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     

Coherent States for Systems of L 2-Supercritical Nonlinear Schrödinger Equations

We consider the propagation of wave packets for a nonlinear Schrödinger equation, with a matrix-valued potential, in the semi-classical limit. For a matrix-valued potential, Strichartz estimates are available under long range assumptions. Under these assumptions, for an initial coherent state polarized along an eigenvector, we prove that the wave function remains in the same eigenspace, at leading order, in a scaling such that nonlinear effects cannot be neglected. We also prove a nonlinear superposition principle for these nonlinear wave packets.     

Nodal Type Bound States for Nonlinear Schrödinger Equations with Decaying Potentials

In this paper, we are concerned with the existence of nodal type bound state for the following stationary nonlinear Schrödinger equation $$-Δu(x)+V(x)u(x)=|u|^{p-1}u, x∈ R^N, N ≥ 3,$$ where 1 < p < (N+2)/(N-2) and the potential V(x) is a positive radial function and may decay to zero at infinity. Under appropriate assumptions on the decay rate of V(x), Souplet and Zhang [1] proved the above equation has a positive bound state. In this paper, we construct a nodal solution with precisely two nodal domains and prove that the above equation has a nodal type bound state under the same conditions on V(x) as in [1].     

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.     

Modified Wave Operator for a System of Nonlinear Schrödinger Equations in 2d

We consider a system of nonlinear Schrödinger equations with quadratic nonlinearities in two space dimensions. We prove the existence of modified wave operators or wave operators under some mass conditions.     

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.     

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.     

Existence Results for Nonlinear Schrödinger Equations With Electromagnetic Fields

 We consider the nonlinear Schr?dinger equation

Splitting Integrators for Nonlinear Schrödinger Equations Over Long Times

Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie–Trotter splitting in time for cubic Schrödinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schrödinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results are valid in arbitrary spatial dimension.     

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.     

A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

We propose a modification of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation (also other equations by natural generalization) formulated with nonzero boundary conditions at infinity. The purpose is to deal with arbitrary-order poles and potentially severe spectral singularities in a simple and unified way. As an application, we use the modified transform to place the Peregrine solution and related higher-order “rogue wave” solutions in an inverse-scattering context for the first time. This allows one to directly study properties of these solutions such as their dynamical or structural stability, or their asymptotic behavior in the limit of high order. The modified transform method also allows rogue waves to be generated on top of other structures by elementary Darboux transformations rather than the generalized Darboux transformations in the literature or other related limit processes. © 2019 Wiley Periodicals, Inc.     

Small Data Scattering for the Nonlinear Schrödinger Equation on Product Spaces

We consider the cubic nonlinear Schrödinger equation, posed on ? ⁿ  × M, where M is a compact Riemannian manifold and n ≥ 2. We prove that under a suitable smallness in Sobolev spaces condition on the data there exists a unique global solution which scatters to a free solution for large times.     

Inverse Scattering for the Nonlinear Schrödinger Equation with the Yukawa Potential

We study the inverse scattering problem for the three dimensional nonlinear Schrödinger equation with the Yukawa potential. The nonlinearity of the equation is nonlocal. We reconstruct the potential and the nonlinearity by the knowledge of the scattering states. Our result is applicable to reconstructing the nonlinearity of the semi-relativistic Hartree equation.     

Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger Equations

We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schrödinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of “irregularity” for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well–posedness theory of the equations. We develop two different approaches to the analysis of the effects of the modulation. A first approach is based on novel estimates for the regularizing effect of the modulated dispersion on the non-linear term using the theory of controlled paths. A second approach is an extension of a Strichartz estimated first obtained by Debussche and Tsutsumi in the case of the Brownian modulation for the quintic NLS.     

Sign-Changing Multi-Peak Solutions for Nonlinear Schrödinger Equations with Compactly Supported Potential

In this paper, we aim to prove the existence and concentration of sign-changing H ¹(? ^N ) solutions to the following nonlinear Schrödinger equations $$-\varepsilon^2\Delta u_{\varepsilon}+V(x)u_{\varepsilon}=K(x)|u_{\varepsilon}|^{p-1}u_{\varepsilon} $$ with N≥3, $1<p<\frac{N+2}{N-2}$ , and ε>0. When the potential function V(x) has compact support, $V(x)\not\equiv 0$ and V(x)≥0, K(x) is permitted to be unbounded under some necessary restrictions, we will show that one sign-changing H ¹(? ^N ) solution exists and exhibits concentration profile around local minimum points of the ground energy function $G(\xi)\equiv V^{\theta}(\xi) K^{-\frac{2}{p-1}}(\xi)$ with $\theta=\frac{p+1}{p-1}-\frac{N}{2}$ in the semiclassical limit ε→0.     

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.