Related papers: Higher order asymptotics for the nonlinear Schr\"o…
We apply the proper orthogonal decomposition (POD) to the nonlinear Schr\"odinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving…
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…
The methodology of the Riemann-Hilbert (RH) factorisation approach for Lax-pair isospectral deformations is used to derive, in the solitonless sector, the leading-order asymptotics as $t \to \pm \infty$ $(x/t \sim \mathcal{O}(1))$ of…
We prove a dispersive estimate for periodic discrete Schr\"odinger operators on the line with optimal rate of decay. Additionally, by standard methods, we deduce dispersive estimates for the discrete nonlinear Schr\"odinger equation with…
The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{\"o}dinger equation posed on $\mathbb{R}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates…
We present a new parallel numerical method for solving the non-stationary Schr\"odinger equation with linear nonlocal condition and time-dependent potential which does not commute with the stationary part of the Hamiltonian. The given…
We consider a class of nonlinear Schroedinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We show new local $L^p$-smoothing estimates for the Schr\"odinger equation with initial data in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of…
In this paper, we consider the hyperbolic nonlinear Schr\"odinger equations (HNLS) on $\mathbb{R}\times\mathbb{T}$. We obtain the sharp local well-posedness up to the critical regularity for cubic nonlinearity and in critical spaces for…
We propose a class of numerical methods for the nonlinear Schr\"odinger (NLS) equation that conserves mass and energy, is of arbitrarily high-order accuracy in space and time, and requires only the solution of a scalar algebraic equation…
We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr\"odinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function''…
We consider the Gerdjikov--Ivanov type derivative nonlinear Schr\"odinger equation \berr \ii q_{t}+q_{xx}-\ii q^2\bar{q}_{x}+\frac{1}{2}(|q|^4-q_0^4)q=0 \eerr on the line. The initial value $q(x,0)$ is given and satisfies the symmetric,…
The purpose of this article is to give a streamlined and self-contained treatment of the long-time asymptotics of the Toda lattice for decaying initial data in the soliton and in the similarity region via the method of nonlinear steepest…
We give explicit time lower bounds in the Lebesgue spaces for all nontrivial solutions of nonlinear Schr\"odinger equations bounded in the energy space. The result applies for these equations set in any domain of $\R^N,$ including the whole…
We study the large order and infinite order soliton of the coupled nonlinear Schrodinger equation with the Riemann-Hilbert method. By using the Riemann-Hilbert representation for the high order Darboux dressing matrix, the large order and…
We present a new generalization of the steepest descent method introduced by Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the semiclassical limit of the focusing nonlinear Schroedinger equation with real analytic,…
The mapping of the Nonlinear Schroedinger Equation with a random potential on the Fokker-Planck equation is used to calculate the localization length of its stationary states. The asymptotic growth rates of the moments of the wave function…
This is a sequel to the paper "Large time asymptotics for a cubic nonlinear Schr\"odinger system in one space dimension" by the same authors. We continue to study the Cauchy problem for the two-component system of cubic nonlinear…
In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same…
In this paper we consider the stabilization of non-fundamental unstable stationary solutions of the cubic nonlinear Schrodinger equation. Specifically we study the stabilization of radially symmetric solutions with nodes and asymmetric…