The Effect of Gain and Strong Dissipative Structures on Nonlinear Schrödinger Equations in Optical Fiber

Xintong Yuan,Chunhua Li

doi:10.1155/2019/7297090

Abstract

We consider the initial value problem for the nonlinear Schrödinger equation satisfying the strong dissipative condition Iλ<0 and Iλ>p-1/2pRλ in one space dimension. Our purpose in this paper is to study how the gain coefficient μ(t) and strong dissipative nonlinearity λvp-1v affect solutions to the nonlinear Schrödinger equation for large initial data. We prove global existence of solutions and present some time decay estimates of solutions for large initial data.

Highlights

Introduction and Main ResultsWe consider the Cauchy problem of nonlinear Schrodinger equation: i∂tV + ∂x2V = λ |V|p−1 V iμ (t) V, (1)V (0, x) = V0 (x), where V = V(t, x) is a complex valued unknown function, t ≥ 0, x ∈ R, p > 1, the gain coefficient μ(t) is a real valued function, and λ ∈ C
Our purpose in this paper is to study how the gain coefficient μ(t) and strong dissipative nonlinearity λ|V|p−1V affect solutions to the nonlinear Schrodinger equation for large initial data
We prove global existence of solutions and present some time decay estimates of solutions for large initial data

Summary

Introduction and Main Results

We study the global existence and investigate time decay estimates of solutions to (1) with the gain coefficient μ(t) and the strong dissipative nonlinearity λ|V|p−1V satisfying Iλ < 0 and |Iλ| > ((p − 1)/2√p)|Rλ| for large initial data, where Iλ and Rλ are the imaginary and real part of λ, respectively. = λ λ(1 + ∈ R, bt)(nα−4)/2|u|αu, was derived to study the rapid decay solutions and scattering properties of the equation i∂tV + ΔV = λ|V|αV by letting u(t, x) = (1 + bt)−n/2V(t/(1 + bt), x/(1 + bt))ei(b|x|2/4(1+bt)) in [12]. We have global existence and time decay estimates of solutions to (27) for large initial data by Theorem 4. Theorems 6 and 7 say how the strong dissipative nonlinearity and gain coefficient of the nonlinear Schrodinger equation (27) affect decay estimates of solutions under different initial conditions.

Proofs of Theorems 1 and 4

Proofs of Theorems 6 and 7