Time Decay for Nonlinear Dissipative Schrödinger Equations in Optical Fields

Abstract

We consider the initial value problem for the nonlinear dissipative Schrödinger equations with a gauge invariant nonlinearity of order for arbitrarily large initial data, where the lower bound is a positive root of for and for Our purpose is to extend the previous results for higher space dimensions concerning -time decay and to improve the lower bound of under the same dissipative condition on : and as in the previous works.

1. Introduction and Main Results

We consider the initial value problem for the following nonlinear Schrödinger equations: where and From the physical point of view, (1) is applied to investigate the light traveling in optical fibers (see, e.g., [1, 2]). Some new nonlinear works concerning the related equations appear in [3, 4]. We note that implies the dissipation of by nonlinear Ohm’s law (see, e.g., [5]). Therefore, the dissipative nonlinearity causes the transportation of the energy. The condition on the coefficient appeared in the previous papers [1, 2, 6, 7] and so forth. It yields a dissipative property of solutions in one space dimension for arbitrarily large initial data, which was shown in [2]. In this paper, we prove -time decay estimate of solutions in any space dimension under the condition of critical or subcritical power order nonlinearity such that where In [2], -time decay of solutions was studied in one space dimension under the condition on such that . More precisely, in [2] it was shown that the estimates of solutions to (1), hold for any in the case of . For small solutions, in [1] under the conditions such that and is close to , the same -time decay as stated above was obtained for . Below we show that if we restrict our attention to the -time decay of solutions to (1) for arbitrarily large initial data, then we can reduce the low bound of the power and consider the problem in higher space dimensions. To investigate the effect of the dissipative nonlinearity, we consider the estimates about the norm of solutions to (1). In [2] the -time decay of the solution to (1) satisfying that was obtained, when and . We develop our problem to the space dimension . Moreover the scope of is extended, when we focus on one space dimension case. Another point in this paper is to prove results directly comparing to the contradiction argument in [2]. More precisely, our strategy of the proof is as follows. We first transform our target equation (1) into the following nonlinear ordinary equation with remainder terms : where (see (48) and (69)). Next by using the positive solution of the separate equation with , we translate it into the separate equation such that by using the condition and the Young inequality (see (56)). The desired estimate of is obtained by integration in time through the estimate of .

By we denote the usual Lebesgue space with the norm if andFor , weighted Sobolev space is defined by where We write and for simplicity.

Let us introduce some notations. We define the dilation operator by and define for Free evolution group is written as where the Fourier transform of is We also have where the inverse Fourier transform These formulas were used in [8] first to study the asymptotic behavior of solutions to nonlinear Schrödinger equations. We denote by the same letter various positive constants.

The standard generator of Galilei transformation is given by We have commutation relations with and such that To prove Theorem 1, we introduce the function space where

In Theorems 1 and 2, we consider the subcritical and critical cases, respectively. Define

Theorem 1. Assume that , and , if , and , if Then (1) has a unique global solution Moreover if we assume that then we have for , where and the identity holds.

The lower bound in Theorem 1 is approximated as Since holds, where was introduced in [2] to study the -estimates of the solution for (1), the lower bound of is improved.

We next consider the critical case .

Theorem 2. Let be the solution of (1) with stated in Theorem 1. Then we have for , where and the identity (24) holds.

2. Proof of Theorem 1

Under the assumptions we have a dissipative property of solutions. Indeed we have by the usual energy method Nonlinear terms are represented as if and which is equivalent to Therefore we have for , where In the same way as in the proof of (29), we have from which it follows that for . We also have Therefore by (32), (35), and (36), we have an a priori estimate of solutions which implies the global in time existence of solutions to (1) in the function space for and This completes the first part of the proof of the theorem.

Our concern is to estimate the time decay rate in the subcritical case since it is expected that the decay rate of solutions is different from that of solutions to linear problem. Denote , and Using the factorization formula , we multiply both sides of (1) by to get where

For the remainder term , we have by (37) the following.

Lemma 3. Let be a solution of (1) in the function space Then the estimate for is true, where

Proof. By Hölder’s and Sobolev’s inequalities, we get where for for , and since we can apply the Sobolev embedding theorem with In the same manner with the same as above, we obtain In view of (37) the result of the lemma follows. This completes the proof of the lemma.

We continue to prove Theorem 1. We let Then we have the ordinary differential equation Multiplying both sides of (46) by and taking the imaginary part of the resulting equation, we obtain from which it follows that Let us consider the case Namely, we consider the separate equation It is well known that a solution of the separate equation (49) containing an arbitrary constant is obtained by integration. Then we have the solution where Therefore we have the estimate for Hence for the solution of (49) we have the estimate if It suggests to us that the solution of (47) has the same asymptotic profile, when is considered as the remainder term. We now change (48) into a separate form by using the solution of (47). Multiplying both sides of (48) by we obtain By the Young inequality with , we get We apply this estimate to the first term of the right-hand side of (54) to have This is the separable form of (48). Integrating in time and using (50), we arrive at where , which implies Hence by we have for Taking norm and applying Lemma 3, we get for We haveif satisfiesTherefore we find that This is the nonlinear version of estimate (53). We now prove the time decay of solutions. We have the formula where . By a direct computation Hence by (64) and the estimate which follows from (35), we have since , and we find for Therefore, identity (24) follows from (68) and (36).

3. Proof of Theorem 2

We start with Note that the solution of the separable equation is represented as Multiplying both sides of (69) by , we obtain Applying the Young inequality to the first term of the left-hand side of the above inequality, we obtain Hence we obtain the separate form of (69) such that Integrating in time, we obtain from which it follows that Since we havefor any We obtain by Lemma 3for . The rest of the proof is the same as that of Theorem 1. This completes the proof of Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work of Nakao Hayashi is partially supported by JSPS KAKENHI Grant Nos. 25220702 and 15H03630. The work of Chunhua Li is partially supported by the Education Department of Jilin Province ( No. 34) and NNSFC under Grant No. 11461074. The work of Pavel I. Naumkin is partially supported by CONACYT and PAPIIT Project IN100113.