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

We consider the initial value problem for the nonlinear Schrödinger equation satisfying the strong dissipative condition Iλ < 0 and |Iλ| > ((p − 1)/2√p)|Rλ| in one space dimension. 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 Schrödinger equation for large initial data. We prove global existence of solutions and present some time decay estimates of solutions for large initial data.


Introduction and Main Results
We consider the Cauchy problem of nonlinear Schrödinger equation: where V = V(, ) is a complex valued unknown function,  ≥ 0,  ∈ R,  > 1, the gain coefficient () is a real valued function, and  ∈ C. Equation ( 1) is applied in problems of dispersion-managed optical fibers and soliton lasers (see [1]).
The coefficients  and () are, respectively, nonlinearity and amplification.In this work, we study the global existence and investigate time decay estimates of solutions to (1) with the gain coefficient () and the strong dissipative nonlinearity |V| −1 V satisfying I < 0 and |I| > (( − 1)/2√)|R| for large initial data, where I and R are the imaginary and real part of , respectively.Over the past few decades, the field of fiber optics has made rapid progress.The damped Schrödinger equation where V = V(, ) is a complex valued unknown function,  ∈ R,  ∈ R, , and  ∈ C, is one of the simplest nonlinear Schrödinger equations for studying cubic nonlinear effects in optical fibers (see [1]).Equation ( 2) is applied in several different aspects of optics (see, e.g., [2]).It has been studied extensively in the context of solitons (see [1]).In the case of () ≡ 0, (1) is reduced to where  ≥ 0,  ∈ R,  > 1, and  ∈ C. The nonlinearity |V| −1 V with I < 0 and |I| > (( − 1)/2√)|R| is called strong dissipative.In [3,4], the large initial problem for (3) with the strong dissipative nonlinearities was investigated.
The question (5) has been studied by some mathematicians from the mathematical point (see, e.g., [10,11]).Let If  1 (0) ≤ 0, nonexistence of global solutions to (5) was studied under some assumptions in [10].It showed that the damping term in (5) cannot prevent blowing up of solutions.In [11], some blow up and global existence of solutions to (5) was investigated.The authors showed that the size of the damping coefficient  affected the solutions.As far as we know, there are not any results about the time decay estimates of solutions to (1) for large initial data.Our question is how the term ()V and the nonlinearity |V| −1 V with strong dissipative condition affect solutions to (1) For ,  ∈ R, weighted Sobolev space  , is defined by We write   (R) =   for 1 ≤  ≤ ∞ and  ,0 =   for simplicity.
Let us introduce some notations.We define the dilation operator by and define  =  (/2) 2 for  ̸ = 0. Evolution operator () is written as where the Fourier transform of  is We also have where the inverse Fourier transform of  is We denote by the same letter  various positive constants.And we write V() for the spatial function V(, ⋅).
If  = 0, we have  * * () = (19 + √ 145)/12, which is a lower bound of  shown in [4].Theorems 6 and 7 say how the strong dissipative nonlinearity and gain coefficient of the nonlinear Schrödinger equation ( 27) affect decay estimates of solutions under different initial conditions.The rest of this paper is organized as follows.In Section 2, we give proofs of Theorems 1 and 4. Theorems 6 and 7 are proven in Section 3.
First, we consider the equation where () ≥ 0, () ∈ We note that and Calculating the right part of (39), we obtain under the strong dissipative condition.
We are now in a position to prove time decay estimates of solutions to (1).By the Sobolev inequality the factorization formula () =   F, and (45), we have for  > 1.Using the transform  =  − ∫  0 () V, we obtain for  > 1.By (46), we have for  > 1.
Multiplying both sides of (51) by F(−), we get where Therefore, we have where the remainder term () is given in (53).Substituting F(−) by , we have the following equation about : We have the following estimates of ().Since the proof of these estimates is similar to that in [4,6], we omit the proof.

Proof of Theorem 7.
We have global existence of solutions V ∈  1,∞ to (27) by Theorem 4. We consider the decay estimates of solutions to (27) by using the method of [4,5] in the following steps.
By using a similar method to that in [4], we have the estimate of () as follows.Here we omit the proof.