SUPPRESSION OF THE FAST AND SLOW MODULATED WAVES MIXING IN THE COUPLED NONLINEAR DISCRETE LC TRANSMISSION LINES

Transcription

1 Available at: off IC/2006/035 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SUPPRESSION OF THE FAST AND SLOW MODULATED WAVES MIXING IN THE COUPLED NONLINEAR DISCRETE LC TRANSMISSION LINES David Yemélé 1 Laboratoire de Mécanique et de Modélisation des Systèmes Physiques, Faculté des Sciences, Université de Dschang, B.P. 067, Dschang, Cameroun and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and Timoléon C. Kofané Laboratoire de Mécanique, Faculté des Sciences, Université de Yaoundé I, B.P. 812, Yaoundé, Cameroun. Abstract The conditions of propagation of fast and slow-modes of modulated waves on the two coupled discrete nonlinear LC transmission lines are examined, each line of the network containing a finite number of cells. It is found that the use of an appropriate unit-cell, a band-pass filter, associated to a convenient choice of the intermediate coupling capacitor between the two lines allows to avoid the crucial problem of mixing of waves of different modes in the network. Good qualitative and quantitative agreements are found between analytical predictions and numerical results. MIRAMARE TRIESTE May Junior Associate of ICTP.

2 1 Introduction In nonlinear dispersive media, the propagation of modulated waves, such as envelope (bright) solitons or hole (dark) solitons, has been the subject of considerable interest for many decades [1]. In nonlinear optics [2, 3, 4, 5, 6], the permanent progress about fiber loss, birefringent fibers, Raman Effect and other undesirable effects allow important improvement of practical results concerning the distortion-less signal transmission in ultrahigh-speed communication and are potentially promising for a number of other applications. In general, the propagation of modulated waves in nonlinear dispersive media is characterized by a series of complex changes in its structure. A theoretical interpretation of these phenomena is crucial in order to understand the nonlinear properties of the medium and the complex dynamics of such nonlinear systems. As an example, the unstable propagation of slowly modulated plane waves (known as modulational instability (MI)) in the nonlinear dispersive medium exhibits the capacity of this medium to support the propagation of envelope soliton in this parameter regime [7]. Similarly, nonlinear discrete electrical transmission lines are convenient tools to study wave propagation in nonlinear dispersive media (see [7, 8] and references therein). In particular, they provide a useful way to check how the nonlinear excitations behave inside the nonlinear medium and to model the exotic properties of new systems [9]. These are the reasons why, since the pioneering works by Hirota and Suzuki [10] and Nagashima and Amagishi [11] on a single electrical line simulating the Toda lattice [12], a growing interest has been devoted to the use of the nonlinear discrete electrical transmission lines [13, 14, 15, 16, 17, 18]. However, the above cited studies are limited to a single mode soliton that the single LC transmission line adequately describes in certain parameter regimes, whereas there are many physical phenomena which can be investigated by the use of more than a single electrical transmission line. Indeed, to cite just two examples, the single electrical transmission line gives a good description of nonlinear deep water waves while it fails when considering nonlinear wave in the long gravity wave region [19, 20]. Similarly, if the single electrical line shares many phenomena with certain optical fibers, it cannot model the birefringent fiber which allows two modes of propagation, i.e. the fast and slow modes [6]. There are only a few works, as far as we know, which report the study of soliton in the coupled nonlinear transmission lines. Yoshinaga and Kakutani [19, 20, 21] have investigated theoretically and experimentally the Korteweg de Vries (KdV) solitons on a coupled LC transmission line consisting of two nonlinear LC ladder lines connected by identical intermediary capacitors, and have shown that the network admits two different modes (a fast-mode and a slow-mode), in each direction of wave propagation. Similarity of this line with a two-layer fluid in the long gravity wave region has also been discussed. Next, the extension of these studies 2

3 to envelope solitons has been made by Essimbi et al. [22, 23] who concentrated efforts to excitation modes, for the wave vector close to the limit k = π/2 [22] and in the vicinity of the gap of the linear spectrum [23]. The soliton propagation and interaction on two-dimensional nonlinear transmission lines have also been studied (see [24] and references therein). However, experiments and analytical studies on the basic coupled NLTL show that whenever the network is excited by an electrical wave, two modes of propagation (slow and fast-modes) are generated in each line and enters unavoidably into play with the wave-coupling behavior causing qualitatively different phenomena compared with the ordinary process of MI, such as the annihilation of both modes. This slow and fast-mode mixing effect is undesirable for the nonlinear modulated wave s experiments in the coupled NLTL as well as for its practical applications. Therefore it is useful to find a way to suppress this undesirable effect. This is the main objective of the paper. For this purpose, we first present carefully the analytical and numerical investigations concerning the nonlinear modulated waves and the possible propagation of envelope solitons on the two coupled LC transmission lines, depending on the appropriate choice of the carrier wave frequency (or angular frequency) and the capacitance of the intermediary coupling capacitor. The paper is organized as follows: In Sec.2 we present the basic characteristics of the coupled nonlinear discrete electrical transmission line under consideration. In Sec.3, in the low-amplitude limit, we derive the coupled nonlinear Schrödinger (NLS) equations governing the propagation of slowly modulated waves in the network and show that they can reduce to the single NLS equation. The frequency domain where the network allows the propagation of envelope solitons is also determined. In Sec.4, by introducing a linear inductor in parallel with each nonlinear capacitor of the two lines, we show that it is possible to avoid the mixing of the fast- and slow-modes of propagation in the network. Numerical experiments are considered in Sec.5 in order to verify the validity of the theoretical predictions, namely the MI phenomenon and the propagation of fast- and slow-envelope solitons. Finally, concluding remarks are devoted to Sec.6. 2 Basic coupled nonlinear discrete LC transmission lines The basic model usually used consists of a nonlinear network with two coupled nonlinear LC transmission lines. Each line contains a finite number of cells which consist of two elements: a linear inductor of inductance L j in the series branch and a nonlinear capacitor of capacitance C j in the shunt branch, where the subscript j designates the line number and can take the values 1 and 2. From now on we shall use this notation. The two lines are connected by an intermediary linear capacitor C, as shown in Figure 1. The capacitance C j are voltage dependent and are biased by constant voltages V bj [7]: C j (V bj + V jn ) dq j,n dv j,n = C 0j ( 1 2αj V j,n + 3β j V 2 j,n), (j = 1,2), (1) 3

4 with C 0j = C j (V bj ) and where α j and β j are the nonlinear positive coefficients of the electrical charge q j,n stored in the n th capacitor of line j. Denoting by V j,n the voltage across the n th capacitor of line j and using the Kirchhoff s laws, the circuit equations are then given by: d 2 q j,n dt 2 = 1 ( d 2 ) V 3 j,n (V j,n 1 2V j,n + V j,n+1 ) + C L j dt 2 d2 V j,n dt 2, n = 1,2,...N, (2) By inserting the above expansion of the capacitance (1) into Eq.(2), we obtain the following coupled equations governing wave propagation in this nonlinear network: d 2 V j,n dt 2 + Ω 2 0j (2V j,n V j,n 1 V j,n+1 ) γ j ( d 2 V j,n dt 2 d2 V 3 j,n dt 2 ) = α j d 2 V 2 j,n d 2 Vj,n 3 dt 2 β j dt 2, n = 1,2,...N, (3) where the characteristic frequencies of each line, Ω 0j, and the coupling coefficients γ j verify the following relations: form: Ω 0j = 1/L j C 0j, γ j = C/C 0j. (4) The linear properties of the network can be studied by assuming a sinusoidal wave of the V j,n = V j exp[i(kn ωt)] + c.c., (5) where V j is the constant amplitude, k and ω are, respectively, the wave number and the angular frequency, and c.c. stands for complex conjugate. Substituting Eq.(5) into Eq.(3) yields the following linear dispersion relation ω 2 = ω (l)2 c sin 2 (k/2), ω (l) c = 2ω 0l, (l = 1,2) (6) where the characteristic frequencies of the network ω 0l are given by: [(1 ω 0l = (1 + γ 2)Ω (1 + γ 1)Ω ( 1)l + γ2 )Ω 2 01 (1 + γ 1)Ω02] γ1 γ 2 Ω 2 01 Ω2 02. (7) 2(1 + γ 1 + γ 2 ) Relations (6) explain that there are two elementary waves (modes) which coexist on each line at the same frequency ω but with different wave vectors. The mode corresponding to l = 2 has a higher group velocity compared to the group velocity of the mode l = 1: it s the fast-mode. Accordingly, the mode l = 1 is called slow-mode. Figure 2 shows, for different situations, the linear dispersion of these two modes of propagation. When the two coupled lines have identical linear characteristic parameters, i.e. Ω 01 = Ω 02 Ω 0 and γ 1 = γ 2 γ, the fast mode reduces to the standard mode of propagation of an isolated single line with the characteristic frequency ω 02 = Ω 0, while the characteristic frequency of the slow mode reduces to ω 01 = Ω 0 /(1+2γ) 1/2. The amplitudes of the signal voltage propagating along the two coupled lines are linearly dependent and verify the following relation (obtained by inserting Eq.(5) into Eq.(3)): V (l) 2 = λ (l) V (l) 1, with λ (l) = ( ) 1 + Ω2 01 γ 1 ω0l 2, (l = 1,2), (8) 4

5 where the superscript l stands for the mode of propagation. For the sake of convenience, hereafter we shall use this notation where the case l = 1 stands for the slow-mode while the case l = 2 corresponds to the fast-mode. Due to the fact that the coefficient λ (2) > 0 and λ (1) < 0, the signal voltages in the two lines are always in phase for the fast-mode whereas they are always 180 out of phase for the slow-mode. Furthermore, it appears from (7) and (8) that when the two lines have identical linear characteristic parameters, the amplitudes of the signal voltages on both lines are equal. 3 Coupled NLS equations: fast-and slow-nonlinear modulated waves 3.1 Coupled NLS equations To describe modulated waves in the network, we consider waves with a slowly variation of envelope in time and space with respect to a given carrier with angular frequency ω = ω p and wave vector k = k p. Then, in order to use the reductive perturbation method in the semi-discrete limit [24, 25, 26, 27, 28], we introduce the slow envelope variables x = ǫ(n v g t) and τ = ǫ 2 t where ǫ is a small parameter and v g a constant. Hence, the solution of Eq.(3) is assumed to have the following general form: V j,n (t) = ǫa j (x,τ)e iθ + ǫ 2 [ φ j (x,τ) + B j (x,τ)e 2iθ] + c.c., (9) with θ = k p n ω p t. Substituting Eq.(9) into the coupled equations (3) and keeping terms of order ǫ 3 proportional to e iθ, one obtains the following coupled NLS equations: ia j,τ + iγ j A (3 j),τ + P j,j A j,xx + P j,(3 j) A (3 j),xx + + Q j,j A j A j 2 +Q j,(3 j) A j A 3 j 2 + Q j A j A2 3 j = 0, (10) where the coefficients verify the following relations: Γ j = γ j /(1 + γ j ), P j,(3 j) = Γ j v 2 g/2ω p, (11a) Q j,(3 j) = ω p α j ξ (j) 3 j /(1 + γ j), P j,j = (1 + γ j)vg 2 Ω2 0j cos k p, Q j,j = α jω p 2ω p (1 + γ j ) (1 + γ j ) Qj = ω p α j η (j) 3 j /(1 + γ j), [ ξ (j) j + η (j) j 3β ] j 2α j (11b) (11c) with η (j) j ξ (j) j = α j δ [ Ω 2 03 j v 2 g ω 4 0l (1 + γ 3 j ) ], η (j) 3 j = α 3 j γ j, δ (12a) = 2 α j [ (1 + γ3 j ) Ω 2 03 j g] /v2 (j), ξ 3 j = 2γ j α 3 j, (12b) = [ (1 + γ 1 ) Ω 2 01/vg][ 2 (1 + γ2 ) Ω 2 02/vg] 2 γ1 γ 2, (12c) δ = [ (1 + γ 1 ) Ω 2 01 v2 g /ω4 0l][ (1 + γ2 ) Ω 2 02 v2 g /ω4 0l] γ1 γ 2. (12d) 5

6 Furthermore, the dc and the second harmonic terms φ j (x,τ) and B j (x,τ) are, respectively, related to the fundamental terms A j (x,τ) by: φ j = ξ (j) j A j 2 +ξ (j) 3 j A 3 j 2, B j = η (j) j A 2 j + η(j) 3 j A2 3 j, (13) obtained by keeping terms proportional to ǫ 4 (e iθ ) 0 and ǫ 2 (e iθ ) 2, respectively. The group velocity appearing in Eq.(11) has two distinct analytical expressions since there are two different modes of propagation: v g = dw dk p = ω 0l ( 1 ω 2 p /ω(l)2 c ). (14) Accordingly, Eqs.(10) constitutes two sets of two coupled NLS equations corresponding to the two different modes of propagation of the network. However, the two modes are entirely independent each other. This allows us to consider, separately, each mode of propagation. Let us mention that coupled NLS equations of different forms have been obtained in various domains of physics. It can be used to describe: interactions of solitons in a baroclinic atmosphere [29], the superposition of forward and backward propagating modulated waves in a single electrical transmission line [30], the evolution of two linear polarization components in nonlinear birefringent optical fibers [6], to cite just a few. On the contrary to the above cited studies, in the context of coupled electrical transmission lines, the coupled NLS equations are easily unlashed by means of the linear dependence relation between A (l) 1 and A (l) 2 : A (l) 2 = λ (l) A (l) 1 (15) obtained from the terms proportional to ǫ 2 e iθ in the series expansion of Eq.(2), where λ (l) verifies Eq.(7). Because of this linear dependence of envelopes A (l) 1 and A (l) 2, the two sub-equations of each coupled NLS equations have to be equivalent and lead to the same NLS equation. This condition is satisfied for different frequencies, if and only if the two coupled transmission lines have the same linear and nonlinear characteristic parameters, i.e. Ω 01 = Ω 02 Ω 0, α 1 = α 2 α, and β 1 = β 2 β (16) and then γ 1 = γ 2 = γ. In this limiting case, the signal voltages propagate along the two coupled lines with identical amplitudes and phase for the fast-mode (i.e. A (2) 1 = A (2) 2 ), and with identical amplitudes but with opposite phases for the slow-mode (i.e. A (1) 1 = A (1) 2 ). Setting u(l) = A (l) 1, we obtain the NLS equation describing the propagation of each mode in the network: iu (l) + Pu (l) xx + Qu (l) u (l) 2 = 0, (l = 1,2) (17) where the dispersive and the nonlinear coefficients are, for the slow-mode ( ) Q (1) = α Γω p b (1) 0 + b (1) 2 a, P (1) = ω p /8, (18) with b (1) 0 = 2 [ 1 1/ Γ(1 ωp/ω 2 c (1)2 ) ], b (1) 2 = Γ Γ 1 + ωp/ω 2 c (1)2, (19) 6

7 and for the fast-mode with ( ) Q (2) = αω p b (2) 0 + b (2) 2 a, P (2) = ω p /8, (20) b (2) 0 = 2(1 ω (2)2 c /ω 2 p), b (2) 2 = ω (2)2 c /ω 2 p. (21) The parameters a and Γ denote the nonlinear coefficients of the coupled NLTL and the normalized coupling coefficients between the two lines, respectively, and whose expressions are given by: a = 3β/2α 2, Γ = 1/(1 + 2γ). (22) Note that, in the absence of the coupling between the two lines, i.e. γ = 0 and then Γ = 1, the two modes of propagation reduce to the standard single mode of propagation of modulated waves in an isolated NLTL [30]. 3.2 Fast- and Slow-mode envelope solitons It is well known that the Benjamin-Feir instabilities exhibited by a dispersive nonlinear medium constitute the proof of its capacity to support envelope solitons in certain domains of propagation. From Eq.(17), it is easy to show that, a continuous slowly modulated plane wave should be unstable if P (l) Q (l) > 0. This instability leads to the formation of small wave packets or envelope pulse solitons train, solution of the NLS equation (17) and whose explicit expression is given by [2]: [ u (l) = A 0 sech (x P (l) v e τ)/l (l) s ] [ ] exp iv e (x P (l) v c τ)/2 where v e and v c are respectively the envelope and phase velocities while L (l) s = 2P (l) /Q (l) /(ǫa 0 ) designates the spatial soliton extension. The superscript l can take the values 1 and 2, corresponding to the two modes of propagation of the network, i.e., the fast and slow-mode envelope solitons. According to the above mentioned, our electrical network exhibits the propagation of the fast-mode envelope soliton if P (2) Q (2) > 0, i.e., when the carrier angular frequency ω p belongs to the domain [ω (2) ] with 1,ω(2) c (23) ω (2) c = 2Ω 0, ω (2) 1 = ω (2) c (2 a) 1/2. (24) With the help of relations (9), (20) and (23), the general expressions of the signal voltage, in the two lines, in terms of the fundamental dc and second harmonics can be readily obtained. For the slow-mode envelope soliton, several frequency domains for which P (1) Q (1) > 0 may exist and depend on the numerical values of the following characteristic angular frequencies, ω a, ω b, ω +, and ω defined as follows: ω a = ω c (1) 1/ Γ 1, ω b = ω (1) c 1 Γ, ω ± = ω c (1) Γ a(1 + Γ 2 )/ Γ ± ˆ, (25) 2(2 a)

8 with ˆ = [3 Γ a(1 + Γ 2 )/ Γ] 2 4(1 a)(2 a) and the coefficient ( a 2 3a + 4 ) ± 2 2(2 a)(1 a) γ c1 = (1/ Γ + 1)/2, Γ± = (3 a) 2. (26) Using the numerical values described in Sec.5, the following classification is obtained: for γ ]0,γ c1 ], there are two domains allowing the propagation of envelope soliton: Domain I corresponds to ω p [ω b,ω + ] and Domain II to ω p [ω,ω (1) c ]. For ]γ c1, ], we have only one domain allowing envelope soliton propagation: ω p [ω b,ω (1) c ]. The above expressions achieve the study of modulated nonlinear waves in the basic model, hereafter referred to Model I. The simultaneous presence of the fast- and slow-modes in the network is at the origin of the crucial problem of the wave mixing, since each input wave at the frequency ω p generates in each line two waves with different wave vectors. This situation renders more complex the study of modulated waves in the network and eventually its applications. In the next section we give the answer to this problem. 4 Suppression of the wave mixing: Influence of parallel self 4.1 Model description and NLS equations In this section we modify the coupled electrical network of Figure 1 by introducing, in each unit cell, the self L 2 in parallel with the nonlinear capacitor C j, j = 1,2. Thus, each unit cell of the new network hereafter referred to Model II contains three elements: two self and a nonlinear capacitor as illustrated in Figure 3. The basic linear properties of this unit-cell are analogous to those of the band-pass filter. In the following, we restrict our analysis on the case where the two lines have identical linear and nonlinear characteristics. This restriction is based on the results of the preceding section for which the coupled electrical network exhibits NLS solitons only in this limiting case. Equations governing wave propagation in this nonlinear network are then given by: d 2 V j,n dt 2 γ + Ω 2 0 (2V j,n V j,n 1 V j,n+1 ) + u 2 0 V j,n ( d 2 ) V j,n dt 2 d2 V 3 j,n dt 2 = α d2 Vj,n 2 dt 2 β d2 Vj,n 3 dt 2, n = 1,2,...N, (27) where Ω 2 0 = 1/L 1C 0, u 2 0 = 1/L 2C 0 and γ = C/C 0 are characteristic parameters of the network. We should have also the occasion to use the constant ω c = (u Ω2 0 )1/2. For the linear waves of the network whose dynamics is governed by Eq.(27), we have identified two modes of propagation whose angular frequency and wave number k are described by the dispersion relation of a typical band-pass filter. For the fast-mode, we have: ω 2 = u Ω 2 0 sin 2 (k/2) (28) 8

9 with the upper cut-off frequency ω c (2) = ω c, and for the slow-mode with the upper cut-off angular frequency ω (1) c ω 2 = Γ [ u Ω2 0 sin2 (k/2) ], Γ = 1/(1 + 2γ), (29) = Γ 1/2 ω c. In the following, as in the preceding section, we use the superscript l = 1 to designate the slow-mode and the superscript l = 2 refers to the fast-mode. Similarly, the subscript j = 1 designates line 1 while j = 2 stands for the second line. For the fast-mode propagation, the waves on the two coupled lines are in phase and have equal amplitudes while for the slow-mode propagation they are in opposite phase and propagate with equal amplitudes. Figure 4 displays the dispersion relation of the two modes of propagation for the coupling coefficient γ = 8. It follows that, when the value of the coupling coefficient exceeds the critical value γ cr = 2Ω 2 0 /u2 0, (30) the band-pass of the two modes of propagation are entirely separated. As a consequence, each input signal at frequency ω generates one and only one mode of propagation in each line. In addition, it is also possible to create the gap between the linear spectrums of the two modes of propagation. As we shall see below this result is a first step toward the suppression of the slowand fast- nonlinear modulated wave mixing. Let us come to nonlinear excitations in the network. In order to describe the slowly modulated nonlinear waves, we use the same reductive perturbation method in the semi-discrete limit as presented in Sec.3. Thus the insertion of Eq.(9) into the coupled differential equation (27) leads to the following standard NLS equation for the envelope evolution: iu (l) + Pu (l) xx + Qu (l) u (l) 2 = 0, (31) where the variable u (l) is related to A (l) 1 and A (l) 2 as follows: u (l) = A (l) 1 = ( 1) (l) A (l) 2. The dispersion coefficient of the NLS equation is given by: ( ) ( ) ω (l)2 P (l) cr ωp 2 ω cr (l)2 + ωp 2 = (32) 8ω 3 p with ω (1) cr = ( Γu 0 ω c ) 1/2, ω (2) cr = (u 0 ω c ) 1/2, (33) while the nonlinear coefficients are given by with a defined by Eq.(22) and b (1) = Q (1) = Γα 2 ω p (a b (1) ), Q (2) = α 2 ω p (a b (2) ) (34) 4Ω 2 0 ω2 p Ω 2 0 [3u2 0 4(1 Γ)ω 2 p / Γ] + (u 2 0 ω2 p / Γ) 2, b(2) = 9 4Ω 2 0 ω2 p 3u 2 0 Ω2 0 + (u2 0 ω2 2, (35) p)

10 The functions B 1 (x,τ) and φ 1 (x,τ) are related to u (l) as B (l) 1 (x,τ) = B(l) 2 (x,τ) = αb(l) u (l)2 (x,τ) while the dc terms vanish due to the existence of low-frequency forbidden band, i.e. φ (l) 1 (x,τ) = φ (l) 2 (x,τ) = Fast-mode envelope soliton As pointed out above, the existence of the envelope soliton in the network is conditioned by the positive value of the product P (2) Q (2). Thus, using Eqs.(32) and (34) we are able to predict the frequency range of the existence of the fast-mode envelope soliton. Accordingly, the fast-mode should propagate on the network if the angular frequency of the carrier wave lies within the interval [ω (2) ]. The width of the band-pass of the fast-mode envelope soliton is then given by: cr,ω c (2) ω (2) = ω c ( 1 u 0 /ω c ) By comparing the band-pass of the fast-mode envelope soliton for the two models of coupled electrical transmission lines (Model I and Model II), we note that the band-pass of soliton in Model II is larger than that of the Model I. Consequently, the introduction of the self in parallel with the nonlinear capacitor enhances the frequency range of the envelope soliton propagation. Note also that the coupling coefficient has no effect on the characteristic parameters of the fast-mode soliton. The explicit expression of the signal voltage corresponding to the fast mode envelope soliton can be easily obtained by means of the relations (9), (28) and (35). 4.3 Slow-mode envelope soliton In this regime of propagation, the product P (1) Q (1) takes positive values in two domains of frequencies depending on the numerical values of ω (1) c and ω (1) cr, and also on the following characteristic frequencies ω a, ω b, ω +, and ω, defined by: ω 2 a = Γu 2 0 [ 1 + 2(Ω 2 0/u 2 0)(1 Γ) δ 1/2], ωa 2 = Γu [ (Ω 2 0/u 2 0)(1 Γ) ˆδ 1/2], (37) (36) and ω 2 ± = Γu 2 0 [ 1 + 2(Ω 2 0/u 2 0)(1 Γ + Γ/a)± ˆ 1/2], (38) where [ ˆδ = Γ)] Ω2 0 u 2 (1 0 ( ) [ Ω2 0 u 2, ˆ = Ω2 0 u 2 (1 Γ + Γ ] 2 ( 0 a ) Ω2 0 u 2 0 ). (39) It depends also on the normalized coupling coefficient Γ ± = ] [Ω (Ω 20 2 /u20 ± 20 /u20 )(3 + Ω20 /u20 ), (40) from which we define the parameter γ c1 = (1/Γ 1)/2. Using the numerical values of the preceding section in addition with L 2 = 0.470mH, we obtain the following hierarchy: 10

11 For γ ]0,γ c1 ], we have one domain of propagation: ω p [ω (1) cr,ω (1) c ]. For γ [γ c1,γ cr ], we have two domains of propagation: ω p [ω a,ω b ] and [ω (1) cr,ω (1) c ]. For γ >> γ cr, there are also one domain of propagation: [ω 0,ω (1) cr ]. On the contrary to the fast-mode envelope soliton, the existence of the slow-mode is strongly dependent on the combined effects of the adding self L 2 and the coupling capacitor. This dependence allows to manage in our convenience the frequency range where the slow-mode soliton can propagate. For example for γ > γ cr, the network exhibits slow-mode envelope soliton out of the frequency-band of the fast-mode. Consequently, the two modes are completely separated and the wave mixing is then avoided. Apart the envelope soliton, the network described in this section may exhibit the hole soliton (fast-mode and slow-mode), in the frequency domain where the product P (l) Q (l) has negative values. Its explicit expression could be found by mean of the hole soliton solution of the NLS equation (31) for different modes of propagation. To conclude this section, we point out that, the merits of the introduction of the additional self in parallel with the nonlinear capacitor in each unit cell of the network results in (i) the creation of the possibility to dissociate the two modes of propagation (fast and slow-mode) by a convenient choice of the coupling capacitance of the two electrical transmission lines and then to avoid the wave mixing in the network; (ii) the enlargement of the band-pass width of the fast-mode envelope soliton. 5 Numerical experiments According to the analytical calculations presented in the preceding sections, it is possible to determine in the spectrum of the coupled NLTL the frequency range for which the network supports the propagation of envelope solitons. In order to check the validity of these predictions, we first present the numerical experiments on the propagation of slowly modulated waves in the network since their unstable propagation for certain parameter regime allows us to conclude about the possibility of the network to support envelope soliton in this frequency domain and then experiments on the fast- and slow-envelope solitons propagation are presented in Model II in order to verify the achievement of the suppression of the wave mixing. 5.1 Slowly modulated plane waves: Modulational Instability Basic Model: Model I The numerical experiments are carried out on the exact equation (3) describing the propagation of waves in the two coupled electrical transmission lines of Figure 1. The parameters of the network are chosen to be L = L 1 = L 2 = 0.220mH, C 0 = C 01 = C 02 = 320pF and γ = 1/2 which implies the cut-off frequency f c (1) = 848kHz for the slow-mode and f c (2) = 1200kHz 11

12 for the fast-mode of propagation. The characteristic parameters of the reversed biased diode are: α 1 = α 2 = α = 0.21V 1 and β 1 = β 2 = β = 0.197V 2 for a constant biased voltage V b1 = V b2 = V b = 2V [26, 27, 28]. The fourth-order Runge-Kutta scheme is used with normalized integration time steep t = corresponding to the sampling period T s = s. Similarly, the number of cells is variable in order to avoid wave reflection at the end of the line and also to run the experiments with sufficiently large time (here t max = 4ms). The input of line 1 (cell n = 0) is supplied by a slowly modulated signal of the form V 1,0 (t) = V m [1 + m cos(2πf m t)] cos(2πf p t) (41) where V m is the amplitude of the unperturbed plane wave (carrier wave) with frequency f p, and where m and f m are respectively the rate and the frequency of the modulation. The input of line 2 is matched by a resistor with variable resistance. The experiments have been made over the whole carrier wave frequency range, i.e f p < f c (2) and for different values of the modulation frequency less than 20kHz. The result is sketched in this table: Mode Modulational Instability Analytical Predictions (khz) Numerical Results(kHz) Slow-mode Mixing of waves Fast-mode Figure 5 shows an example of the MI exhibited by the network where the corresponding mode of propagation is the fast-mode. The input signal parameters are V m = 0.5V, f p = 1180kHz, f m = 4.8kHz and m = This sine wave with a small modulation applied at n = 0 on line 1, exhibits some nonlinear distortions of the envelope as one can easily observe in the Figure 5.1 at cell number 1500 of both lines. As time goes on and the wave propagates along the network, the modulation increases and the sine wave breaks into a periodic envelope pulse train. It is a typical example of modulational instability phenomenon. It is also found that the amplitude of the resulting sine wave is equal to 0.5V (see Figure 5.2: at cell 20 of lines 1 and 2), the algebraic sum of the amplitudes of each sine wave in the two lines which would be 0.25V. This result is in accordance with the analytical treatment of Sec.2 for linear waves. However when instability occurs in the system, the amplitude of the resulting wave packet is larger than the amplitude of the input wave 0.5V, the envelope pulse wave in each line behaves with the amplitude slightly equal to 0.6V (see Figures 5.2 c1 and c2). Let us mention that, it is not possible to observe, separately, the propagation of the slow-mode of modulated waves because of the simultaneous presence of the fast-mode Suppression of the fast and slow modulated waves mixing: Model II The numerical experiments are carried out on the exact equation (41) describing the propagation of waves in the two identical coupled electrical transmission lines of Figure 3. The 12

13 parameters of the network are chosen to be L 1 = 0.220mH and L 2 = 0.470mH which implies the critical coupling coefficient γ cr = The characteristic parameters of the reversed biased diode are the same as in Model I. The corresponding cut-off frequencies are f c (2) = 1268kHz and f (2) 0 = ω 0 /2π = 411kHz for the fast-mode of propagation. As in the preceding paragraph, the fourth-order Runge-Kutta scheme is used with the same normalized integration time steep and with the same number of cells. Similarly, at the input of the line 1, we apply a slowly modulated signal (41). Two particular cases are investigated according to the magnitude of the coupling normalized coefficient. For the first case γ = 8 corresponding to these two cut-off frequencies of the slow-mode f c (1) = 307kHz and f (1) 0 = Γ 1/2 u 0 /2π = 100kHz, while for the second case, γ = 40 and the cut-off frequencies are f c (1) = 141kHz and f (1) 0 = Γ 1/2 u 0 /2π = 46kHz. The theoretical predictions and the corresponding numerical results are shown in the following tables: For γ = 8 Mode Modulational Instability Analytical Predictions (khz) Numerical Results (khz) Slow-mode and and Fast-mode For γ = 40 Mode Modulational Instability Analytical Predictions (khz) Numerical Results (khz) Slow-mode Fast-mode Figure 6 shows an example of MI developed by the network with coupling coefficient γ = 8 by a slowly modulated wave with carrier frequency f p = 150kHz and initial amplitude V m = 0.8V. Frequency modulation and modulation rate are: f m = 3.47kHz and m = 0.01, respectively. In this domain of frequency only the slow-mode can propagate. It appears that, the mechanism of development of this instability is different to the well-known mechanism of MI described by Benjamin-Feir [31] and presented in Figure 5 for the fast-mode. 5.2 Envelope solitons In the preceding paragraph, we have shown that, in certain range of frequency, if a sine wave applied at one end of the coupled lines is slowly modulated, we may expect modulation growth and formation of wave packets which propagates along the network. This transformation of sine wave into envelope soliton is interpreted by the fact that, in this domain of frequencies, the modulated plane wave is not solution of the wave equations (3). When this wave is applied as the input voltage, as the wave propagates along the network, the nonlinear system improves their profiles and adapts them to the exact profile solution of the wave equations. In this paragraph we show that this exact profile is given by the solution of the NLS equation (31). For this 13

14 purpose, we take as the input voltage, the profile of a modulated soliton given by [ ] V (l) 1,0 (t) = V msech v g (l) t/l (l) s cos (2πf p t) (42) where f p and v g (l) are the carrier frequency and the group velocity of the wave packet, respectively, and where L (l) s is the soliton width defined in (23). The upper-script (l) stands for the two modes of propagation. The parameters of the signal voltage are f p = 1180kHz belonging to the domain of MI and to the frequency range of the fast-mode, and V m = 2ǫA 0 = 0.4V. With these values, the group velocity is v g (2) = 1367cells/ms and the soliton width L (2) s = 15cells. However, since the amplitude of the signal generated in each line is the initial one divided by 2, we take L (2) s = 30cells which is the best estimation of the soliton generated in the network. Figure 7 shows the propagation of this fast-modulated soliton along the line 1. The same behavior (not presented) is observed on line 2. The phase plane plot diagnostic sketched in Figure 9(a) is 45 o line confirming the fact that the waves on the two lines are in phase. In addition, this result confirms the fact that the modulated soliton generated in the network corresponds to the fast-mode. When we consider the following numerical values for the parameter of the signal voltage, f p = 150kHz, V m = 2ǫA 0 = 0.4V and γ = 8, implying that v (1) g = 631cells/ms and L (1) s = 58cells, we obtain the picture of Figure 8 and the diagnostic phase plane plot of Figure 9(b) which produces the 45 o line, which evidences the propagation of the slow-modulated soliton. These numerical experiments confirm the fact that the use of the coupled NLTL described by model II allows to avoid the wave mixing due to the existence of two modes of propagation in the network. 6 Conclusion In this paper, we have investigated the dynamics of modulated waves in two coupled discrete nonlinear electrical transmission lines (NLTL). More precisely, we have first shown that, in the limiting case where the two coupled lines are identical, the network may support the propagation of envelope solitons. The modes of propagation of these modulated solitons have been detected; the fast- and the slow-mode. The fast mode corresponds to the mode of propagation of a single isolated line while the slow mode results in the coupling between the two lines. However, the simultaneous presence of these two modes in each line of the coupled discrete NLTL is at the origin of the undesirable wave mixing effects. We have shown that, the use of the coupled discrete NLTL in which the unit-cells have the characteristics of the band-pass filter allows to avoid any wave mixing. Next, the analytical studies are completed by the numerical experiments preformed in the network. The obtained results confirm the validity of the analytical approach. In fact, to each 14

15 input signal in either line of the network, one and only one mode of propagation is created in each line; the nature of the generated mode (fast or slow) depends on the magnitude of the carrier frequency. Finally, we mention that, while this study is crucial for the best understanding and interpretation of experimental results of modulated waves in the coupled discrete NLTL, it can also be viewed as the first step toward the study of modulated waves interactions on two-dimensional discrete NLTL where the low-frequency regime case has been issued by Dinkel et al. [32]. This work is now under consideration. Acknowledgments. David Yemélé is grateful to the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy, for hospitality, where part of this work was done during his visit under the Associate Federation Scheme and to the Swedish International Developing Cooperation Agency (SIDA) for financial support. He also thanks Pr. Jean Marie Bilbault and Pr. Patrick Marquié for helpful discussions and for the critical reading of the manuscript. References [1] A. C. Scott, Nonlinear Science: Emergency Dynamics of Coherent Structures (Oxford University Press, New York, 1999). [2] A. Hasegawa, Optical Solitons in Fibers, Second Enlarged Edition, (Springer-Verlag, Berlin, 1989). [3] P. Tchofo Dinda, G. Millot, E. Seve and M. Haelterman, Opt. Lett. 21, 1640 (1996). [4] E. Seve, P. Tchofo Dinda, G. Millot, M. Remoissenet, J.M. Bilbault and M. Haelterman, Phys. Rev. A 54, 3519 (1996). [5] P. Tchofo Dinda, G. Millot, and S. Wabnitz, Opt. Lett. 22, 1595 (1997). [6] D.C. Hutchings and J.M. Arnold, J. Opt. Soc. Am. B. 16, 513 (1999). [7] M. Remoissenet, Waves Called Solitons, (3rd ed. Springer-Verlag, Berlin 1999). [8] A. C. Scott, Active and Nonlinear Wave Propagation in Electronics,(Wiley-Interscience, New York, 1970). [9] K. E. Lonngren, Solitons in Action, ed. by K. E. Lonngren and A. C. Scott (Academic, New York, 1978). [10] R. Hirota and K. Suzuki: J. Phys. Soc. Jpn. 28, 1366(1970); Proc. IEEE 61, 1483 (1973). [11] H. Nagashima and Y. Amagishi, J. Phys. Soc. Jpn. 45, 680 (1978). [12] M. Toda, J. Phys. Soc. Jpn. 23, 501 (1967). 15

16 [13] A. Noguchi, Electron. Commun. Jpn. 57-A, 9 (1974). [14] J. A. Kolosick, D. K. Landt, H. C. Hsuan and K. E. Lonngren, Proc. IEEE 62, 578 (1974). [15] T. Yagi and A. Noguchi, Electron. Commun. Jpn. 59A, 1 (1976). [16] J. Sakai and T. Kawata, J. Phys. Soc. Jpn. 41, 1819 (1976). [17] P. Marquié, J.M. Bilbault and M. Remoissenet, Phys. Rev. E 49, 828 (1994). [18] P. Marquié, J.M. Bilbault and M. Remoissenet, Phys. Rev. E 51, 6127 (1995). [19] T. Kakutani and N. Yamasaki, J. Phys. Soc. Jpn. 45, 674 (1978). [20] T. Yoshinaga and T. Kakutani, J. Phys. Soc. Jpn. 49, 2072 (1980). [21] T. Yoshinaga and T. Kakutani, J. Phys. Soc. Jpn. 56, 3447 (1987). [22] B. Z. Essimbi, A. M. Dikand, T. C. Kofané and A. A. Zibi, J. Phys. Soc. Jpn. 64, 2361 (1995). [23] B. Z. Essimbi, L. Ambassa, and T. C. Kofané, J. Physica D 106, 207 (1997). [24] T. Taniuti and N. Yajima, J. Math. Phys. 10, 1369 (1969). [25] E. J. Parkes, J. Phys. A, Math. Gen. 20, 2025 (1987). [26] D. Yemélé, P. Marquié and J. M. Bilbault, Phys. Rev. E 68, (2003). [27] D. Yemélé and P. Marquié Chaos, Solitons Fractals 15, 465(2003). [28] D. Yemélé, P. K. Talla and T. C. Kofané, J. Phys. D: Appl. Phys. 36, 1429 (2003). [29] B. Tan and S. Liu, J. atmos. Sci. 52, 1501 (1995). [30] P. Marquié and J.M. Bilbault, Phys. Lett. A 174, 250 (1993). [31] T. B. Benjamin and J. E. Feir, J. Fluid Mech. 27, 417 (1967). [32] J. N. Dinkel, C. Setzer, S. Rawal, and K. E. Lonngren, Chaos, Solitons Fractals 12, 91 (2001). 16

17 Figure Captions I). Figure 1: Schematic representation of the coupled nonlinear LC transmission lines (Model Figure 2: Linear dispersion curves of the two coupled LC transmission lines for the fastmode (1) and slow-mode (2) of propagation, for two particular cases: a) The two coupled lines are different: C = 160pF, C 01 = C 02 = 320pF, L 1 = 0.220mH and L 2 = 0.470mH. curves (3) and (4) are the dispersion curves of the two lines taking separately. It appears that the coupling capacitance lower the cut-off frequency. b) The two coupled lines are identical: C = 160pF, C 01 = C 02 = 320pF, L 1 = L 2 = 0.220mH. In this case, the fast-mode reduces to the mode of propagation of a single line. Each curve is divided in two regions concerning the modulational instability (MI) described by the NLS equation. Figure 3: Schematic representation of the unit cell of two coupled LC transmission lines with an additional linear inductor in parallel with the nonlinear capacitor (Model II). Figure 4: Linear dispersion curves of the network (Model II): a) For the two modes of propagation; fast-mode (2) and slow-mode (1). b) Zoom on the dispersion curve of the slow-mode. The coupling coefficient is taken to be γ = 8 > γ cr. One note that the intersection between the curves of the two modes observed in Model I has disappeared, indicating the entire separation of the two modes of propagation on the network. Figure 5: Example of MI in the coupled network. The initial condition corresponding to the input wave is given by Eq.(41) with carrier frequency f p = 1180kHz and initial amplitude V m = 0.5V. The frequency modulation and modulation rate are, f m = 4.81kHz and m = 0.01, respectively. 5.1 Signal voltage (in Volts) in the network as a function of cell number n, (a) for line 1 and (b) for line 2, at given arbitrary times t 1, t 2 and t 3 (with t 1 < t 2 < t 3 ), respectively. At this carrier frequency, the fast-mode only is generated by the network. As time goes on the wave exhibits a modulation of its amplitude and phase, on the two lines, which lead to the formation of wave packets. 5.2 Signal voltage (in Volts) as a function of time (the time unit is t 0 = 0.6ms) at different cells (20, 1500 and 2250). Figure 6: MI exhibits by the propagation of the slow-mode in the Network. The input wave 17

18 is given by Eq.(41) with carrier frequency f p = 150kHz belonging to the MI domain and initial amplitude V m = 0.8V, frequency modulation f m = 4.81kHz and modulation rate m = Signal voltage (in Volts) in the network as a function of cell number n, (a) for line 1 and (b) for line 2, at given arbitrary times t 1, t 2 and t 3 (with t 1 < t 2 < t 3 ), respectively. The plane wave on the two lines exhibits a MI. 6.2 Signal voltage (in Volts) as a function of time and at different cells (100, 500), showing also the MI. Figure 7: Propagation of fast-mode envelope soliton on the network in line 1. Signal voltage (in Volts), (a) at a given cell as a function of time (in ms), (b): at a given time (t 1 = 1.50ms, t 2 = 2.29ms and t 3 = 3.06ms) as a function of cell number n. The input signal is given by Eq.(42) with carrier frequency f p = 1180kHz belonging to the MI domain and initial amplitude V m = 0.4V. Figure 8: Propagation of slow-mode envelope soliton on the network in line 1. Signal voltage (in Volts), (a) at a given cell as a function of time (in ms), (b): at a given time (t 1 = 1.50ms, t 2 = 2.29ms and t 3 = 3.06ms) as a function of cell number n. The input signal is given by Eq.(42) with carrier frequency f p = 150kHz belonging to the MI domain and initial amplitude V m = 0.4V and γ = 8. Figure 9: Signal voltage at a given cell in line 1 as a function of the signal voltage of the same cell in line 2,(a) fast-mode and (b) slow-mode, with the initial conditions described in Figure 7 and 8 for the fast and slow mode, respectively. 18

19 Figure 1 19

20 Figure 2 20

21 Figure 3 21

22 Figure 4 22

23 Figure

24 Figure

25 Figure

26 Figure

27 Figure 7 27

28 Figure 8 28

29 Figure 9 29