Modified Ginzburg-Landau equation and Benjamin-Feir instability

In this paper the modulated wave train in nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg – Landau Equation (MGLE). Benjamin – Feir inst...

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Дата:	2003
Автор:	Kengne, E.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2003
Назва видання:	Нелінійні коливання
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/176944
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Цитувати:	Modified Ginzburg-Landau equation and Benjamin-Feir instability / E. Kengne // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 346-356. — Бібліогр.: 28 назв. — англ.

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spelling	irk-123456789-1769442021-02-10T01:25:51Z Modified Ginzburg-Landau equation and Benjamin-Feir instability Kengne, E. In this paper the modulated wave train in nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg – Landau Equation (MGLE). Benjamin – Feir instability for the MGLE is analyzed Вивчається проходження модульованих хвиль у нелiнiйному моноiндуктивному LC-ланцюзi. З використанням методу кратних шкал отримано, що еволюцiя нелiнiйних збуджень описується за допомогою рiвняння, яке ми називаємо модифiкованим рiвнянням Гiнзбурга – Ландау (МРГЛ). Аналiзується стабiльнiсть МРГЛ у сенсi Бенджамiна – Фейра. 2003 Article Modified Ginzburg-Landau equation and Benjamin-Feir instability / E. Kengne // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 346-356. — Бібліогр.: 28 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176944 517.9 en Нелінійні коливання Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In this paper the modulated wave train in nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg – Landau Equation (MGLE). Benjamin – Feir instability for the MGLE is analyzed
format	Article
author	Kengne, E.
spellingShingle	Kengne, E. Modified Ginzburg-Landau equation and Benjamin-Feir instability Нелінійні коливання
author_facet	Kengne, E.
author_sort	Kengne, E.
title	Modified Ginzburg-Landau equation and Benjamin-Feir instability
title_short	Modified Ginzburg-Landau equation and Benjamin-Feir instability
title_full	Modified Ginzburg-Landau equation and Benjamin-Feir instability
title_fullStr	Modified Ginzburg-Landau equation and Benjamin-Feir instability
title_full_unstemmed	Modified Ginzburg-Landau equation and Benjamin-Feir instability
title_sort	modified ginzburg-landau equation and benjamin-feir instability
publisher	Інститут математики НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/176944
citation_txt	Modified Ginzburg-Landau equation and Benjamin-Feir instability / E. Kengne // Нелінійні коливання. — 2002. — Т. 5, № 4. — С. 346-356. — Бібліогр.: 28 назв. — англ.
series	Нелінійні коливання
work_keys_str_mv	AT kengnee modifiedginzburglandauequationandbenjaminfeirinstability
first_indexed	2025-07-15T14:53:53Z
last_indexed	2025-07-15T14:53:53Z
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fulltext	UDC 517.9 MODIFIED GINZBURG – LANDAU EQUATION AND BENJAMIN – FEIR INSTABILITY МОДИФIКОВАНЕ РIВНЯННЯ ГIНЗБУРГА – ЛАНДАУ I НЕСТАБIЛЬНIСТЬ У СЕНСI БЕНДЖАМIНА – ФЕЙРА E. Kengne Univ. Dschang P.O. Box 173, Dschang, Cameroon e-mail: ekengne6@yahoo.fr In this paper the modulated wave train in nonlinear monoinductance LC circuit is studied. Using the method of multiple scales in general form, we establish that the evolution of nonlinear excitations is governed by what we called the Modified Ginzburg – Landau Equation (MGLE). Benjamin – Feir instabi- lity for the MGLE is analyzed. Вивчається проходження модульованих хвиль у нелiнiйному моноiндуктивному LC-ланцюзi. З використанням методу кратних шкал отримано, що еволюцiя нелiнiйних збуджень описується за допомогою рiвняння, яке ми називаємо модифiкованим рiвнянням Гiнзбурга – Ландау (МРГЛ). Аналiзується стабiльнiсть МРГЛ у сенсi Бенджамiна – Фейра. 1. Introduction. Considering nonlinear transmission line as a convenient tool to examine wave propagations in dispersive media, various physical systems have been studied [1 – 3]. Since the pioneering works of Hirota and Suzuki [4, 5] in order to stimulate the integrable Toda lattice [6] by electric circuits there has been an increased interest in the propagation of wave trains in nonlinear-dispersive transmission lines, involving the phenomena such as Benjamin – Feir instability [7 – 9], the formation of stationary localized waves, that is, the envelope solitons [10, 11] and the dark solitons [12, 13]. The Benjamin – Feir (or, as it is sometimes called, the modulational) instability is widespread and plays an important role in various nonlinear wave phenomena. Simply put, if dispersion and nonlinearity act against each other, monochromatic wave trains do not wish to remain monochromatic. The sidebands of the carrier wave can draw on its energy via a resonance mechanism with the result that the envelope becomes modulated. In one space dimension, this envelope modulation continues to grow until the soliton shape is reached. At this point, nonli- nearity and dispersion are in exact balance and no further distortion occurs [14, 15]. It is well known that the self-modulation of one space dimension waves in nonlinear di- spersive systems can be described by the so-called Ginzburg – Landau equation (GLE) [16 – 18], iut + Puxx +Q \|u\|2 u = iγ, (1.1) where the subscripts t and x denote the partial differentiation with respect to t and x, respecti- vely. If PQ < 0, a plane wave in this system is stable for the modulation and, otherwise, is unstable. Especially in the later case there exist special families of solutions, which are called envelope solitons and show various interesting phenomena [19, 20]. c© E. Kengne, 2003 346 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 MODIFIED GINZBURG – LANDAU EQUATION AND BENJAMIN – FEIR INSTABILITY 347 Fig. 1. Asection for a distributed nonlinear-dispersive transmission line. Recently, there has been a progress towards a mathematical understanding of equation (1.1). Kirchgassner [21] and Mielke [22 – 24] restrict attention to steady-state equations and view the single unbounded spatial direction as an evolution variable. In this paper, we give a rigorous derivation of the full time-dependent Modified Ginzburg – Landau Equation (MGLE). The Benjamin – Feir (modulational) instability for the obtained MGLE is investigated. 2. Basic equations. In this section we derive a nonlinear wave equation for the electromag- netic wave propagation in a nonlinear-dispersive transmission line shown in Fig. 1. By using the method of multiple scales, we derive a MGLE. 2.1. The model equations. In the considered transmission line, Fig. 1, CN is a nonlinear capacitor such as a ”VARICAP” or a reverse-biased p − n junction diode, the capacitance of which depends on the voltage applied across it. By applying the Kirchhoff’s voltage theorem and the current theorem we obtain ∂I ∂x + ∂ρ(V ) ∂t = 0, ∂V ∂x + L ∂I1 ∂t = 0, ∂2V ∂x∂t + 1 Cs (I − I1) = 0 (2.1) where the current through the nonlinear capacitor is given by ∂ρ(V )/∂t. From equations (2.1) we can eliminate I and I1 and write Cs ∂4V ∂x2∂t2 + 1 L ∂2V ∂x2 − ∂2ρ ∂t2 = 0. (2.2) With no loss of generality, we may regard ρ(0) = 0 and expand ρ(V ) to obtain ρ(V ) ≈ ρ′(0)V + + ρ′′(0) 2 V 2. For bounded solutions, we must have ρ′(0) ≥ 0. Hence we have ρ(V ) ≈ C0 ( V − β′V 2 ) = C0V − CNV 2. (2.3) Substituting (2.3) into (2.2), we obtain the following partial differential equation for the voltages: C0 ∂2V ∂t2 − 1 L ∂2V ∂x2 − CS ∂4V ∂x2∂t2 − CN ∂2V 2 ∂t2 = 0. (2.4) ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 348 E. KENGNE 2.2. Derivation of the generalized complex Ginzburg – Landau equation. If we introduce the notations α = −1/L, β = −CN , λ = CS , equation (2.4) takes the form C0 ∂2V ∂t2 + α ∂2V ∂x2 − Cs ∂4V ∂x2∂t2 + β ∂2V 2 ∂t2 = 0. (2.5) We follow Taniuti and Yajima [25, 26] and seek a first-order uniform expansion by using the method of multiple scales in the form V = ε1/2v11 exp [i(kX0 − ωT0)] + εV22 exp [2i(kX0 − ωT0)] + + ε3/2v33 exp [3i(kX0 − ωT0)] + ε2 [v42 exp [2i(kX0 − ωT0)] + + v44 exp [4i(kX0 − ωT0)]] + cc+ . . . , (2.6) where cc stands for the complex conjugate, ε is a small, dimensionless parameter related to the amplitudes (0 < ε � 1), vij = vij(X1, T1, T2), Tn = εnt, and Xn = εnx. Substituting (2.6) into (2.5) and equating coefficients of like powers of ε and exp [iθ] (here θ = kX0 − ωT0), we obtain the following: for order ε1/2, exp [iθ], [ C0ω 2 + αk2 + λk2ω2 ] v11 = 0, (2.7) for order ε3/2, exp [iθ], −2iω [ C0 + λk2 ] ∂v11 ∂T1 + 2ik [ α+ λω2 ] ∂v11 ∂X1 − 2βω2v∗11v22 = 0, (2.8) for order ε, exp [2iθ], −4 ( C0ω 2 + αk2 + 4λk2ω2 ) − 4ω2βv2 11 = 0, (2.9) for order ε3/2, exp [3iθ], −9 ( C0ω 2 + αk2 + 9λk2ω2 ) − 18ω2βv11v22 = 0, (2.10) for order ε2, exp [2iθ],[ −4 ( C0ω 2 + αk2 + 4λk2ω2 )] v42 − 8βω2v∗11v33 = 0, (2.11) for order ε2, exp [4iθ],[ −16 ( C0ω 2 + αk2 + 16λk2ω2 )] v44 − βω2 [ 32v11v33 + 16v2 22 ] = 0, (2.12) ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 MODIFIED GINZBURG – LANDAU EQUATION AND BENJAMIN – FEIR INSTABILITY 349 Fig. 2. The dispersion curve for the linearized version of the above transmission line. for order ε5/2, exp [iθ], C0 [ ∂2v11 ∂T 2 1 − 2iω ∂v11 ∂T2 ] + α ∂2v11 ∂X2 1 − λ [ −k2∂ 2v11 ∂T 2 1 + 2iωk2∂v11 ∂T2 + 4kω ∂2v11 ∂T1∂X1 − − ω2∂ 2v11 ∂X2 1 ] + β [ −4iω ∂v∗11v22 ∂T1 − 2ω2v∗11v42 − 2ω2v∗22v33 ] = 0. (2.13) For the nontrivial solution we must have v11 6= 0. Then (2.7) gives C0ω 2 + αk2 + λk2ω2 = 0. (2.14) Equation (2.14) is the dispersion relation illustrated in Fig. 2 for the line parameters Cs = 5C0 = 1200pF, L = 14µH, CN = 38, 4pF, 0 ≤ k ≤ 1, 58, ε = 0, 1. (2.15) Using the dispersion relation (2.14), equations (2.9) – (2.12) give v22 = − β 3λk2 v2 11, (2.16) v33 = β2 12λ2k4 v3 11, (2.17) v42 = − β3 108λ2k8ω2 \|v11\|2 v2 11, (2.18) ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 350 E. KENGNE v44 = − β3 54λ3k6 v4 11, (2.19) respectively. Solving for ∂v11/∂T1, from (2.9) and (2.16) we obtain ∂v11 ∂T1 = C0 α ω3 k3 ∂v11 ∂X1 − iβω3 αk2 v∗11v22, where Vg = −C0 α (ω k )3 = C0 √ −α (C0 + λk2)3/2 is the group velocity. Hence, ∂2v11 ∂T 2 1 = C2 0 α2 ω6 k6 ∂2v11 ∂X2 1 − C0 iβω6 α2k5 ∂ ∂X1 (v∗11v22)− iβω3 αk2 ∂ ∂T1 (v∗11v22) , (2.20) ∂2v11 ∂T1∂X1 = C0 α ω3 k3 ∂2v11 ∂X2 1 − iβω3 αk2 ∂ ∂X1 (v∗11v22) . Combining (2.20) and (2.13), and using (2.16) – (2.18), we obtain, in terms of the original vari- ables t and x, i ∂v11 ∂t + P ∂2v11 ∂x2 + iQ1 ∂ ∂t ( \|v11\|2 v11 ) + iQ2 ∂ ∂x ( \|v11\|2 v11 ) +Q3 \|v11\|4 v11 = 0, (2.21) where P = P (k) = −1 2 ω′′ = − 3C0λ √ −αk 2 (C0 + λk2)5/2 , (2.22) Q1 = Q1(k) = − β3ε 2λk2 (C0 + λk2) , (2.23) Q2 = Q2(k) = β2ε ( C0 + 4λk2 ) 3λk3 (C0 + λk2)2 , (2.24) Q3 = Q3(k) = − β4ε2 √ −α 12λ3k5 (C0 + λk2)3/2 . (2.25) Using the transformation ξ = x−Q2Q −1 1 t, τ = t, v11(ξ, τ) = u(ξ, τ) exp [ iQ2P −1Q−1 1 ξ/2 ] , (2.26) we write (2.21) in the final form i ∂u ∂τ + P ∂2u ∂ξ2 + iQ1 ∂ ∂τ ( \|u\|2 u ) − γu+Q3 \|v11\|4 v11 = 0, (2.27) ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 MODIFIED GINZBURG – LANDAU EQUATION AND BENJAMIN – FEIR INSTABILITY 351 where γ = −Q2 2P −1Q−2 1 /4. Thus the resulting equation (2.27) that describes the evolution of a wavepacket is a complex envelope equation that involves higher order nonlinearities. We call this equation the MGLE. For the line parameters (2.15) we plot the following coefficient of the spatial dispersion curve. Fig. 3. The coefficient of the spatial dispersion curve. It is seen from Fig. 3 that the coefficient of the spatial dispersion is always negative when 0 ≤ ≤ k ≤ 1, 58. In the next section we study the Benjamin – Feir instability of the monochromatic wave solutions. 3. The Benjamin – Feir instability. To study the Benjamin – Feir instability of the monochro- matic wave solutions, we first express u in the polar form u(ξ, τ) = a(ξ, τ) exp [ib(ξ, τ)] . (3.1) Substituting (3.1) into (2.27) and separating imaginary and real parts we obtain ( 1 + 3Q1a 2 ) ∂a ∂τ + P ( 2 ∂a ∂ξ ∂b ∂ξ + a ∂2b ∂ξ2 ) = 0, (3.2) ( a+Q1a 3 ) ∂b ∂τ + P ( a ( ∂b ∂ξ )2 − ∂2a ∂ξ2 ) + γa−Q3a 5 = 0. (3.3) If the wave has a fixed, single wavenumber, then P = −1 2 d2ω dk2 = 0 and system (3.2), (3.3) reduces to ( 1 + 3Q1a 2 ) ∂a ∂τ = 0, ( a+Q1a 3 ) ∂b ∂τ + γa−Q3a 5 = 0 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 352 E. KENGNE whose solutions are a = a0, b = Q3a 4 0 − γ 1 +Q1a2 0 τ + const, (3.4) where a0 is constant. It natural to define the local wavenumber k as the ξ derivative of the total phase and the local frequency as the negative of the τ derivative of the total phase θ = k0ξ − ω0τ + b(ξ, τ), k = k0 + bξ, ω = ω0 − bτ . Note that kτ + ωξ = bξτ − bτξ = 0, (3.5) which expresses the conservation of the number of waves. We will write the change in the wavenumber bξ as K. Now equation (3.2) gives ∂ ∂τ ( 2a2 + 3Q1a 4 ) + 4P ∂ ξ ( a2K ) = 0, (3.6) which is an equation of conservation of the wave action. On the other hand, equation (3.3) gives a ( 1 +Q1a 2 ) bτ + P ( aK2 − aξξ ) + γa−Q3a 5 = 0, which when differentiating with respect to ξ gives a2(1 +Q1a 2)2Kτ + aξ ( 1 + 3Q1a 2 ) [ Q3a 5 − γa+ P ( aξξ − aK2 )] + + Pa ( 1 +Q1a 2 ) ( aξK 2 + 2aKKξ − aξξξ ) + a ( 1 +Q1a 2 ) ( γ − 5Q3a 4 ) aξ = 0. (3.7) Equation (3.7) is a relation for conservation of waves, since ω = ω0 + γ + P ( K2 − aξξ a ) −Q3a 4 1 +Q3a2 . Next, the monochromatic wave solution (3.4) means that k = k0, ω = ω0 − Q3a 4 0 − γ 1 +Q1a2 0 . This is the Stokes wave. Test it linear stability by setting a = a0 + ã, K = K̃, (3.8) ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 MODIFIED GINZBURG – LANDAU EQUATION AND BENJAMIN – FEIR INSTABILITY 353 where ã is assumed to be infinitesimal. Substituting (3.8) into (3.6) and (3.7) and keeping only linear terms in perturbation quantities, we obtain ãτ = − Pa0 1 + 3Q1a2 0 K̃ξ, Kτ = P a0 ( 1 +Q1a2 0 ) ãξξξ + 2 ( Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ ) a2 0 a0 ( 1 +Q1a2 0 )2 ãξ, or ãττ = − P ( 1 +Q1a 2 0 )−2( 1 + 3Q1a2 0 ) [ P ( 1 +Q1a 2 0 ) ãξξξξ + 2a2 0 ( Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ ) ãξξ ] . (3.9) Therefor if ã ∝ exp [ilξ + Ωτ ] , Ω2 = − ( 1 +Q1a 2 0 )−2 k2( 1 + 3Q1a2 0 ) [( 1 +Q1a 2 0 ) P 2l2 − 2a2 0P ( Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ )] . (3.10) Because β = −CN < 0, it follows from (2.22) – (2.25) that P (k) < 0, Q1(k) > 0, Q3(k) < 0, and γ = −Q2 2P −1Q−2 1 /4 > 0. Therefore we have the following results. Theorem 3.1. If Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ < 0, (3.11) the monochromatic wave solution (3.4) will be unstable to long waves in the range 0 < l2 < 2a2 0 ( Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ ) P ( 1 +Q1a2 0 ) . (3.12) Inequality (3.11) is the Benjamin – Feir instability criterion to the MGLE in the electrical monoinductance transmission line. This new result is different from the Lange and Newell cri- terion for the Stokes wave [27, 28] by the presence of the amplitude a0 of the monochromatic wave. For the line parameters (2.15) we plotQ1Q3a 4 0 +2Q3a 2 0 +Q1γ as a function of the amplitude a0 or/and a function of the wavenumber k. ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 354 E. KENGNE Fig. 4. The dependence of Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ on a0 with k = 0, 1. Fig. 5. The dependence of Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ on k, 0, 09 ≤ k ≤ 0, 624, for a0 = 1000. Fig. 4 shows that for the wavenumber k = 0, 1, condition (3.11) holds for all a0 > a0c ' ' 0, 084. For these values of a0, the monochromatic wave solutions corresponding to the fi- xed wavenumber k = 0, 1 are modulational unstable. All the monochromatic wave solutions associated to k = 0, 1 with any amplitude a0 < a0c are stable. It is seen from Figures 5, 6, and 7 that for any fixed wavenumber 0 < k < 0, 624, the corresponding monochromatic wave solution with the amplitude a0 = 1000 is modulational unstable, while any monochromatic wave solution corresponding to the wavenumber 0, 625 ≤ ≤ k ≤ 1, 58 with the amplitude a0 = 1000 is modulational stable. Figures 4 – 7 show that Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ, as a function of k or/and a0, changes its sign for particular value of k or/and a0. ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 MODIFIED GINZBURG – LANDAU EQUATION AND BENJAMIN – FEIR INSTABILITY 355 Рис. 6. The dependence of Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ on k, 0, 624 < k ≤ 0, 625, for a0 = 1000. Рис. 7. The dependence of Q1Q3a 4 0 + 2Q3a 2 0 +Q1γ on k, 0, 625 ≤ k ≤ 1, 58, for a0 = 1000. 4. Conclusion. In this paper monoinductance LC circuit is considered and envelope modulati- on is reduced to the MGLE. Benjamin – Feir instability for the MGLE is analyzed. As far as we know there have been no such Stokes wave analysis related to LC circuit. As in most cases, the linear part of the modulation equation (the coefficient of the spatial dispersion) is fixed, that is, P = −1 2 d2ω dk2 . We also have that Q1Q3a 4 0 + 2Q3a 2 0 + Q1γ < 0 is a necessary but not sufficient condition for the instability. It should be noted that Q1Q3a 4 0 + 2Q3a 2 0 + Q1γ > 0 is a sufficient condition for the stability. In fact, if this last condition is satisfied then for every real l, Ω will always be pure imaginary and ã ∝ exp [ikξ + Ωτ ] will be bounded. In most cases the criterion of the instability does not depend on the amplitude of the monochromatic wave. But for our MGLE, the said criterion depends on the amplitude a0. This fact allows us to construct an unstable monochromatic wave for a given wavenumber. ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 3 356 E. KENGNE 1. Scott A. Active and nonlinear wave propagation in electronics. — New York: Wiley-Intersci., 1970. 2. Solitons in action / Eds. K.E. Lonngren, A.C. Scott. — New York: Acad. Press, 1978. 3. Ostrovski L.A., Gorshkov K.A., andPapko V.V. Phys. scr. — 1979. — 20. — P. 357. 4. Hirota R., Suzuki K. Direct methods in soliton theory // J. Phys. Soc. Jap. — 1970. — 28. — P. 1366. 5. Hirota R., Suzuki K. Direct methods of finding exact solutions of nonlinear evolution equations // Backlund Transformation: Proc. IECE. — 1973. — 61. — P. 1438. 6. Toda M. Theory of nonlinear Lattices // J. Phys. Soc. Jap. — 1967. — 22. — P. 413; 23. — P. 501. 7. Benjamin T.B., Feir J.F. The disintegration of wave trains on deep water // J. Fluid Mech. — 1967. — 27. — P. 417 – 430. 8. 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