scholarly journals The initial value problem, scattering and inverse scattering, for Schrödinger equations with a potential and a non-local nonlinearity

2006 ◽  
Vol 39 (37) ◽  
pp. 11461-11478 ◽  
Author(s):  
María de los Ángeles Sand Romero ◽  
Ricardo Weder
Keyword(s):  
Inverse Scattering ◽  
Initial Value Problem ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Non Local ◽  
Local Nonlinearity

Inverse scattering transforms for non-local reverse-space matrix non-linear Schrödinger equations

2021 ◽  
pp. 1-21
Author(s):  
WEN-XIU MA ◽  
YEHUI HUANG ◽  
FUDONG WANG
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Scattering Problems ◽  
Plemelj Formula ◽  
Non Linear ◽  
Non Local ◽  
Space Matrix ◽  
Scattering Transforms

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.

scholarly journals A SYSTEM OF COUPLED SCHRÖDINGER EQUATIONS WITH TIME-OSCILLATING NONLINEARITY

2012 ◽  
Vol 23 (11) ◽  
pp. 1250119 ◽  
Author(s):  
X. CARVAJAL ◽  
P. GAMBOA ◽  
M. PANTHEE
Keyword(s):  
Nonlinear Optics ◽  
Initial Data ◽  
Periodic Function ◽  
Initial Value Problem ◽  
Coupled System ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Periodic Functions ◽  
Initial Value ◽  
Physical Problems

This paper is concerned with the initial value problem (IVP) associated to the coupled system of supercritical nonlinear Schrödinger equations [Formula: see text] where θ1 and θ2 are periodic functions, which has applications in many physical problems, especially in nonlinear optics. We prove that, for given initial data φ, ψ ∈ H1(ℝn), as |ω| → ∞, the solution (uω, vω) of the above IVP converges to the solution (U, V) of the IVP associated to [Formula: see text] with the same initial data, where I(g) is the average of the periodic function g. Moreover, if the solution (U, V) is global and bounded, then we prove that the solution (uω, vω) is also global provided |ω| ≫ 1.

scholarly journals Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Nakao Hayashi ◽  
Chunhua Li ◽  
Pavel I. Naumkin
Keyword(s):  
Lower Bound ◽  
Positive Root ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Time Decay ◽  
Large Initial Data ◽  
Initial Value ◽  
Optical Fields ◽  
Gauge Invariant ◽  
Dissipative Condition

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearityλup-1uof orderpn<p≤1+2/nfor arbitrarily large initial data, where the lower boundpnis a positive root ofn+2p2-6p-n=0forn≥2andp1=1+2forn=1.Our purpose is to extend the previous results for higher space dimensions concerningL2-time decay and to improve the lower bound ofpunder the same dissipative condition onλ∈C:Im⁡ λ<0andIm⁡ λ>p-1/2pRe λas in the previous works.

Regularity properties of Schrödinger equations in vector-valued spaces and applications

2019 ◽  
Vol 31 (1) ◽  
pp. 149-166
Author(s):  
Veli Shakhmurov
Keyword(s):  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
The Cauchy Problem ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

Abstract In this paper, regularity properties and Strichartz type estimates for solutions of the Cauchy problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E, in which by choosing E and A, we can obtain numerous classes of initial value problems for Schrödinger equations, which occur in a wide variety of physical systems.

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