International Journal of Differential Equations
Volume 2018 (2018), Article ID 8236942, 13 pages
https://doi.org/10.1155/2018/8236942

Research Article

High-Speed Transmission in Long-Haul Electrical Systems

Beatriz Juárez-Campos,¹ Elena I. Kaikina,² Pavel I. Naumkin,² and Héctor Francisco Ruiz-Paredes¹

¹Instituto Tecnológico de Morelia, Avenida Tecnológico No. 1500, Lomas de Santiaguito, 58120 Morelia, MICH, Mexico
²Centro de Ciencias Matemáticas, UNAM Campus Morelia, AP 61-3 Xangari, 58089 Morelia, MICH, Mexico

Correspondence should be addressed to Elena I. Kaikina; ekaikina@matmor.unam.mx

Received 29 January 2018; Accepted 12 March 2018; Published 18 April 2018

Academic Editor: Sining Zheng

Copyright © 2018 Beatriz Juárez-Campos et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the equations governing the high-speed transmission in long-haul electrical systems , , , where , and is the Fourier transformation. Our purpose in this paper is to obtain the large time asymptotics for the solutions under the nonzero mass condition

1. Introduction

We study the equations governing the high-speed transmission in long-haul electrical systems

where , , and is the Fourier transformation defined by Note that we have the relation , so we can only consider the case . For the regular solution of (1) we have the conservation law We are interested in the case of nonzero mass condition By (1) we get the conservation of the mass for all

This equation arises in the context of high-speed soliton transmission in long-haul optical communication system [1]. Also it can be considered as a particular form of the higher order nonlinear Schrödinger equation introduced by [2] to describe the nonlinear propagation of pulses through optical fibers. This equation also represents the propagation of pulses by taking higher dispersion effects into account than those given by the Schrödinger equation (see [3–11]).

The higher order nonlinear Schrödinger equations have been widely studied recently. For the local and global well-posedness of the Cauchy problem we refer to [12–14] and references cited therein. The dispersive blow-up was obtained in [15]. The existence and uniqueness of solutions to (1) were proved in [16–25] and the smoothing properties of solutions were studied in [18–21, 24, 26–31]. The blow-up effect for a special class of slowly decaying solutions of Cauchy problem (1) was found in [32].

As far as we know the question of the large time asymptotics for solutions to Cauchy problem (1) is an open problem. We develop here the factorization technique originated in our previous papers [33–38].

We denote the Lebesgue space by , where the norm for and . The weighted Sobolev space is , where , , , and We also use the notations , shortly, if it does not cause any confusion. Let be the space of continuous functions from an interval to a Banach space Different positive constants might be denoted by the same letter . We denote by or the Fourier transform of the function , then the inverse Fourier transformation is given by

We are now in a position to state our result.

Theorem 1. Assume that the initial data have a sufficiently small norm . Then there exists a unique global solution of Cauchy problem (1). Furthermore the estimate

is true, where

Next we prove the existence of the self-similar solutions .

Theorem 2. There exists a unique solution of Cauchy problem (1) in the self-similar form , such that

where is sufficiently small number and

Furthermore the estimate

is true, where

Now we state the stability of solutions to Cauchy problem (1) in the neighborhood of the self-similar solution

Theorem 3. Suppose that

Let and be the solutions constructed in Theorems 1 and 2, respectively. Then there exists small such that the asymptotics

are true for .

Our approach is based on the factorization techniques. Define the free evolution group and write

where is the dilation operator. There is a unique stationary point in the integral , which is defined as the root of the equation for all . Define the scaling operator Hence we find the following decomposition , where the multiplication factor and the deformation operator

where the phase function Denote , We have , and also , ; therefore we obtain the commutator Since , then we get Also we need the representation for the inverse evolution group , where the inverse dilation operator , the inverse scaling operator , and the inverse deformation operator

We have Hence the commutator . Define the new dependent variable . Since with , applying the operator to (1), substituting , and using the factorization techniques, we get

since the nonlinearity is gauge invariant. Finally we mention some important identities. The operator plays a crucial role in the large time asymptotic estimates. Note that commutes with , that is, To avoid the derivative loss we also use the operator Note the commutator relation with Thus using , we get Also we have the identity and holds.

2. Estimates in the Uniform Norm

2.1. Kernels

Define the kernel

for , where the cutoff function is such that for or and for We change , then we get , where and , To compute the asymptotics of the kernel for large we apply the stationary phase method (see [39], p. 110)

for , where the stationary point is defined by the equation By virtue of formula (13) with , , and , we get

for Also we have the estimate

In the same manner changing , we get for the kernel

for , where with , Then by virtue of formula (13) with , , and , we obtain

for Also we have the estimate

2.2. Asymptotics for the Operator

In the next lemma we estimate the operator in the uniform norm. Define the cutoff function such that for and for and . Consider two operators

so that we have for Define the norm

Lemma 4. The following estimates if and if are valid for all ,

Proof. We write

for For the summand we integrate by parts via identity

with , to get

We find the estimates

in the domain Therefore we obtain

By the Hardy inequality and by the Cauchy-Schwarz inequality, changing we find

To estimate the integral we integrate by parts via the identity

with , to get

We find the estimates and

Then by the Hardy inequality we obtain

We have and Thus we have for all , .
To estimate we integrate by parts via identity (24)

We find the estimates and in the domain Then by the Hardy inequality we obtain

We have and if . Thus we have for all if Lemma 13 is proved.

2.3. Asymptotics for the Operator

We next consider the operator . Since and , then by the Riesz interpolation theorem (see [40], p. 52) we have

for In the next lemma we find the asymptotics of Denote Also define the norm

Lemma 5. Let , Then the estimate is valid for all

Proof. We write

for In the integral we use the identity

with , , and integrate by parts

Then apply the estimates in the domain . If for all then we find the Hardy inequality

Hence Therefore changing , we have

if In the integral using the identity

with , , we integrate by parts

Then using the estimates in the domains and , we get

Therefore by the Hardy inequality

We have

if , Therefore we get Lemma 5 is proved.

3. Commutators with

First we estimate the Fourier type integral

in the -norm. In the particular factorized case , with estimate , we find

Next we obtain a more general result.

Lemma 6. Suppose that for all , , where Then the estimate is true for all

Proof. We write

where the kernel Integrating two times by parts via the identity , with we get

Since we get

Then by the Cauchy-Schwarz inequality and Young inequality we obtain

if Lemma 14 is proved.

Next we estimate

Lemma 7. Suppose that for all , Then the estimate is true.

Proof. As in the proof of Lemma 13 we decompose

for In the first summand we integrate by parts via identity (19), to get

Using the estimate then changing , we obtain

Thus we get To estimate the integral we integrate by parts via identity (24), to get

We find the estimates , then we obtain

Thus as above we get for all if To estimate we integrate by parts via identity (24)

We find the estimates in the domain , and then we obtain

if Thus we find

for all Lemma 7 is proved.

In the next lemma we estimate the commutator Define the norm

Lemma 8. Let , Then the estimate is true for all

Proof. For we integrate by parts

where , , and Since and , we have

and similarly for all in the domain Hence we have for all , , and , where , and by Lemma 14

and by the Hardy inequality and Lemma 14

Also we have

for all , in the domain Hence we get for all , Therefore applying Lemma 7 we obtain Lemma 8 is proved.

In the next lemma we estimate the operator

Lemma 9. Let , Then the estimate is true for all

Proof. We integrate by parts

where , , and We find

and for all , in the domain if Then for all , , and Hence by Lemma 14 we find

and by the Hardy inequality

Also we have for all , in the domain , if Hence for all , , and by Lemma 7

Lemma 9 is proved.

In the next lemma, we estimate the derivative

Lemma 10. Let , Then the estimate is true for all

Proof. Since with , , then we obtain the commutator Also Hence

By Lemma 8 we find for all , if , Also by Lemma 9 we get for all , if , Hence the result of the lemma follows. Lemma 10 is proved.

4. A Priori Estimates

Local existence and uniqueness of solutions to Cauchy problem (1) were shown in [19, 20] when . By using the local existence result, we have the following.

Theorem 11. Assume that the initial data Then there exists a unique local solution of Cauchy problem (1) such that .

We can take if the data are small in and we may assume that . To get the desired results, we prove a priori estimates of solutions uniformly in time. Define the following norm

where , , and .

First we obtain the large time asymptotic behavior of the nonlinearity

Lemma 12. The asymptotics is true for all and , where and is small.

Proof. In view of factorization formula (11) we find , where Then by Lemma 5 with , , and small, we get

in the case of and

in the case of Via identity , we consider the remainder terms

where By Lemma 10 with , we have

Using Lemma 13 we get and Hence and Also we find and Therefore we obtain for and for Next by Lemma 13 we have for Then we get and Lemma 12 is proved.

Next we estimate the solutions in the norm

Lemma 13. Assume that holds. Then there exists such that the estimate is true for all .

Proof. By continuity of the norm with respect to , arguing by the contradiction we can find the first time such that To prove the estimate for the norm we use (11). Then in view of Lemma 12, we get

For the case of we can integrate For the case of multiplying by and taking the real part of the result we obtain Integrating in time we obtain

Therefore Applying estimate of Lemma 4 we find , , and

if Consider a priori estimates for . Using the identity , we get Applying the operator to (1), in view of the commutators , , we get Then by the energy method we obtain

from which it follows that Therefore for all . Thus we obtain Lemma 13 is proved.

5. Proof of Theorem 1

By Lemma 13 we see that a priori estimate is true for all . Therefore the global existence of solutions of Cauchy problem (1) satisfying the estimate follows by a standard continuation argument by local existence Theorem 11.

6. Proof of Theorem 2

In this section we prove the existence of a unique self-similar solution for (1), which is uniquely determined by the total mass condition Define the operators

Then for the self-similar solutions , where , we find that have a self-similar form, that is, with Using the relation by factorization formula (11) we get Therefore Note that is not in . Therefore we need the approximate equation. Define for and for , and denote . Also define the approximate equation

Let us show a priori estimate uniformly in . Applying Lemma 12 with we get

Integrating with respect to , we obtain Also multiplying by and integrating with respect to we get Hence . Thus we obtain for some small . Taking the limit , we find that there exists a unique solution of equation in . By the definition of , we obtain and In the same way as in the proof of (73) we have estimate of for .

7. Proof of Theorem 3

Define the norm with a small .

Lemma 14. Suppose that , , where is sufficiently small. Let for , , where Let be a self-similar solution. Then the estimate is true for all .

Proof. By the continuity of the norm with respect to , arguing by the contradiction we can find for the first time such that We denote and . Applying estimate of Lemma 4 we find

Thus we need to estimate the norm . For the difference we get from (1) Hence by the energy method

Next we get

In the same manner

Note that for the case of self-similar solution . Hence

By (78) we have To estimate we use the above estimates to get

In view of Lemma 4

Since , we get by the Hardy inequality and by a direct calculation Hence Therefore by (83) . Thus we obtain from (82) which implies Therefore Lemma 14 is proved.

Now we turn to the proof of asymptotic formula (7) for the solutions of Cauchy problem (1). Let be the self-similar solution with the total mass condition Note that by Theorem 2 and by Theorem 1. Also for Then by Lemma 14 we find Thus asymptotics (7) follows. Theorem 3 is proved.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work is partially supported by CONACYT 252053-F and PAPIIT Project IN100817.

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