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

     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.     

An optimal control problem for two-dimensional Schrödinger equation

In this paper we consider an optimal control problem controlled by three functions which are in the coefficients of a two-dimensional Schrödinger equation. After proving the existence and uniqueness of the optimal solution, we get the Frechet differentiability of the cost functional using Hamilton-Pontryagin function. Then we state a necessary condition to an optimal solution in the variational inequality form using the gradient.     

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.     

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

In this paper, one-dimensional (1D) 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).     

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.     

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.     

On the two-dimensional stationary Schrödinger equation with a singular potential

We study the equation

−△u(x,y)+ν(x,y)u(x,y)=0     

The use of Sinc-collocation method for solving multi-point boundary value problems

Multi-point boundary value problems have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of these problems by using the Sinc-collocation method. Some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of multi-point boundary value problems to some algebraic equations. It is well known that the Sinc procedure converges to the solution at an exponential rate. Numerical examples are included to demonstrate the validity and applicability of the new technique.     

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

     

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.     

A counter example to Strichartz estimates for the inhomogeneous Schrödinger equation

We disprove Strichartz estimates for the solution of the inhomogeneous Schrödinger equation in a certain range of the Lebesgue exponents values.     

A local smoothing estimate for the Schrödinger equation

We show that the Schrödinger operator eitΔ is bounded from Wα,q(Rn) to Lq(Rn×[0,1]) for all α>2n(1/2−1/q)−2/q and q?2+4/(n+1). This is almost sharp with respect to the Sobolev index. We also show that the Schrödinger maximal operator sup0<t<1|eitΔf| is bounded from Hs(Rn) to when s>s₀ if and only if it is bounded from Hs(Rn) to L²(Rn) when s>2s₀. A corollary is that sup0<t<1|eitΔf| is bounded from Hs(R²) to L²(R²) when s>3/4.     

The general quasilinear ultrahyperbolic Schrödinger equation

We consider the general quasilinear Schrödinger equation whose second order coefficients are given by a real symmetric non-degenerate matrix. We deduce conditions which guarantee that the associated initial value problem is locally well posed.     

A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

A family of predictor-corrector exponential Numerov-type methods is developed for the numerical integration of the one-dimensional Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. The new methods are very simple and integrate more exponential functions than both the well-known fourth-order Numerov-type exponentially fitted methods and the sixth algebraic order Runge-Kutta-type methods. Numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above.