Related papers: Dimension reduction for Nonlinear Schr\"odinger eq…
In this paper, we take ODE reductions of the general nonlinear Schr\"odinger equation (NLS) AKNS system, and reduce them to Painlev\'e type equations. Specifically, the stationary solution is solved in terms of elliptic functions, and the…
The modulational stability of the nonlinear Schr{\"o}dinger (NLS) equation is examined in the cases with linear and quadratic external potential. This study is motivated by recent experimental studies in the context of matter waves in…
We demonstrate the systematic derivation of a class of discretizations of nonlinear Schr{\"o}dinger (NLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic condition. We…
We consider multiple solutions to the nonlinear Schr\"odinger equation (NLS) with a partial confinement, which is physically relevant to dynamics of the Bose-Einstein condensate. Our study not only verifies the existence of positive ground…
In this paper, we introduce the reverse-space and reverse-space-time nonlocal discrete derivative nonlinear Schr\"odinger (DNLS) equations through the nonlocal symmetry reductions of the semi-discrete Gerdjikov-Ivanov equation. The…
We obtain a self-dual formulation of the conventional nonlinear Schr\"odinger equation (NLSE) in the 1+1 dimension by studying the dimensional reduction of the self-dual Chern-Simons nonlinear Schr\"odinger model (NLSM) in the 2+1…
We consider a nonlinear Klein-Gordon equation with a quasilinear quadratic term. The Nonlinear Schr\"odinger (NLS) equation can be derived as a formal approximation equation describing the evolution of the envelopes of slowly modulated…
For nonlinear dispersive systems, the nonlinear Schr\"odinger (NLS) equation can usually be derived as a formal approximation equation describing slow spatial and temporal modulations of the envelope of a spatially and temporally…
We consider the cubic nonlinear Schr\"odinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness…
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a…
We present an open-source Python implementation of an idealized high-order pseudo-spectral solver for the one-dimensional nonlinear Schr\"odinger equation (NLSE). The solver combines Fourier spectral spatial discretization with an adaptive…
The aim of this paper is to develop the inverse scattering transform (IST) for multi-component generalisations of nonlocal reductions of the nonlinear Schrodinger (NLS) equation with PT-symmetry related to symmetric spaces. This includes:…
In this paper, developing a new approach based on Fourier analysis methods for dispersive PDEs, we establish a low regularity NLS approximation for the one-dimensional cubic Klein-Gordon equation. Our main result includes energy class…
The nonlinear Schr\"{o}dinger (NLS) equation can be derived as a formal approximation equation describing the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation.…
This paper is devoted to a comprehensive study of the nonlinear Schr\"odinger equations with combined nonlinearities of the power-type and Hartree-type in any dimension n\ge3. With some structural conditions, a nearly whole picture of the…
In dimension two, we investigate a free energy and the ground state energy of the Schr\"odinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling…
In this paper we develop a bilinearisation-reduction approach to derive solutions to the classical and nonlocal nonlinear Schr\"{o}dinger (NLS) equations with nonzero backgrounds. We start from the second order Ablowitz-Kaup-Newell-Segur…
As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type equations are considered. Through the bilinearization reduction method, we construct double…
We consider the 3D cubic nonlinear Schr\"odinger equation (NLS) with a strong toroidal trap. In the first part, we show that as the confinement is strengthened, a large class of global solutions to the time-dependent model can be described…