The analysis on the single particle model of CDW

Abstract

Abstract. Grüner put forward a single particle model of charge-density wave, which is a typical nonlinear differential equation, and also a mathematical model of pendulum. This Letter analyzes the solution of equation by the rotated vector fields theory, providing the relation between the applied field E and the periodic solution, and the conclusion that the critical value of E for the periodic solution is fixed in the over-damped situation. With these conclusions, it derives the formulae of nonlinear conductivity, narrow-band noise, which are consistent with the empirical ones given by Fleming.

The Analysis of Rotated Vector Field for the Pendulum

“…By the idea of rotated vector fields, this paper proves the relation between β and the periodic solution of the equation, and find that if the Γ is bigger enough (over damping), the critical value of β, with which this equation will have a critical periodic solution, will be a fixed value all the time, namely, β 0 ≡ 1. These conclusions serve as essential roles in [11].…”

“…These conclusions above are employed in the research of CDW. The derivations in detail was presented in other paper [11]. It is unnecessary to go into details in this paper.…”

“…Although, an article gave the description for its solutions with the analysis of vector fields rigorously [6], it is merely an scenario, considering β as fixed values, to describe the behavior of its solutions with the Γ changing. By the idea of rotated vector fields, it is the purpose of this E-mail addresses: liliank@tju.edu.cn paper, to elucidate the behavior of its solutions with the β changing, which just meets the needs for clarifying the mechanism of CDW in [11].…”

“…Let Γ = 0, β = 0, thus equation (2) degenerates into a integrable system, sine-Gordon's Equation, namely, d 2 φ dt 2 + sin φ = 0. It is easy to give a solution, z = 2(cos φ + 1) (11). As shown in Figure2, it corresponds a trajectory L 0 connecting the saddle A 0 with A 1 .…”