scholarly journals Stability of multi-solitons in the cubic NLS equation

2014 ◽  
Vol 11 (02) ◽  
pp. 329-353 ◽  
Author(s):  
Andres Contreras ◽  
Dmitry Pelinovsky
Keyword(s):  
Initial Data ◽  
Inverse Scattering ◽  
Orbital Stability ◽  
Inverse Scattering Transform ◽  
Nls Equation ◽  
Transform Methods ◽  
Nonlinear Schrödinger ◽  
Scattering Transform ◽  
The Stability

We address the stability of multi-solitons for the cubic nonlinear Schrödinger (NLS) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we establish the orbital stability of multi-solitons in the L2(ℝ) space when the initial data is in a weighted L2(ℝ) space.

scholarly journals Prolongation Structures and N-Soliton Solutions for a New Nonlinear Schrödinger-Type Equation via Riemann-Hilbert Approach

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Yuxin Lin ◽  
Yong Fang ◽  
Huanhe Dong
Keyword(s):  
Spectral Problem ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Type Equation ◽  
Lax Pair ◽  
Inverse Scattering Transform ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Scattering Transform ◽  
Matrix Spectral Problem

In this paper, a new integrable nonlinear Schrödinger-type (NLST) equation is investigated by prolongation structures theory and Riemann-Hilbert (R-H) approach. Via prolongation structures theory, the Lax pair of the NLST equation, a 2×2 matrix spectral problem, is derived. Depending on the analysis of red the spectral problem, a R-H problem of the NLST equation is formulated. Furthermore, through a specific R-H problem with the vanishing scattering coefficient, N-soliton solutions of the NLST equation are expressed explicitly. Moreover, a few key differences are presented, which exist in the implementation of the inverse scattering transform for NLST equation and cubic nonlinear Schrödinger (NLS) equation. Finally, the dynamic behaviors of soliton solutions are shown by selecting appropriate spectral parameter λ, respectively.

Unique continuation properties of the nonlinear Schrödinger equation

1997 ◽  
Vol 127 (1) ◽  
pp. 191-205 ◽  
Author(s):  
Bing-Yu Zhang
Keyword(s):  
Open Subset ◽  
Function Theory ◽  
Unique Continuation ◽  
Inverse Scattering Transform ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Scattering Transform ◽  
Hardy Function ◽  
Continuation Problem ◽  
Image Position

Consider the unique continuation problem for the nonlinear Schrödinger (NLS) equationBy using the inverse scattering transform and some results from the Hardy function theory, we prove that if u ∈ C(R; H1(R)) is a solution of the NLS equation, then it cannot have compact support at two different moments unless it vanishes identically. In addition, it is shown under certain conditions that if u is a solution of the NLS equation, then u vanishes identically if it vanishes on two horizontal half lines in the x–t space. This implies that the solution u must vanish everywhere if it vanishes in an open subset in the x–t space.

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