearity and electromagnetic fields

In this paper, we consider the following perturbed nonlinear Schrödinger system with electromagnetic fields where N ≥ 3, 2 ^蜧 = 2N ∕ (N ? 2) is the Sobolev critical exponent; A is the real vector magnetic potential; and V (x), K(x), and H(s,t) are continuous functions. Under certain conditions on V, H, and K, we establish some new results on the existence of the least‐energy solutions (u_?,v_?) for small ? by using variational method. Copyright © 2012 John Wiley & Sons, Ltd.     

An inverse scattering problem for the Schrödinger equation in a semiclassical process

We study a direct and an inverse scattering problem for a pair of Hamiltonians (H(h),H₀(h)) on , where H₀(h)=−h²Δ and H(h)=H₀(h)+V, V is a short-range potential and h is the semiclassical parameter. First, we show that if two potentials are equal in the classical allowed region for a fixed non-trapping energy, the associated scattering matrices coincide up to O(h^∞) in . Then, for potentials with a regular behaviour at infinity, we study the inverse scattering problem. We show that in dimension n3, the knowledge of the scattering operators S(h), , up to O(h^∞) in , and which are localized near a fixed energy λ>0, determine the potential V at infinity.