Semiclassical limit of the nonlinear Schrödinger equation in small time

We study the semi-classical limit of the nonlinear Schrödinger equation for initial data with Sobolev regularity, before shocks appear in the limit system, and in particular the validity of the WKB method.

     

Existence and mapping properties of the wave operator for the Schrödinger equation with singular potential

We consider the Schrödinger equation in three-dimensional space with small potential in the Lorentz space and we prove Strichartz-type estimates for the solution to this equation. Moreover, using Cook's method, we prove the existence of the wave operator. In the last section we prove the equivalence between the homogeneous Sobolev spaces and in the case .

     

Divergent Solution to the Nonlinear Schr\"{o}dinger Equation with the Combined Power-Type Nonlinearities

In this paper, we consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with combined power-type nonlinearities, which is mass-critical/supercr-itical, and energy-subcritical. Combing Du, Wu and Zhang'' argument with the variational method, we prove that if the energy of the initial data is negative (or under some more general condition), then the $H^1$-norm of the solution to the Cauchy problem will go to infinity in some finite time or infinite time.     

Bi-solitons, breather solution family and rogue waves for the (2+1)-dimensional nonlinear Schr\"{o}dinger equation

In this paper, bi-solitons, breather solution family and rogue waves for the (2+1)-Dimensional nonlinear Schr\"{o}dinger equations are obtained by using Exp-function method. These solutions derived from one unified formula which is solution of the standard (1+1) dimension nonlinear Schr\"{o}dinger equation. Further, based on the solution obtained by other authors, higher-order rational rogue wave solution are obtained by using the similarity transformation. These results greatly enriched the diversity of wave structures for the (2+1)-dimensional nonlinear Schr\"{o}dinger equations     

Local exact controllability of Schr\"{o}dinger equation with Sturm- Liouville boundary value problems

In this paper, we investigate the controllability of 1D bilinear Schr\"{o}dinger equation with Sturm-Liouville boundary value condition. The system represents a quantumn particle controlled by an electric field. K. Beauchard and C. Laurent have proved local controllability of 1D bilinear Schr\"{o}dinger equation with Dirichlet boundary value condition in some suitable Sobolev space based on the classical inverse mapping theorem. Using a similar method, we extend this result to Sturm-Liouville boundary value proplems.     

Convex solutions of the Schröder equation in Banach spaces

The problem of the existence and uniqueness of increasing and convex solutions of the Schröder equation, defined on cones in Banach spaces, is examined on a base of the Krein-Rutman theorem.

     

稳态Schr\"{o}dinger方程解的Liouville型定理

给出了锥中稳态Schr\"{o}dinger方程解的Liouville型定理, 推广了邓冠铁在半空间中关于拉普拉斯方程解的相关结论.     

Rate of -concentration of blow-up solutions for critical nonlinear Schrödinger equation

This paper concerns the rate of -concentration of the blow-up solutions for the critical nonlinear Schrödinger equation. The result of Tsutsumi is improved in terms of Merle and Raphaël's recent arguments.

     

Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

This paper adresses the construction and study of a Crank-Nicolson-type discretization of the two-dimensional linear Schrödinger equation in a bounded domain with artificial boundary conditions set on the arbitrarily shaped boundary of . These conditions present the features of being differential in space and nonlocal in time since their definition involves some time fractional operators. After having proved the well-posedness of the continuous truncated initial boundary value problem, a semi-discrete Crank-Nicolson-type scheme for the bounded problem is introduced and its stability is provided. Next, the full discretization is realized by way of a standard finite-element method to preserve the stability of the scheme. Some numerical simulations are given to illustrate the effectiveness and flexibility of the method.

     

Exact travelling wave solutions for nonlinear Schr\"{o}dinger equation with variable coefficients

In this paper, two nonlinear Schr\"{o}dinger equations with variable coefficients in nonlinear optics are investigated. Based on travelling wave transformation and the extended $(\frac{G''}{G})$-expansion method, exact travelling wave solutions to nonlinear Schr\"{o}dinger equation with time-dependent coefficients are derived successfully, which include bright and dark soliton solutions, triangular function periodic solutions, hyperbolic function solutions and rational function solutions.     

Large time behavior of small solutions to subcritical derivative nonlinear Schrödinger equations

We consider the derivative nonlinear Schrödinger equations

where the coefficient satisfies the time growth condition

is a sufficiently small constant and the nonlinear interaction term consists of cubic nonlinearities of derivative type

where and . We suppose that the initial data satifsfy the exponential decay conditions. Then we prove the sharp decay estimate , for all , where . Furthermore we show that for there exist the usual scattering states, when and the modified scattering states, when

     

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.     

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).     

Solutions to a class of Schrödinger equations

We establish existence and multiplicity of solutions to a class of nonlinear Schrödinger equations with, e.g., ``atomic' Hamiltonians, via critical point theory.

     

Dynamical behaviour and exact solutions of thirteenth order derivative nonlinear Schr\"{o}dinger equation

In this paper, we considered the model of the thirteenth order derivatives of nonlinear Schr\"{o}dinger equations. It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system. By using bifurcation theory of planar dynamical systems, in different parametric regions, we determined the phase portraits. In each of these parametric regions we obtain possible exact explicit parametric representation of the traveling wave solutions corresponding to homoclinic, hetroclinic and periodic orbits.     

Mass concentration phenomena for the -critical nonlinear Schrödinger equation

In this paper, we show that any solution of the nonlinear Schrödinger equation which blows up in finite time, satisfies a mass concentration phenomena near the blow-up time. Our proof is essentially based on Bourgain's (1998), which has established this result in the bidimensional spatial case, and on a generalization of Strichartz's inequality, where the bidimensional spatial case was proved by Moyua, Vargas and Vega (1999). We also generalize to higher dimensions the results in Keraani (2006) and Merle and Vega (1998).

     

Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrödinger operators

For the general one dimensional Schrödinger operator with real we study some analytic aspects related to order-one trace formulas originally due to Buslaev-Faddeev, Faddeev-Zakharov, and Gesztesy-Holden-Simon-Zhao. We show that the condition guarantees the existence of the trace formulas of order one only with certain resolvent regularizations of the integrals involved. Our principle results are simple necessary and sufficient conditions on absolute summability of the formulas under consideration. These conditions are expressed in terms of Fourier transforms related to .

     

Beurling-Nevanlinna inequality for subfunctions of the stationary Schrödinger operator

The classical Beurling-Nevanlinna upper bound for subharmonic functions is extended to subsolutions of the stationary Schrödinger equation.