High-Speed Transmission in Long-Haul Electrical Systems

International Journal of Differential Equations, Apr 2018

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

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High-Speed Transmission in Long-Haul Electrical Systems

Hindawi International Journal of Differential Equations Volume 2018 High-Speed Transmission in Long-Haul Electrical Systems Beatriz Ju?rez-Campos 0 Elena I. Kaikina 1 Pavel I. Naumkin 1 H?ctor Francisco Ruiz-Paredes 0 Academic Editor: Sining Zheng 0 Instituto Tecnolo ?gico de Morelia, Avenida Tecnolo ?gico No. 1500, Lomas de Santiaguito , 58120 Morelia, MICH , Mexico 1 Centro de Ciencias Matema ?ticas, UNAM Campus Morelia , AP 61-3 Xangari, 58089 Morelia, MICH , Mexico Correspondence should be addressed to Elena I. Kaikina; We study the equations governing the high-speed transmission in long-haul electrical systems ? (1/3)| |3 = (|| 2), (, ) ? R+ ? R, (0, ) = 0(), ? R, where ? R, | | = F?1|| F, and F is the Fourier transformation. Our purpose in this paper is to obtain the large time asymptotics for the solutions under the nonzero mass condition ? 0() =? 0. - 1. Introduction We study the equations governing the high-speed transmission in long-haul electrical systems ? 13 3 2 (| | ) , (, )? R+ ? R, (1) (0, )= 0 ( ), ?R, where ? R, | |3 = F?1|| 3F, and F is the Fourier transformation defined by F = (1/ ?2) ?R ? . Note that we have the relation (?, ) = (, ?), so we can only consider the case > 0. For the regular solution of (1) we have the conservation law ?()? L2 = ? 0?L2 . We are interested in the case of nonzero mass condition ?R 0() ? = 0. By (1) we get the conservation of the mass ?R ( , ) = ? R 0() =? 0 for all > 0. 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 Schro?dinger 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 Schro?dinger equation (see [ 3?11 ]). The higher order nonlinear Schro?dinger equations have been widely studied recently. For the local and global wellposedness 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 L = { ? S ; ? ? L < ?}, where the norm ?? L = (? | ( ))| 1/ for 1 ? < ? and ?? L? = sup?R |()|. The weighted Sobolev space is H, = { ? S ; ? ? H, = ?? ?? ? ? L < ?}, ? where , ? R, 1 ? ? ?, ?? = ?1 + 2, and ? = ?1 ? 2. We also use the notations H, = H2, , H,0 (?) = (1/?2) ?R H = shortly, if it does not cause any confusion. Let C(I;B)be the space of continuous functions from an interval I to a Banach space B. Different positive constants might be denoted by the same letter . We denote by F or ? () the Fourier transform of the function , then the inverse Fourier transformation is given by F?1 = (1/ ?2) ?R (). We are now in a position to state our result. Theorem 1. Assume that the initial data 0 ? H ? H0,1 have 1 a suf f iciently small norm? 0?H1?H0,1 ? . Then there exists a |3 unique global solution F ?(/3)| ? C([0, ?)L;? ? H0,1) of Cauchy problem (1). Furthermore the estimate is sufficiently small number and ( ) =? 0, (3) Furthermore the estimate Now we state the stability of solutions to Cauchy problem (1) in the neighborhood of the self-similar solution V (, ). Theorem 3. Suppose that 1 ?2 are true for ? 1. (2) (4) (5) (7) Our approach is based on the factorization techniques. Define the free evolution group U() =F?1 ?(/3)|| 3 F and write U ()F?1 = D ? | | 2 where the phase function (, ) = (1/3)|| 3 ||( ? ). Denote A = (1/2||) , = 0, 1. We have A1 = A0 + , and also A1V = V, [, V] = ?A0V; therefore we obtain the commutator V = ?2||[, V]. Since (, ) = || ? ||, then we get [||, V] = ?V . Also we need the representation for the inverse evolution group FU(?) = V? B?1D?1, where the inverse dilation operator D?1 = | 1|/2(), the inverse scaling operator (B?1)() = (||) , and the inverse deformation operator V? () = ? 2 | | (,) since the nonlinearity is gauge invariant. Finally we mention some important identities. The operator J = U()U(?) = + | | plays a crucial role in the large time asymptotic estimates. Note that J commutes with L, that is, [J, L] = 0. To avoid the derivative loss we also use the operator P = 3 + . Note the commutator relation [P?, ?(/3)|| 3 ] = 0 (9) (10) (11) with P? D V we have the identity J = holds. = 3 ? . Thus using () = ?= F ?1 ?(/3)|| 3 ,?we get P = U()F ?1P?.? Also ?1P ? 3 ?1L and [L, P] = 3L 2. Estimates in the Uniform Norm 2.1. Kernels. Define the kernel (, ) = ? ? R ?(,) ? ( ?1 ) (12) (, ) = || 3 (, ) = || () and () = (1/3)( + 2)( ? 1)2, > 0. To compute the asymptotics of the kernel (, ) for large we apply the stationary phase method (see [ 39 ], p. 110) for ? +? , where the stationary point 0 is defined by the equation ( 0) = 0.By virtue of formula (13) with () = ?(), () = ?() , and 0 = 1, we get (, ) = 1/2 ?2 ? 3? + ( 1/2 1+ ? 3 ?1 ? ) (14) 3 for 1/2|| +1 ? ?. Also we have the estimate | ? 3??1/2. In the same manner changing = , we get for the kernel (, )| ? ? (, )= ? 2 | | = 2? 2 | | ? ?. Also we have the estimate | ?(, )| ? ? 3??1/2. Therefore we obtain 1 ? 1/2 1/3</<3 ? ? + ( )? ( ) ? 1/2 ? ? 1 + ( ? ) ( ) (18) (19) (20) (21) (22) 2.2. Asymptotics for the Operator V. In the next lemma we estimate the operator V in the uniform norm. Define the cutoff function 1() ? C2(R)such that 1() = 0for || ? 3 and 1() = 1for || ? 2 and 2() = 1 ? 1(). Consider two operators ) ) ? 1 + ( ? ) 2 for all ? 1. for =? 0. In the integral we use the identity 1 (,) = 3 (( ? ) ) (,) with and integrate by parts 3 = (1 + ( ? ) (, )) ?1 , (, ) = ?2||( ? ), 1 = + 1? ?1 T hen apply the estimates |( ? ) 3|| ??? ?( )|+|(? 2 1? ?1 1? ? ) ??? ( 3|| ?( ))| ? || ?? |?|/(1+||(? 2 ) )in the domain 1/3 ? / ? 3. If |()| ? |()| ? (0, 1)then we find the Hardy inequality for all . Therefore changing = , we have ? (,) ) (, ) = ?2||( ? ) ? ? the estimate ??? (V ? ? )? L? ? (?1)/3 ?? I , is valid (35) (36) , we (37) (38) (39) ( 4 (1 ? ? ( )) + Therefore ?FU(?)()?L? < Lemma 4 we find | (, )| ? ? |(, ) (, )| ? 2 ?1, and ({ }?? ?? ?]??? 3Y) and Next we estimate the solutions in the norm X . Lemma 13. Assume that ?? X1 ? > 0 such that the estimate ?? X < holds. Then there exists is true for all > 1 . Proof. By continuity of the norm ?? X with respect to , arguing by the contradiction we can find the first time > 0 such that ?? X = . To prove the estimate for the norm we use (11). Then in view of Lemma 12, ?= = FU (? ) 2 (| | ) ?1 ? ? 5 ??? ?6 2 ? ?+ ( 3 ?1 { }?? ?? ?] ) . For the case of || < ?2/3 ?? 3Y ?1 ||? case of || ? of the result we obtain ((|?, )| 2) = Integrating in time we obtain ?1/3 we can integrate |(?, )| ? | (?1, )|+ 1/3 3 3. For the ? + ? + ?1/3 multiplying by ?and taking the real part (?1{}?? ?? ?]??? 4Y). ?(, ) 2 ? ?( ?3 , ) 2 + ? 4Y ? { ?3 ? 2 + 4 ? 1 1/3 ?? 1/3 } ? ?1?] ? 5. Proof of Theorem 1 By Lemma 13 we see that a priori estimate ?? X ? is true for all > 0 . Therefore the global existence of solutions of Cauchy problem (1) satisfying the estimate ?? X? ? 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 selfsimilar solution V (, ) ? ?1/3 ( ?1/3) for (1), which is uniquely determined by the total mass condition = (1/?2) ?R V (, ) =? 0. Define the operators V? = ? 1 ? 2 R V?? = ? 2 ? R ?(,) (,) ( ), Then for the self-similar solutions V (, ) = ?1/3 ( ?1/3) = D BV ? , where ? (, ) ? FU(?)V (), we f ind that? have a self-similar form, that is, ? (, ) ? ()? with ? = 1/3. Using the relation ? (, ) = (1/3)?1 () by factorization formula (11) we get (1/3) () V??|V? |2V? . Therefore () = 3 V??|V? ? ( ).Note that ( ) is not in L2. Therefore we need the approximate equation. Define ?() = 1 for || ? 1 and ?() = 0 for || > 2, and denote ? () = ?(/). Also define the approximate equation , () = 3 V??? V?? 2 , V?? ( , ()) . Let us show a priori estimate ? , ?Z = ? , ?L? + ? , ?L2 ? 3| | uniformly in . Applying Lemma 12 with = 1 we get , () 2 ? Integrating with respect to , we obtain ? , ?L2 ? ? , ?3Z. Also multiplying by , and integrating with respect to ?1? ? , ?3Z ). we get | , ()| = + (? 0 ?? Hence ? , ?L? ? | | + ? , ?3Z. T hus we obtain? , ?Z ? 2|| + ? , ?3Z ? 3| | for some small . Taking the limit ? ?, we find that there exists a unique solution 1/2 ?1/6 ?1 ??|| ? ? L2 ? 2 2 ?| 1| 1 ? | 2| 2?L2 ? 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. [10] 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. 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Beatriz Juárez-Campos, Elena I. Kaikina, Pavel I. Naumkin, Héctor Francisco Ruiz-Paredes. High-Speed Transmission in Long-Haul Electrical Systems, International Journal of Differential Equations, 2018, DOI: 10.1155/2018/8236942