Related papers: On the continuous resonant equation for NLS: I. De…
We consider the dynamics of nonlinear Schr\"odinger equations with strong constant magnetic fields. In an asymptotic scaling limit the system exhibits a purely magnetic confinement, based on the spectral properties of the Landau…
We consider the Cauchy problem for the focusing nonlinear Schr\"odinger equation with initial data approaching two different plane waves $A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as $x\to\pm\infty$. Using…
We consider the derivation of the defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$-body dynamics. We reformat the hierarchy approach with Klainerman-Machedon theory and prove a bi-scattering…
Nonequilibrium reaction networks (NRNs) underlie most biological functions. Despite their diverse dynamic properties, NRNs share the signature characteristics of persistent probability fluxes and continuous energy dissipation even in the…
In this paper, we determine the spectral instability of periodic odd waves for the defocusing fractional cubic nonlinear Schr\"odinger equation. Our approach is based on periodic perturbations that have the same period as the standing wave…
We consider the asymptotic behavior of the solutions of a nonlinear Schr\"odinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for…
In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schr\"odinger equations to higher-order Schr\"odinger equations with a more general nonlinearity. More precisely, we consider a…
A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to classical…
We study the one-dimensional nonlinear Schr\"odinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a…
We consider the long-time dynamics of focusing energy-critical Schr\"odinger equation perturbed by the $\dot{H}^\frac{1}{2}$-critical nonlinearity and with inverse-square potential(CNLS$_a$) in dimensions $d\in\{3,4,5\}$…
We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…
We establish the full asymptotic stability of solitary wave solutions for the 1D focusing cubic Schr\"odinger equation on the line under small perturbations in weighted Sobolev spaces, building upon our results in [58]. The proof integrates…
In this paper we study the following class of fractional relativistic Schr\"odinger equations: \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in…
We investigate the global well-posedness and modified scattering for the one-dimensional Schr\"odinger equation with gauge-invariant polynomial nonlinearity. For small localized initial data of finite energy in a low-regularity class, we…
We introduce a class of symplectic resonance based schemes for Schr\"odinger's equation in dimension one, building on the work in [1] wherein resonance based numerical schemes were developed in the context of dispersive PDE driven by time…
We consider the cubic and quintic nonlinear Schr\"{o}dinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for…
We suggest a new derivation of a kinetic equation of Kolmogorov-Zakharov (KZ) type for the spectrum of the weakly nonlinear Schr\"odinger equation with stochastic forcing. The kynetic equation is obtained as a result of a double limiting…
This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schr\"odinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency…
In this article we study the one-dimensional, asymptotically linear, non-linear Schr\"odinger equation (NLS). We show the existence of a global smooth curve of standing waves for this problem, and we prove that these standing waves are…
In this paper, we investigate the asymptotic behavior of small solutions to the initial value problem for a system of cubic nonlinear Schrodinger equations (NLS) in one spatial dimension. We identify a new class of NLS systems for which the…