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

Abstract

We consider the initial value problem for the nonlinear Schrödinger equation satisfying the strong dissipative condition and in one space dimension. Our purpose in this paper is to study how the gain coefficient and strong dissipative nonlinearity 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.

1. Introduction and Main Results

We consider the Cauchy problem of nonlinear Schrödinger equation:where is a complex valued unknown function, , , , the gain coefficient is a real valued function, and . 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 satisfying and for large initial data, where and 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 equationwhere is a complex valued unknown function, , , , and , 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 , (1) is reduced towhere , , , and . The nonlinearity with and is called strong dissipative. In [3, 4], the large initial problem for (3) with the strong dissipative nonlinearities was investigated. The global solution to (3) decays like in the sense of for , when . Moreover for , if in [3] and in [4], respectively. To study the time decays of solutions to (3), the estimate of is useful. To get a better estimate of , the contradiction argument was used in [3] and the method of [5] was applied in [4]. In [6], the time decay estimates of global solutions to (3) with in the sense of were considered under the strong dissipative condition. It showed that for , where . In the case of , the a priori - bound for is not satisfied, since There are some results about the lifespan of small solutions to (3) with (see, e.g., [7]). When optical pluses propagate inside a fiber, nonlinearities in (1) affect optical pluses’ shapes. Some related nonlinear Schrödinger equations have been studied (see, e.g., [8, 9]). In the case of and , (1) becomesin one space dimension, where , , and . The question (5) has been studied by some mathematicians from the mathematical point (see, e.g., [10, 11]). Let . If , 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 and the nonlinearity with strong dissipative condition affect solutions to (1) for large initial data.

Let denote the usual Lebesgue space with the norm if and For , weighted Sobolev space is defined byWe write for and for simplicity.

Let us introduce some notations. We define the dilation operator by and define for 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 for the spatial function .

The standard generator of Galilei transformations is given as We have

We also have commutation relations with and such thatwhere .

Before stating our main theorem, we introduce the function space where and . We have the following global existence of solutions to (1) for large initial data.

Theorem 1. Let , , , and . We assume that . Then (1) has a unique global solution satisfying , where is the solution to (19).

Let . Multiplying both sides of (1) by , integrating over , and taking the imaginary parts, we have We could not ensure the sign of by the assumptions in Theorem 1. This case is interesting. We have the equationfrom (1) by using the transformation . A straightforward calculation shows that solves (1) if and only if solves (19). Thus the transformation provides an effective tool to study the global existence of solutions to (1).

Remark 2. Let , , , , and for all , where . We assume that . Then (1) has a unique global solution .

Example 3. We considerin one space dimension, where and . Since , and for , (20) has a unique global solution if .

If for and , we have the result about (1): where . Time decay estimates of solutions to (1) for large initial data are shown as follows.

Theorem 4. Let , for , and the strong dissipative condition , hold. We assume that . Then (1) has a unique global solution satisfying the following time decay estimates: and for .

From Theorem 4, we obtain that the solution to (1) is global and bounded in for large initial data. Moreover, we show that determines the time decay rate of the solution, when satisfies the assumptions in Theorem 4. Then we consider a special situation of Theorem 4. Let in (1); we have the following equation:where , , , , and . By Theorem 4, we have the time decay estimates to (24).

Corollary 5. Let and . We assume that and the strong dissipative condition and hold. Then (24) has a unique global solution satisfying the following time decay estimates: and for .

In the last section, we consider the equationwhere , , , , and . From (27), we haveby taking the variable change . A similar nonlinear equation , where , , and , was derived to study the rapid decay solutions and scattering properties of the equation by letting in [12].

Let and . Multiplying both sides of (27) by , integrating over , and taking the imaginary parts, we havewhere . We have global existence and time decay estimates of solutions to (27) for large initial data by Theorem 4. We show a better decay rate of solutions to (27) inspired by the papers [4, 5].

Theorem 6. Let and . We assume that and the strong dissipative condition and hold. Then (27) has a unique global solution satisfying the following time decay estimates:for . Moreover, let , and if and , we havefor .

When , the exponent coincides with , which is the lower bound given by Kita-Shimomura [3] and Jin-Jin-Li [4]. Let Since for , the lower bound in Theorem 6 can be improved by in Theorem 7. The operator plays an important role in achieving the lower bound .

Theorem 7. Let , , and . We assume that and the strong dissipative condition and hold. Then (27) has a unique global solution satisfying the following time decay estimates:for .

If , we have , 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.

2. Proofs of Theorems 1 and 4

2.1. Proof of Theorem 1

We have the equation from (1) by using the variable change and .

Local existence of solutions to (33) can be shown by the standard contraction mapping principle (see, e.g., [13]). Therefore, we have local existence of solutions to (1).

First, we consider the equationwhere , , , and . The following lemma is useful to study global existence and time decay of solutions to (34).

Lemma 8. Let , , , and be the local solution of (34), where . And let the strong dissipative condition and be satisfied. Then we have

Proof. Multiplying both sides of (34) by and taking the imaginary parts, we have, by and the assumptions of ,By (36) and the strong dissipative condition and , we obtainWe note that andCalculating the right part of (39), we obtainunder the strong dissipative condition.
Multiplying both sides of (34) by , we obtain by the assumptions of and . Using (37), (40), and (41), we have

By Lemma 8, we have

From (43), we obtain a unique global solution to (33). Since , we have a unique global solution to (1).

2.2. Proof of Theorem 4

Using the transform , we get the equation about from (1):where . We have local existence of solutions to (44) and to (1), respectively (see, e.g., [13]).

By Lemma 8 and the assumptions in Theorem 4, we haveAndThus, we obtain global-in-time existence of solutions to (44) by (45) and to (1) by (46), respectively.

We are now in a position to prove time decay estimates of solutions to (1). By the Sobolev inequality the factorization formula , and (45), we have for . Using the transform , we obtain for . By (46), we have for .

3. Proofs of Theorems 6 and 7

3.1. Proof of Theorem 6

From Theorem 4, we have global existence of solutions to (27) and the time decay estimates (30). To get a better decay estimates of solutions to (27), we use the method of [4, 5]. Changing a variable such as , we haveand .

Multiplying both sides of (51) by , we get whereTherefore, we have where the remainder term is given in (53). Substituting 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.

Lemma 9. Let be a solution of (51) in the function space Then, we haveandfor .

From (55), we obtainHence, we get for . By Lemmas 8 and 9, we obtain for .

Multiplying both sides of the above by , where , we have

By the Young inequality, we have for . Let for and . Integrating in time from 1 to , we getfor , , , and . To guarantee thatwe need Let Then, from (63), we getfor , , and .

Since and , we have . By (66), we have for . Using the transform , we have our desired result.

3.2. Proof of Theorem 7

We have global existence of solutions 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.

Since holds, from (51), we haveBy the strong dissipative condition and and using for , we havefor . Thus, we obtainfor

By using a similar method to that in [4], we have the estimate of as follows. Here we omit the proof.

Lemma 10. Let be a solution of (51) in the function space Then, we have for .

(58) showswhere . Let . By Lemma 10 and (58), we get for . Multiplying both sides of the above by , where , we have By the Young inequality, we have for . Let for and . Integrating in time from 1 to , we getfor , , , and . To obtainwe need Let Then, from (76), we getfor , , and .

When and , we have . By (79), we have for . By using the relation , we have our desired result.

Data Availability

All data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank Professor Hideaki Sunagawa for some helpful comments. The work is partially supported by the Education Department of Jilin Province (JJKH20180892KJ) and NNSFC (Grant no. 11461074).