Modulated harmonic wave in series connected discrete Josephson transmission line: the discrete calculus approach

TL;DR

Problem: modulation of harmonic waves in a discrete Josephson transmission line (JTL). Approach: develop a discrete-calculus (DC) framework for modulation of discrete wave equations and validate it on the FPUT problem before applying to the JTL. The main result is that the modulation amplitude in the discrete JTL is governed by the defocusing nonlinear Schrödinger equation, $i(\partial a/\partial τ + v_g ∂a/∂x) - D ∂^2 a/∂x^2 + 2D |a|^2 a = 0$ (or in a moving frame, $i\partial a/\partial τ' - \partial^2 a/\partial ξ^2 + 2|a|^2 a = 0$). The work also derives a dark-soliton solution and cross-validates the results with a Fourier-integral approach in the Appendix and with FPUT results, highlighting the universality and applicability of NLS as an envelope model.

Abstract

We consider the modulated harmonic wave in the discrete series connected Josephson transmission line (JTL). We formulate the approach to the modulation problems for discrete wave equations based on discrete calculus. We check up the approach by applying it to the Fermi-Pasta-Ulam-Tsingou type problem. Applying the approach to the discrete JTL, we obtain the equation describing the modulation amplitude, which turns out to be the defocusing nonlinear Schrödinger (NLS) equation. We compare the profile of the single soliton solution of the NLS with that of the soliton obtained in our previous publication.

Figures (3)

• Figure 1: Discrete JTL.
• Figure 2: Dark soliton as given by Eq. (\ref{['rho']}. We have chosen $A_0=.5A_1$.
• Figure 3: The soliton profile calculated in Ref. kogan. ($\Lambda$ is the period of the JTL, which in the present publication was chosen to be equal to one.)