Space-time scattering for the Schrödinger equation

Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL ^r (R;L ^q (R ^d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.     

Eigenvalue asymptotics and a non-linear Schrödinger equation

The equation 1 $$ - \Delta u + qu + f(x,u) = \lambda u, u \in W^{1,2} (\mathbb{R}^N )$$ is considered, whereq is bounded below andq(x)→∞ as |x|→∞. Under appropriate conditions on the perturbation termf(x, u) it is shown that given anyr>0, (*) has an infinite sequence (λ _{n, r} ) _{n ∈ N} of eigenvalues, eachλ _{n, r} being associated with an eigenfunctionu _n,r which satisfies \(\smallint _{R^N } \left| {u_{n,r} } \right|^2 = r^2 \) . Information about the behaviour ofλ _{n, r} for largen is provided. The proofs rely on the compactness of the embedding of a certain weighted Sobolov space in anL ^p space; this is proved in §2.     

The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.     

An asymptotic expression of the Schrödinger equation

The problem of solving the time–independent Schr?dinger equation for the motion of an electron of mass μ and charge –e (e > 0) in the field of two fixed Coulomb centers has been the subject of extensive studies in theoretical physics and quantum computation. In the present paper, after making a series of coordinate transformations, we apply the qualitative theory of nonlinear differential equations to the study of the Schr?dinger equation under certain parametric conditions, and obtain an asymptotic formula. The work has been presented at the International Conference on Quantum Computation and Quantum Technology, Texas A&M University, College Station, Texas, November 13-16, 2005. The author would like to thank the organizer Professor Goong Chen for his generous support. This work is also partly supported by UTPA Faculty Research Council Grant 119100.     

Radial Schrödinger equation: The spectral problem

Using the integral transformation method involving the investigation of the Laplace transforms of wave functions, we find the discrete spectra of the radial Schrödinger equation with a confining power-growth potential and with the generalized nuclear Coulomb attracting potential. The problem is reduced to solving a system of linear algebraic equations approximately. We give the results of calculating the discrete spectra of the S-states for the Schrödinger equation with a linearly growing confining potential and the nuclear Yukawa potential.     

Breathers in a damped-driven nonlinear Schrödinger equation

We construct time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation as solutions of a boundary value problem for the space-dependent Fourier coefficients.     

Theory of bifurcations of the Schrödinger equation

We obtain solvability conditions and a representation of solutions for a boundary value problem for a linear nonstationary Schrödinger equation in a Hilbert space as well as sufficient conditions for the bifurcation of solutions of this equation.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.     

Maximum decay rate for the nonlinear Schrödinger equation

In this paper, we consider global solutions for the following nonlinear Schrödinger equation in with and We show that no nontrivial solution can decay faster than the solutions of the free Schrödinger equation, provided that u(0) lies in the weighted Sobolev space in the energy space, namely or in according to the different cases.     

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.     

The structure of solutions of the Schrödinger equation

The goal of the paper is to study the structure of the eigenfunctions of the one-dimensional Schrödinger equation from the point of view of the Euler theorem. It turns out that analog of exponent is exponentially increasing solution. Sometimes linear combinations of such solutions cancel each other at infinity and then we obtain an eigenfunction from L₂(R¹).     

Variable sinh-Gaussian solitons in nonlocal nonlinear Schrödinger equation

Based on the nonlocal nonlinear Schrödinger equation that governs phenomenologically the propagation of laser beams in nonlocal nonlinear media, we theoretically investigate the propagation of sinh-Gaussian beams (ShGBs). Mathematical expressions are derived to describe the beam propagation, the intensity distribution, the beam width, and the beam curvature radius of ShGBs. It is found that the propagation behavior of ShGBs is variable and closely related to the parameter of sinh function (PShF). If the PShF is small, the transverse pattern of ShGBs keeps invariant during propagation for a proper input power, which can be regarded as solitons. If the PShF is large, it varies periodically, which is similar to the evolution of temporal higher-order solitons in nonlinear optical fiber. Numerical simulations are carried out to illustrate the typical propagation characteristics.     

Hirota bilinearization of coupled higher-order nonlinear Schrödinger equation

In this paper, we present the Hirota bilinearization of the coupled Sasa–Satsuma equation. The procedure employed here generates a more general solution than the one reported earlier. We also discuss the soliton solutions of the equation and show that the solutions found earlier are only special cases of the solution discussed here.     

Attractors for discrete nonlinear Schrödinger equation with delay

In this paper,we first prove the existence of the global attractor Aν ∈ C（[-ν,0],2）（ν 0） for a weak damping discrete nonlinear Schrdinger equation with delay.Then we consider an upper semi-continuity of Aν as ν → 0＋.     

Strichartz estimates for the periodic non-elliptic Schrödinger equation

The purpose of this Note is to prove sharp Strichartz estimates with derivative losses for the non-elliptic Schrödinger equation posed on the 2-dimensional torus.