Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if

     

Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation

     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

Exact solutions of the nonlinear Schrödinger equation by the first integral method

The first integral method is an efficient method for obtaining exact solutions of some nonlinear partial differential equations. This method can be applied to nonintegrable equations as well as to integrable ones. In this paper, the first integral method is used to construct exact solutions of the nonlinear Schrödinger equation.     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).     

Exact solutions to nonlinear Schrödinger equation with variable coefficients

According to Ma-Fuchsseiter’s idea, a trial equation method was proposed to find the exact envelop traveling wave solutions to some nonlinear differential equations with variable coefficients. As an application, combining with the complete discrimination system for polynomial, some exact envelop traveling wave solutions to Schrödinger equation with variable coefficients were obtained. At the same time, the physical meanings of the obtained solutions are discussed, and the problem needed to further study is pointed out.     

A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

In this paper, one-dimensional (1D) nonlinear Schrödinger equation

     

Improved inhomogeneous Strichartz estimates for the Schrödinger equation

We study inhomogeneous Strichartz estimates for the Schrödinger equation for dimension n?3. Using a frequency localization, we obtain some improved range of Strichartz estimates for the solution of inhomogeneous Schrödinger equation except dimension n=3.     

On the blow up phenomenon of the critical nonlinear Schrödinger equation

In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the L² norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and L²-concentration are proved. The proof relies on linear and nonlinear profile decompositions.     

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.     

Stability of small periodic waves for the nonlinear Schrödinger equation

The nonlinear Schrödinger equation possesses three distinct six-parameter families of complex-valued quasiperiodic traveling waves, one in the defocusing case and two in the focusing case. All these solutions have the property that their modulus is a periodic function of x−ct for some c∈R. In this paper we investigate the stability of the small amplitude traveling waves, both in the defocusing and the focusing case. Our first result shows that these waves are orbitally stable within the class of solutions which have the same period and the same Floquet exponent as the original wave. Next, we consider general bounded perturbations and focus on spectral stability. We show that the small amplitude traveling waves are stable in the defocusing case, but unstable in the focusing case. The instability is of side-band type, and therefore cannot be detected in the periodic set-up used for the analysis of orbital stability.     

Parametric resonance of ground states in the nonlinear Schrödinger equation

We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t^-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.     

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.     

Dynamics for the energy critical nonlinear Schrödinger equation in high dimensions

In [T. Duyckaerts, F. Merle, Dynamic of threshold solutions for energy-critical NLS, preprint, arXiv:0710.5915 [math.AP]], T. Duyckaerts and F. Merle studied the variational structure near the ground state solution W of the energy critical NLS and classified the solutions with the threshold energy E(W) in dimensions d=3,4,5 under the radial assumption. In this paper, we extend the results to all dimensions d?6. The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W.     

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.     

Evolution of solitary waves for a perturbed nonlinear Schrödinger equation

Soliton perturbation theory is used to determine the evolution of a solitary wave described by a perturbed nonlinear Schrödinger equation. Perturbation terms, which model wide classes of physically relevant perturbations, are considered. An analytical solution is found for the first-order correction of the evolving solitary wave. This solution for the solitary wave tail is in integral form and an explicit expression is found, for large time. Singularity theory, usually used for combustion problems, is applied to the large time expression for the solitary wave tail. Analytical results are obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks, in the solitary wave tail. Two examples, the near-continuum limit of a discrete NLS equation and an explicit numerical scheme for the NLS equation, are considered in detail. For the discrete NLS equation it is found that three qualitatively different types of solitary wave tail can occur, while for the explicit finite-difference scheme, only one type of solitary wave tail occurs. An excellent comparison between the perturbation solution and numerical simulations, for the solitary wave tail, is found for both examples.     

A note on blow-up phenomena for mass critical nonlinear Schrödinger equation

In this paper, we mainly discuss the radial case for L² critical nonlinear Schrödinger equation with finite blow-up time. We describe that the solution may concentrate some points with different speeds. Furthermore, we give further research to the conjecture given by F. Merle and P. Raphael (2005) in [13] and we proved the conjecture for some cases.     

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces Hs, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the Hs norm of the solution, in terms of the Hs norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in Hs.