High-Speed Transmission in Long-Haul Electrical Systems

Juárez-Campos, Beatriz; Kaikina, Elena I.; Naumkin, Pavel I.; Ruiz-Paredes, Héctor Francisco

doi:https://doi.org/10.1155/2018/8236942

International Journal of Differential Equations

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Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 8236942 | https://doi.org/10.1155/2018/8236942

High-Speed Transmission in Long-Haul Electrical Systems

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

Academic Editor: Sining Zheng

Received29 Jan 2018

Accepted12 Mar 2018

Published18 Apr 2018

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 [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 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 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 , , andif 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 equationLet 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 findThus 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 . HenceBy (78) we have To estimate we use the above estimates to getIn 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.

References

F. J. Diaz-Otero and P. Chamorro-Posada, “Interchannel soliton collisions in periodic dispersion maps in the presence of third order dispersion,” Journal of Nonlinear Mathematical Physics, vol. 15, pp. 137–143, 2008.
View at: Google Scholar
J. R. Taylor, Generation and compression of femtosecond solitons in optical fibers, in Optical Solitons-Theory and Experiment, J. R. Taylor, Ed., Cambridge Studies in Modern Optics, Cambridge University Press, Cambridge, UK, 1992.
View at: Publisher Site
S. Cui and S. Tao, “Strichartz estimates for dispersive equations and solvability of the Kawahara equation,” Journal of Mathematical Analysis and Applications, vol. 304, no. 2, pp. 683–702, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
S. B. Cui, D. G. Deng, and S. P. Tao, “Global existence of solutions for the Cauchy problem of the Kawahara equation with L2 initial data,” Acta Mathematica Sinica, vol. 22, no. 5, pp. 1457–1466, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE Journal of Quantum Electronics, vol. 23, no. 5, pp. 510–524, 1987.
View at: Publisher Site | Google Scholar
A. T. Il'ichev and A. Y. Semenov, “Stability of solitary waves in dispersive media described by a fifth-order evolution equation,” Theoretical and Computational Fluid Dynamics, vol. 3, no. 6, pp. 307–326, 1992.
View at: Publisher Site | Google Scholar
T. Kawahara, “Oscillatory solitary waves in dispersive media,” Journal of the Physical Society of Japan, vol. 33, no. 1, pp. 260–264, 1972.
View at: Publisher Site | Google Scholar
Y. Kodama, “Optical solitons in a monomode fiber,” Journal of Statistical Physics, vol. 39, no. 5-6, pp. 597–614, 1985.
View at: Publisher Site | Google Scholar | MathSciNet
A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Taki, “Impact of the third-order dispersion on the modulation instability gain of pulsed signals,” Optics Expresss, vol. 35, no. 8, pp. 1194–1196, 2010.
View at: Publisher Site | Google Scholar
I. Naumkin and R. Weder, “High-energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and application,” Mathematical Methods in the Applied Sciences, vol. 38, no. 12, pp. 2427–2465, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
M. Taki, A. Mussot, A. Kudlinski, E. Louvergneaux, M. Kolobov, and M. Douay, “Third-order dispersion for generating optical rogue solitons,” Physics Letters A, vol. 374, no. 4, pp. 691–695, 2010.
View at: Publisher Site | Google Scholar
X. Carvajal, “Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices,” Electronic Journal of Differential Equations, vol. 13, pp. 1–13, 2004.
View at: Google Scholar | MathSciNet
X. Carvajal and F. Linares, “A higher order nonlinear Schrödinger equation with variable coefficients,” Differential Integral Equations, vol. 16, no. 9, pp. 1111–1130, 2003.
View at: Google Scholar | MathSciNet
C. Laurey, “The Cauchy problem for a third order nonlinear Schrödinger equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 29, no. 2, pp. 121–158, 1997.
View at: Google Scholar
J. L. Bona, G. Ponce, J.-C. Saut, and C. Sparber, “Dispersive blow-up for nonlinear Schrödinger equations revisited,” Journal de Mathématiques Pures et Appliquées, vol. 102, no. 4, pp. 782–811, 2014.
View at: Google Scholar
J. Ginibre, Y. Tsutsumi, and G. Velo, “Existence and uniqueness of solutions for the generalized Korteweg de Vries equation,” Mathematische Zeitschrift, vol. 203, no. 1, pp. 9–36, 1990.
View at: Publisher Site | Google Scholar | MathSciNet
N. Hayashi, “Analyticity of solutions of the Korteweg-de Vries equation,” SIAM Journal on Mathematical Analysis, vol. 22, no. 6, pp. 1738–1743, 1991.
View at: Publisher Site | Google Scholar | MathSciNet
T. Kato, “On the Cauchy problem for the (generalized) Korteweg-de Vries equation,” in Studies in Applied Mathematics, vol. 8 of Adv. Math. Suppl. Stud., pp. 93–128, Academic Press, New York, 1983.
View at: Google Scholar | MathSciNet
C. E. Kenig, G. Ponce, and L. Vega, “On the (generalized) Korteweg-de Vries equation,” Duke Mathematical Journal, vol. 59, no. 3, pp. 585–610, 1989.
View at: Publisher Site | Google Scholar | MathSciNet
C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness of the initial value problem for the Korteweg-de Vries equation,” Journal of the American Mathematical Society, vol. 4, no. 2, pp. 323–347, 1991.
View at: Publisher Site | Google Scholar | MathSciNet
C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,” Communications on Pure and Applied Mathematics, vol. 46, no. 4, pp. 527–620, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
S. N. Kruzhkov, A. V. Faminskiĭ, and J. R. Schulenberger, “Generalized Solutions of the Cauchy Problem for the Korteweg-de Vries Equation,” Mathematics of the USSR - Sbornik, vol. 48, no. 2, pp. 391–421, 1984.
View at: Publisher Site | Google Scholar
J.-C. Saut, “Sur quelques généralizations de l'équation de Korteweg- de Vries,” Journal de Mathématiques Pures et Appliquées, vol. 58, no. 1, pp. 21–61, 1979.
View at: Google Scholar | MathSciNet
G. Staffilani, “On the generalized Korteweg-de Vries-type equations,” Differential Integral Equations, vol. 10, no. 4, pp. 777–796, 1997.
View at: Google Scholar | MathSciNet
M. Tsutsumi, “On global solutions of the generalized Kortweg-de Vries equation,” Publications of the Research Institute for Mathematical Sciences, vol. 7, pp. 329–344, 1971/72.
View at: Publisher Site | Google Scholar | MathSciNet
A. De Bouard, N. Hayashi, and K. Kato, “Gevrey regularizing effect for the (generalized) Korteweg - de Vries equation and nonlinear Schrödinger equations,” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, vol. 12, no. 6, pp. 673–725, 1995.
View at: Google Scholar
T. Cazenave and I. Naumkin, “Modified scattering for the critical nonlinear Schrödinger equation,” Journal of Functional Analysis, vol. 274, no. 2, pp. 402–432, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
P. Constantin and J.-C. Saut, “Local smoothing properties of dispersive equations,” Journal of the American Mathematical Society, vol. 1, no. 2, pp. 413–439, 1988.
View at: Publisher Site | Google Scholar | MathSciNet
W. Craig, K. Kappeler, and W. A. Strauss, “Gain of regularity for solutions of KdV type,” Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, vol. 9, no. 2, pp. 147–186, 1992.
View at: Google Scholar
S. Klainerman, “Long-time behavior of solutions to nonlinear evolution equations,” Archive for Rational Mechanics and Analysis, vol. 78, no. 1, pp. 73–98, 1982.
View at: Publisher Site | Google Scholar | MathSciNet
S. Klainerman and G. Ponce, “Global, small amplitude solutions to nonlinear evolution equations,” Communications on Pure and Applied Mathematics, vol. 36, no. 1, pp. 133–141, 1983.
View at: Publisher Site | Google Scholar | MathSciNet
J. L. Bona and J.-C. Saut, “Dispersive blowup of solutions of generalized Korteweg-de Vries equations,” Journal of Differential Equations, vol. 103, no. 1, pp. 3–57, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
N. Hayashi and E. I. Kaikina, “Asymptotics for the third-order nonlinear Schrödinger equation in the critical case,” Mathematical Methods in the Applied Sciences, vol. 40, no. 5, pp. 1573–1597, 2017.
View at: Publisher Site | Google Scholar | MathSciNet
N. Hayashi, J. A. Mendez-Navarro, and P. Naumkin, “Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case,” Journal of Differential Equations, vol. 261, no. 9, pp. 5144–5179, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
N. Hayashi and P. I. Naumkin, “The initial value problem for the cubic nonlinear Klein-Gordon equation,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 59, no. 6, pp. 1002–1028, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
N. Hayashi and T. Ozawa, “Scattering theory in the weighted L2(Rn) spaces for some Schr ödinger equations,” Annales de l’Institut Henri Poincaré. Section A, Physique Théorique, vol. 48, no. 1, pp. 17–37, 1988.
View at: Google Scholar
I. P. Naumkin, “Sharp asymptotic behavior of solutions for cubic nonlinear Schrödinger equations with a potential,” Journal of Mathematical Physics, vol. 57, no. 5, Article ID 051501, 051501, 31 pages, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
I. P. Naumkin, “Initial-boundary value problem for the one dimensional Thirring model,” Journal of Differential Equations, vol. 261, no. 8, pp. 4486–4523, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
M. V. Fedoryuk, “Asymptotics: integrals and series,” in Mathematical Reference Library, Nauka, vol. 13, p. 544, Springer-Verlag, Moscow, 1987.
View at: Google Scholar
E. M. Stein and R. Shakarchi, “Functional analysis,” in Introduction to Further Topics in Analysis, vol. 4 of Princeton Lectures in Analysis, Princeton University Press, Princeton, NJ, USA, 2011.
View at: Google Scholar | MathSciNet

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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.

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