PHYSICAL REVIEW B VOLUME 41, NUMBER 16 1 JUNE 1990 Emission of waves by moving kinks in a spatially modulated sine-Gordon system Boris A. Malomed P. P. Shirshov Institute for Oceanology of the U. S. S. R. Academy of Sciences, 23 Krasikov Street, Moscow, 117218, Union of Soviet Socialist Republics Michael I. Tribelsky The Moscow Scientific and Production Association NIOPIK, 1-4 B. Sadovaya Street, Moscow, 103787, Union of Soviet Socialist Republics (Received 28 August 1989) Emission of radiation (quasilinear waves) by a periodic array of moving kinks is investigated within the framework of a driven damped sine-Gordon model with the additional perturbing term eF(x)sing. At first, radiative energy losses of a solitary kink are studied in detail; new emission- induced steps on the one-kink velocity-drive characteristic (VDC) are predicted. Next, interference of waves emitted by kinks belonging to a rarefied periodic array is considered. Emission-induced steps on the array's VDC are described in detail. Emission of radiation by a densely packed array (represented by a nearly linear solution of the SG equation) is also considered, and the correspond- ing steps are described for random and periodic inhomogeneities. I. INTRODUCTION The sine-Gordon (SG) model finds many important ap- plications in quasi-one-dimensional solid-state physics (see, e. g. , the recent review paper'). In many cases, the corresponding physical systems are subject to periodic spatial modulation. The modulation gives rise to the per- turbed SG model i.e. , a magnetic Aux quantum. In commensurable charge-density-wave (CDW) systems and in weak or easy-plane ferromagnets (or ferroelectrics), which are other popular solid-state realizations of the SG model, the kink represents, respectively, a charged soliton or a domain wall. A realistic SG model should also include a dissipative term and a drive which makes the kink move: P« — P„„+ sing = P, (b« — P„„+sing+a/, + f =P, (4) where the perturbation P may be a combination of the following three terms: P, =a g (x)sing, P2 = ezF (x)P„„, P3 = e3F(x)p«, (2b) (2c) F(x) being a periodic function. To apply the perturba- tion theory, we wi11 assume a11 the parameters e sma11. In particular, the model (1), (2) describes a long Josephson junction (LJJ) with the thickness of a dielectric layer sub- ject to a weak periodic modulation. In that case, p ts the magnetic ffux in the junction, and the terms (2a) — (2c) take account of, respectively, variation of the maximum Josephson current density, inductance, and capacity of' the junction. In other physical applications of the SG model (a sufficiently comprehensive list can be found in Ref. 1), the same perturbing terms (2a) — (2c) are as well physically meaningfu1. The simplest soliton solution of the unperturbed SG equation is the well-known kink P» =4 tan 'exp I o [x — ((t)]/( I — V2)'~2 j, where cr =+1, g(t)= Vt, and V are the kink's polarity, center-of-mass coordinate, and velocity, respectively. In the theory of LJJ's, the kink represents a so-called Auxoo, a being a dissipative constant, f being the drive. In LJJ s and in the CDW systems, f accounts for dc bias current or dc voltage, respectively. Motion of a kink in the inhomogeneous system is ac- companied by emission of radiation (i.e. , plasma waves in an LJJ, spin waves in ferromagnets and ferroelectrics, and so on). The corresponding radiative energy losses add to the usual dissipative losses, thus aftecting a depen- dence of the kink's velocity V upon the drive f. This dependence is a basic dynamical characteristic of a corre- sponding physical system. For instance, it directly gives a current-voltage characteristic of the corresponding LJJ or CDW system. In the former case, Vis proportional to voltage across the junction, while f is the bias current density; on the contrary, in the latter case f is propor- tional to the voltage, while the electric current carried by the charged solitons is proportional to V. A solitary kink is a rather idealized entity. From the physical standpoint, a more interesting object is a period- ic array of kinks with some spacing I, . As it has been predicted theoretically and revealed experimentally in LJJ s in Ref. 5, radiation emitted by separate kinks be- longing to the array may undergo resonant interference. The constructive interference (superradiance) gives rise to an abrupt increase of an effective radiative braking force acting upon the kinks. This is observed in the form of new well-discernible steps on a current-voltage charac- 1990 The American Physical Society