Low Regularity Exponential-Type Integrators for Semilinear Schrödinger Equations

We introduce low regularity exponential-type integrators for nonlinear Schrödinger equations for which first-order convergence only requires the boundedness of one additional derivative of the solution. More precisely, we will prove first-order convergence in \(H^r\) for solutions in \(H^{r+1}\) (with \(r > d/2\)) of the derived schemes. This allows us lower regularity assumptions on the data than for instance required for classical splitting or exponential integration schemes. For one-dimensional quadratic Schrödinger equations, we can even prove first-order convergence without any loss of regularity. Numerical experiments underline the favorable error behavior of the newly introduced exponential-type integrators for low regularity solutions compared to classical splitting and exponential integration schemes.     

Solitary Waves for Linearly Coupled Nonlinear Schrödinger Equations with Inhomogeneous Coefficients

Motivated by the study of matter waves in Bose–Einstein condensates and coupled nonlinear optical systems, we study a system of two coupled nonlinear Schrödinger equations with inhomogeneous parameters, including a linear coupling. For that system, we prove the existence of two different kinds of homoclinic solutions to the origin describing solitary waves of physical relevance. We use a Krasnoselskii fixed point theorem together with a suitable compactness criterion.     

Some remarks on localization of Schrödinger means

We study localization and localization almost everywhere of Schrödinger means of functions in Sobolev spaces.     

Strichartz Estimates for Schrödinger Equations on Scattering Manifolds

The present article is concerned with Schrödinger equations on non-compact Riemannian manifolds with asymptotically conic ends. It is shown that, for any admissible pair (including the endpoint), local in time Strichartz estimates outside a large compact set are centered at origin hold. Moreover, we prove global in space Strichartz estimates under the nontrapping condition on the metric.     

The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n???1)/n?p?≤?2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation ??Δu?+?Vu?=?0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H ^p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.     

Spherical Perturbations of Schrödinger Equations

We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights and on , for which the a priori inequality holds with constant independent of . Here is the Laplace-Beltrami operator on the sphere . Due to the relation between and the tangential component of the gradient, , we obtain some "Morawetz-type" estimates for on . As a consequence we establish some new estimates for the free Schr?dinger propagator , which may be viewed as certain refinements of the -(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem for certain time dependent first order spherical perturbations of the dimensional Schr?dinger equation.     

Dispersion Estimates for Spherical Schrödinger Equations

We derive a dispersion estimate for one-dimensional perturbed radial Schrödinger operators. We also derive several new estimates for solutions of the underlying differential equation and investigate the behavior of the Jost function near the edge of the continuous spectrum.     

Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

Strichartz Estimates for the Schrödinger Equation on Polygonal Domains

We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.     

Group Classification of Nonlinear Schrödinger Equations

We propose an approach to problems of group classification. By using this approach, we perform a complete group classification of nonlinear Schrödinger equations of the form i _t + + F(, *) = 0.     

Stochastic Nonlinear Equations of Schrödinger Type

In this article, the solution for a stochastic nonlinear equation of Schrödinger type, which is perturbed by an infinite dimensional Wiener process, is investigated. The existence of the solution is proved by using the Galerkin method. Moment estimates for the solution are also derived. Examples from physics are given in the final part of the article.     

Approximate Stabilization of One-dimensional Schr?dinger Equations in Inhomogeneous Media

We present how to control the bilinear 1D infinite-dimensional Schr?dinger equations in inhomogeneous media (with x-dependent coefficients), getting the approximate stabilization around ground state. Our arguments are based on constructing a Lyapunov function and a strategy similar to LaSalle invariance principle.     

Schrödinger Equations on Damek–Ricci Spaces

In this paper we consider the Laplace–Beltrami operator Δ on Damek–Ricci spaces and derive pointwise estimates for the kernel of e ^τΔ, when τ ∈ ?* with Re τ ≥0. When τ ∈i?*, we obtain in particular pointwise estimates of the Schrödinger kernel associated with Δ. We then prove Strichartz estimates for the Schrödinger equation, for a family of admissible pairs which is larger than in the Euclidean case. This extends the results obtained by Anker and Pierfelice [4 Anker , J.-P. , Pierfelice , V. ( 2009 ). Nonlinear Schrödinger equation on real hyperbolic spaces . Ann. Inst. H. Poincaré (C) Non Linear Analysis 26 : 1853 – 1869 . [Google Scholar]] on real hyperbolic spaces. As a further application, we study the dispersive properties of the Schrödinger equation associated with a distinguished Laplacian on Damek–Ricci spaces, showing that in this case the standard L ¹ → L ^∞ estimate fails while suitable weighted Strichartz estimates hold.     

Schemes for Generating Different Nonlinear Schrödinger Integrable Equations and Their Some Properties

Acta Mathematicae Applicatae Sinica, English Series - In the paper, we want to derive a few of nonlinear Schrödinger equations with various formats and investigate their properties, such as...     

Scattering Theory for Radial Nonlinear Schrödinger Equations on Hyperbolic Space

We study the long-time behavior of radial solutions to nonlinear Schr?dinger equations on hyperbolic space. We show that the usual distinction between short-range and long-range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz-type inequalities. The latter are established without symmetry assumptions. Received: July 2006, Revision: April 2007, Accepted: April 2007     

Some results on a class of nonliner Schrödinger equations

By using a Liapunov-Schmidt reduction we prove an existence result for the nonlinear Schr?dinger equation in where satisfies suitable assumptions. We also provide a necessary condition for the existence of solutions. Received June 7, 1999 / in final form November 10, 1999 / Published online July 20, 2000     

Maximal Estimates of Schrödinger Equations on Rearrangement Invariant Sobolev Spaces

We establish the maximal estimates for the solutions of some initial value problems on rearrangement-invariant quasi-Banach function spaces. Our result covers the cases for which the initial value problem is given by the Schrödinger equation.