Related papers: Higher order asymptotics for the nonlinear Schr\"o…
The long-time asymptotic behavior of the focusing nonlinear Schr\"odinger (NLS) equation on the line with symmetric nonzero boundary conditions at infinity is characterized by using the recently developed inverse scattering transform (IST)…
We consider large time asymptotics for damped nonlinear Schr\"{o}dinger equations. It is known that the nonlinear solution asymptotically behaves like a linear solution when time $t$ tends to infinity in the energy space. We prove that its…
We consider the long time dynamics of nonlinear Schr\"odinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate,…
In this paper, under the exponential/polynomial decay condition in Fourier space, we prove that the nonlinear solution to the quasi-periodic Cauchy problem for the weakly nonlinear Schr\"odinger equation in higher dimensions will…
In this paper, we consider the final state problem for the nonlinear Schr\"odinger equation with a homogeneous nonlinearity which is of the long range critical order and is not necessarily a polynomial, in one and two space dimensions. As…
A classification of large-time and finite-time blow-up asymptotics of solutions of the Cauchy problem for higher-order Schr\"odinger equations is performed.
We study the derivative nonlinear Schr\"odinger equation on the real line and obtain global-in-time bounds on high order Sobolev norms.
We consider the initial value problem for a three-component system of quadratic nonlinear Schr\"odinger equations with mass resonance in two space dimensions. Under a suitable condition on the coefficients of the nonlinearity, we will show…
We study the asymptotic behavior in time of solutions to the one dimensional nonlinear Schr\"odinger equation with a subcritical dissipative nonlinearity $\lambda |u|^\alpha u$, where $0<\alpha<2$, and $\lambda $ is a complex constant…
Nonlinear Schr\"odinger equation (with the Schwarzian initial data) is important in nonlinear optics, Bose condensation and in the theory of strongly correlated electrons. The asymptotic solutions in the region $x/t={\cal O}(1)$,…
Multi-soliton and high-order soliton solutions are two type of famous ones in the integrable focusing nonlinear Schr\"odinger equation. The dynamics of multi-soliton was well known to us since 70s of the last century by the determinant…
The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift…
We consider the Cauchy problem for the Gross-Pitaevskii (GP) equation. Using the DBAR generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of the GP in the…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr\"odinger equation $ iq_{t}(x,t)+q_{xx}(x,t)-2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with a step-like…
We implement the Numerical Unified Transform Method to solve the Nonlinear Schr\"odinger equation on the half-line. For so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the…
We consider solutions of the defocusing nonlinear Schr\"odinger equation in the quarter plane whose Dirichlet boundary data approach a single exponential $\alpha e^{i\omega t}$ as $t \to \infty$. In order to determine the long time…
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…
We begin a study of a multi-parameter family of Cauchy initial-value problems for the modified nonlinear Schr\"odinger equation, analyzing the solution in the semiclassical limit. We use the inverse scattering transform for this equation,…
In this short note, we prove a decay estimate for non-linear solutions of 3D cubic defocusing non-linear Schr\"odinger equation.