Table of Contents Author Guidelines Submit a Manuscript
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

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; xm.manu.romtam@anikiake

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 systemswhere , , 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 [311]).

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 [1214] and references cited therein. The dispersive blow-up was obtained in [15]. The existence and uniqueness of solutions to (1) were proved in [1625] and the smoothing properties of solutions were studied in [1821, 24, 2631]. 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 [3338].

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 asymptoticsare 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 getsince 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 identitywith , 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 identitywith , 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 havefor 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 identitywith , , 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 identitywith , , 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