Global well-posedness of the fractional Klein-Gordon-Schr¨odinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R~(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H~(s_1) and wave data in H~(s_2) × H~(s_2-1)for 3/4- α s_1≤0 and-1/2 s_2 3/2, where α is the fractional power of Laplacian which satisfies 3/4 α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 s_2 1/2 by using the conservation law for the L~2 norm of u.     

ON THE MODIFIED NONLINEAR SCHRDINGER EQUATION IN THE SEMICLASSICAL LIMIT:SUPERSONIC,SUBSONIC,AND TRANSSONIC BEHAVIOR

The purpose of this paper is to present a comparison between the modified nonlinear Schro¨dinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr¨odinger (NLS) equation in the semiclassical limit. We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods. The main message is that, depending on initial data, the MNLS equation can behave either like the defocusing NLS equation, like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data), or like an interesting mixture of the two. In the latter case, we identify a feature of the dynamics analogous to a sonic line in gas dynamics, a free boundary separating subsonic flow from supersonic flow.     

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint,we obtain a finite dimensional system from the Lax pair of the nonlinear Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.     

The inviscid limit for the Landau-Lifshitz-Gilbert equation in the critical Besov space

We prove that in dimensions three and higher the Landau-Lifshitz-Gilbert equation with small initial data in the critical Besov space is globally well-posed in a uniform way with respect to the Gilbert damping parameter. Then we show that the global solution converges to that of the Schr¨odinger maps in the natural space as the Gilbert damping term vanishes. The proof is based on some studies on the derivative Ginzburg-Landau equations.     

Reducibility for Schr¨odinger Operator with Finite Smooth and Time-Quasi-periodic Potential

In this paper, the author establishes a reduction theorem for linear Schr¨odinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM(Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schr¨odinger operator possesses the property of pure point spectra and zero Lyapunov exponent.     

The defocusing energy-supercritical Hartree equation

In this paper,we study the global well-posedness and scattering problem for the energysupercritical Hartree equation iut+Δu.(|x|.γ.|u|2)u=0 with γ4 in dimension d γ.We prove that if the solution u is apriorily bounded in the critical Sobolev space,that is,u ∈Lt∞(I;Hxsc(Rd)) with sc:= γ/2.11,then u is global and scatters.The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation(NLW) and nonlinear Schrdinger equation(NLS).We utilize the strategy derived from concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of three scenarios:finite time blowup;soliton-like solution and low to high frequency cascade.Making use of the No-waste Duhamel formula,we deduce that the energy of the finite time blow-up solution is zero and so get a contradiction.Finally,we adopt the double Duhamel trick,the interaction Morawetz estimate and interpolation to kill the last two scenarios.     

The complex 2-sphere in C~3 and Schr?dinger flows Dedicated to Professor Hesheng Hu on the Occasion of Her 90th Birthday

By using holomorphic Riemannian geometry in C~3, the coupled Landau-Lifshitz(CLL) equation is proved to be exactly the equation of Schr¨odinger flows from R~1 to the complex 2-sphere CS~2(1) → C~3.Furthermore, regarded as a model of moving complex curves in C~3, the CLL equation is shown to preserve the PT symmetry if the initial data is of the P symmetry. As a consequence, the nonlocal nonlinear Schr¨odinger(NNLS)equation proposed recently by Ablowitz and Musslimani is proved to be gauge equivalent to the CLL equation with initial data being restricted by the P symmetry. This gives an accurate characterization of the gaugeequivalent magnetic structure of the NNLS equation described roughly by Gadzhimuradov and Agalarov(2016).     

On the concentration properties for the nonlinear Schrödinger equation with a Stark potential

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

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+.     

GLOBAL WEAK SOLUTION TO THE NONLINEAR SCHRDINGER EQUATIONS WITH DERIVATIVE

In this paper, we study the nonlinear Schr¨odinger equations with derivative. By using the Gal¨erkin method and a priori estimates, we obtain the global existence of the weak solution.     

ON THE DECAY AND SCATTERING FOR THE KLEIN-GORDON-HARTREE EQUATION WITH RADIAL DATA

In this paper,we study the decay estimate and scattering theory for the Klein-Gordon-Hartree equation with radial data in space dimension d≥3.By means of a compactness strategy and two Morawetz-type estimates which come from the linear and nonlinear parts of the equation,respectively,we obtain the corresponding theory for energy subcritical and critical cases.The exponent range of the decay estimates is extended to 0<γ≤4 and γ相似文献   

A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth.As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schr¨odinger equations.     

The GLM representation of the two-component nonlinear Schrödinger equation on the half-line

The Gelfand-Levitan-Marchenko representation is used to analyze the initialboundary value problem of two-component nonlinear Schr¨odinger equation on the half-line.It has shown that the global relation can be effectively analyzed by the Gelfand-LevitanMarchenko representation. we also derive expressions for the Dirichlet-to-Neumann map to characterize the unknown boundary values.     

EXACT SOLUTIONS OF (2+1)-DIMENSIONAL NONLINEAR SCHRöDINGER EQUATION BASED ON MODIFIED EXTENDED TANH METHOD

In this paper, the modified extended tanh method is used to construct more general exact solutions of a(2+1)-dimensional nonlinear Schr¨odinger equation.With the aid of Maple and Matlab software, we obtain exact explicit kink wave solutions, peakon wave solutions, periodic wave solutions and their 3D images.     

A RIGOROUS DERIVATION OF THE GROSS-PITAEVSKII HIERARCHY FOR WEAKLY COUPLED TWO-DIMENSIONAL BOSONS

In this article, we consider the dynamics of N two-dimensional boson systems interacting through a pair potential N-1Va(xi-xj) where Va(x) = a-2V (x/a). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrdinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices {k ut, k ≥ 1} solves the GP hierarchy. Denote by ψN,t the solution to the N-particle Schrdinger equation. Under the assumption that a = N-ε for 0 ε 3/4, we prove that as N →∞ the limit points of the k-particle density matrices of ψN,t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V (x) dx.     

POSITIVE SOLUTIONS TO HYBRID SCHR?DINGER EQUATION WITH NORMAL AND FRACTIONAL LAPLACIANS

In this paper, we study the hybrid Schr¨odinger equation involving normal and fractional Laplace operator, and obtain the existence of the solutions to this class of the hybrid partial differential equation. Our main argument is variational methods.     

ON THE CONCENTRATION PROPERTIES FOR THE NONLINEAR SCHRDINGER EQUATION WITH A STARK POTENTIAL

In this paper, we study blow-up solutions of the Cauchy problem to the L2 critical nonlinear Schrdinger equation with a Stark potential. Using the variational characterization of the ground state for nonlinear Schrdinger equation without any potential, we obtain some concentration properties of blow-up solutions, including that the origin is the blow-up point of the radial blow-up solutions, the phenomenon of L2-concentration and rate of L2-concentration of blow-up solutions.     

Energy Scattering Theory for Electromagnetic NLS in Dimension Two

We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schr?dinger equations with scaling critical magnetic potentials in dimension two.In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov–Bohm potential which decays likely the Coulomb potential |x|~(-1).     

The NLS approximation for two dimensional deep gravity waves Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

This article is concerned with in?nite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr¨odinger equation(NLS) on the natural cubic time scale.     

EXISTENCE OF GENERALIZED SOLUTION AND ORBITAL STABILITY OF STANDING WAVES FOR NONLINEAR SINGULAR SCHRDINGER EQUATION

The existence of generalized solution to the initial value problem iu_t △u k/(x_N)u_X_N q(x)u |u|~(p-1)u=0 on R~N is studied, By Galerkin method, we prove that the solution always exists for every initial value in H~1(R~N; k) if 1相似文献