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.

Regularity properties of nonlinear abstract Schrödinger equations and applications

2020 ◽  
Vol 31 (13) ◽  
pp. 2050105
Author(s):  
Veli Shakhmurov
Keyword(s):  
Banach Space ◽  
Initial Value Problems ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Initial Value ◽  
Physical Systems ◽  
Regularity Properties ◽  
Linear And Nonlinear ◽  
Vector Valued Function ◽  
Vector Valued

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator [Formula: see text] defined in a Banach space [Formula: see text], in which by choosing [Formula: see text] and [Formula: see text] we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems.

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.

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