Modulational instability criteria for coupled nonlinear transmission lines with dispersive elements

E. Kengne, S. T. Chui, and W. M. Liu

Phys. Rev. E 74, 036614 – Published 18 September 2006

Abstract

We investigate the modulational instability of Stokes wave solutions on a system of coupled nonlinear electrical transmission lines with dispersive elements. In the continuum limit, and in suitable scaled coordinates, the voltage on the system is described by the two-dimensional coupled nonlinear Schrödinger equations. The set of coupled nonlinear Schrödinger equations obtained is analyzed via a perturbation approach. No assumption is made on the signs of the relevant coefficients such as the coefficients of nonlinearity and the coupling coefficients. A set of explicit criteria of modulational stability and modulational instability is derived and analyzed. It is numerically shown that the effect of the dispersive elements in the line is to decrease the instability region and the instability growth rate.

Received 6 June 2006

DOI:https://doi.org/10.1103/PhysRevE.74.036614

Authors & Affiliations

E. Kengne^1,2,*, S. T. Chui³, and W. M. Liu¹

¹Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
²Department of Mathematics & Computer Science, Faculty of Science, University of Dschang, P. O. Box 4509, Douala, Republic of Cameroon
³Bartol Research Institute, University of Delaware, Newark, Delaware 19716, USA

^*Email address: ekengne6@yahoo.fr

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Vol. 74, Iss. 3 — September 2006

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Images

Figure 1
Part of the system of the nonlinear dispersive transmission line.Reuse & Permissions
Figure 2
Part of the system of the nonlinear dispersive transmission line coupled by a capacitor $C_{2}$ .Reuse & Permissions
Figure 3
Linear dispersion curve defined by the dispersion relations in the special case when $k_{B}^{X_{0}} = - k_{A}^{X_{0}} = k$ and $k_{B}^{Y_{0}} = - k_{A}^{Y_{0}} = - 0.2$ for $k ∊ [0.12844, 0.2]$ . Here we take $L = 200 μ H$ , $C_{0} = 370 pF$ , $C_{2} = 100 pF$ , $C_{S} = 0 pF$ [for curve (a)] and $C_{S} = 50 pF$ [for curve (b)]. Comparing the two figures, we see that the gap zone is larger in the presence of the linear capacitance $C_{S}$ .Reuse & Permissions
Figure 4
(Color online) Dispersion coefficients in terms of the wave number $k$ taken from condition (6). The black curves are the $M_{1}$ curve, while the red curves are the $M_{2}$ curve. The dashed lines correspond to $C_{S} = 0$ and the solid lines correspond to $C_{S} = 50 pF$ , $0.128 44 \leq k \leq 1.5$ . The parameter lines are the same as above. These curves show that the dispersive coefficients $M_{j}$ take both positive and negative values and inform that the dispersion coefficient $M_{1}$ $(M_{2})$ decreases (increases) with the introduction of the linear constant capacitance $C_{S}$ in the line.Reuse & Permissions
Figure 5
(Color online) The perturbation square frequency $Ω_{k -}^{2}$ (a) and the typical structure of the modulational instability growth rate $Γ$ (b) as a functions of squared modulation wave number $k^{2}$ in the special case when $k_{B}^{X_{0}} = - k_{A}^{X_{0}} = - 0.4$ , $k_{B}^{Y_{0}} = - k_{A}^{Y_{0}} = - 0.2$ , and the line parameters $L = 200 μ H$ , $V_{0} = 2 V$ , and $C_{0} = 370 pF$ . (a) The dotted black curve, the solid black curve, and the solid red curve correspond to $C_{S} = 0$ , 50, and $100 pF$ , respectively. (b) Curves $a$ , $b$ , and $c$ correspond to $C_{S} = 0$ , 50, and $100 pF$ , respectively. The region of the modulational instability and the values of the modulational instability growth rate $Γ$ decrease with the introduction of the linear capacitance $C_{S} = 0$ .Reuse & Permissions
Figure 6
(Color online) Field amplitudes of perturbed Stokes wave solutions for different time and for $ε = 0.01$ . The red curve is the Stokes wave amplitude $∣ ψ_{j 0} ∣$ , the solid black curve gives the amplitudes $∣ ψ_{j} (x, y, τ) ∣$ of the perturbed solution at the initial time $τ = 0$ , and the dotted black curve corresponds to the amplitude of the perturbed solution at a given time $τ > 0$ $(τ = 63 \times 10^{9}$ for curves $a$ and $b$ corresponding to $C_{S} = 0$ , $τ = 96 \times 10^{9}$ for curves $c$ and $d$ that correspond to $C_{S} = 50 pF$ , and $τ = 131 \times 10^{9}$ for curves $e$ and $f$ for $C_{S} = 100 pF)$ . The onset of instability occurs at $τ \approx 96 \times 10^{9}$ , or $τ \approx 131 \times 10^{9}$ , or $τ \approx 131 \times 10^{9}$ for $C_{S} = 0$ or $C_{S} = 50 pF$ , or $C_{S} = 100 pF$ , respectively.Reuse & Permissions

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