Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation

We consider the initial value problem for the fractional nonlinear Schrödinger equation with a fractional dissipation. Global existence and scattering are proved depending on the order of the fractional dissipation.     

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H^2,2(?) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H^2,2(?) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.     

Global existence of small solutions to quadratic nonlinear schrödinger equations

Abstract. We study the global existence of small solutions to some quadratic nonlinear Schrödinger equations in three or four space dimensions.     

Exponentially small asymptotics of solutions to the defocusing nonlinear Schrödinger equation

The matrix Riemann-Hilbert factorization approach is used to derive the leading-order, exponentially small asymptotics as t → ± ∞ such that x/t ∼ O(1) of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation, i∂_tu + ∂_x²u − 2(|u|² − 1)u = 0, with finite density initial data u(x,0) = _x→±∞exp(i(1 ∓ 1)φ/2)(1+o(1)), φ ϵ [0, 2π).     

Global solutions of nonlinear Schrödinger equations

We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.     

The elliptic-in-t solutions of the nonlinear Schrödinger equation

Four various anzatzes of the Krichever curves for the elliptic-in-t solutions of the nonlinear Schrödinger equation are considered. An example is given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 188–200, May, 1996.Translated by V. I. Serdobol'skii.     

Asymptotics and existence of bounded solutions of a nonlinear Schrödinger equation on the half-line

We study the asymptotics and existence of nonzero bounded solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear second-order ordinary differential equation. We prove the existence of countably many nonzero bounded solutions on the half-line and derive asymptotic formulas at infinity for these solutions.     

Classification of solutions of the generalized mixed nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We construct a generalized Darboux transformation for a generalized mixed nonlinear Schrödinger equation and consider a complete reduction classification...     

Global existence for a system of weakly coupled nonlinear Schrödinger equations

This paper discusses the weakly coupled nonlinear Schrödinger equations in the supercritical case. With the best constant of the Gagliardo-Nirenberg inequality, we derive a sufficient condition for the global existence of solutions; this condition is expressed in terms of stationary solutions (nonlinear ground state).     

Toward a classification of quasirational solutions of the nonlinear Schrödinger equation

Based on a representation in terms of determinants of the order 2N, we attempt to classify quasirational solutions of the one-dimensional focusing nonlinear Schrödinger equation and also formulate several conjectures about the structure of the solutions. These solutions can be written as a product of a t-dependent exponential times a quotient of two N(N+1)th degree polynomials in x and t depending on 2N?2 parameters. It is remarkable that if all parameters are equal to zero in this representation, then we recover the P_N breathers.     

Riemann-Hilbert approach for multisoliton solutions of generalized coupled fourth-order nonlinear Schrödinger equations

The main purpose of this work is to develop Riemann-Hilbert approach to obtain the soliton solutions for generalized coupled fourth-order nonlinear Schrödinger equations, which describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber. Starting from the spectral analysis of the Lax pair, a Riemann-Hilbert problem is set up. After solving the obtained Riemann-Hilbert problem with reflectionless case, we systematically derive multisoliton solutions for the generalized coupled fourth-order nonlinear Schrödinger equations. In addition, the localized structures and dynamic behaviors of one- and two-soliton solutions are shown by some graphic analysis.