Physics LettersA 159 (1991) 318—322 North-Holland PHYSICS LETTERS A Kink capture by a local impurity in the sine-Gordon model Fei Zhang, Yuri S. Kivshar ~, Boris A. Malomed 2 and Luis Vázquez Departamento de Fisica Teórica I, Facultad de Ciencias FIsicas, Universidad Complutense, E-28040 Madrid, Spain Received 16 June 1991; accepted for publication 14 August 1991 Communicated by AR. Bishop We demonstrate analytically and numerically that the impurity mode plays a very important role in the kink scattering by a local inhomogeneity, and a sine-Gordon kink may be trapped by an impurity due to loss of its kinetic energy to excite the impurity mode. We develop a collective-coordinate approach to describe the kink—impurity interactions and calculate the threshold veloc- ity of kink capture. Our numerical simulation results are in good agreement with the theoretical prediction. 1. Introduction approach, the only inelastic effect is emission of lin- ear waves (radiative losses). Further, it was shown It is well known that nonlinearity may drastically by Malomed [10,11] that a sine-Gordon (sG) kink change transport properties of disordered systems may be trapped by an attractive impurity due to ra- when it contributes to create soliton pulses [1,2]. As diative losses and the threshold velocity was found a first step to understand the soliton transmission analytically. However, the previous theoretical stud- through disordered media, one has to study the ~ ies [7—11]totally ignored the fact that the under- liton scattering by a single impurity. The study of the lying nonlinear system supports a localized impurity so-called dynamical solitons has been carried out in mode which may be excited due to the kink scatter- a number of papers (see, e.g., refs. [3—6]) taking into ing. The spectrum of the linearized sG model has a account one or two impurities. In particular numer- gap, so that the impurity modes in a linear system ical and experimental investigations have shown that are permitted either below or above the frequency due to the interaction with one impurity the ampli- band, the latter is possible for a discrete model when tude of the soliton decreases a little, but the soliton the cut-off frequency exists. In the nonlinear case the can pass the impurity almost freely. In such an in- impurity modes may be treated as breather excita- teraction, the soliton loses its energy to generate a tions, which are trapped by the impurities [12]. reflected wave, and in the case of a light-mass im- The importance of the impurity mode in the kink purity, to excite a localized mode in the vicinity of scattering was firstly observed in numerical simu- the impurity site, the so-called impurity mode [3—5]. lations of the 0” chain [13]. It has been demon- On the other hand, the kink—impurity interactions strated that an incident kink may excite the impurity have been explained, for a long time, by the well- mode, so that, after the reflection or transmission of known model in which a kink moving in an inho- the kink, the impurity continues to vibrate with un- mogeneous medium is considered as an effective diminished amplitude. However, a kink cannot be particle, the kink position being the particle coor- trapped by a heavy mass impurity investigated in ref. dinate (see ref. [7—91 and references therein). In this [1 3]. In the present paper we will demonstrate that the I OnleavefromlnstituteforLowTemperaturePhysicsandEn- sG kink may be trapped by a local impurity due to gineering, 47 Lenin Avenue, 310164 Kharkov, USSR. loss of its kinetic energy to excite the impurity mode, 2 P.P. Shirshov Institute for Oceanology, 117218 Moscow, and in this scattering process the impurity mode ex- USSR. citation is more important than radiative losses. We 318 Elsevier Science Publishers B.V.