PHYSICAL REVIEW B VOLUME 38, NUMBER 7 1 SEPTEMBER 1988 Dynamics of kink-kink collisions in the double-sine-Gordon system R. Ravelo, M. El-Batanouny, and C. R. Willis Department of Physics, Boston University, Boston, Massachusetts 02215 P. Sodano* Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 0213' (Received 24 November 1987) We study the double-sine-Gordon kink-kink collisions in a formalism which employs collective variables to describe the internal oscillations of the kinks and their translational motion in their center-of-mass frame. The equations of motion are solved in the absence of the radiation Aeld and dynamical dressing and the results are compared with numerical molecular-dynamics simulations. We investigate the energy exchange between the translational and internal modes, and a mechanism is proposed to explain the values of the translational velocity at which maximum energy is ex- changed between the two modes. I. INTRODUCTION The soliton or kink concept has been applied as a para- digm to model nonlinear excitations, localized defects, and particles in many areas of physics: condensed matter, ' quantum optics, plasma physics, particle physics, and even biophysics. However, the nonintegra- bility of many soliton c1asses has hampered the explicit treatment of the corresponding multisoliton systems and of the underlying soliton-soliton and soliton-antisoliton interactions. Even in cases of integrable soliton systems, such as the sine-Gordon system, where multisoliton solu- tions can be readily obtained, it has been rather difficult to extract the relevant details about the nature of the pair interaction process. For example, although the statistical mechanics of the sine-Gordon system has been treated exactly within the transfer-integral method (TIM), and can yield successfully a virial expansion of the free energy in terms of the soliton density, it fails to provide a physi- cal interpretation of the virial coefficients in terms of in- tersoliton interactions. It has only been recently that direct investigations of these interactions was undertak- en. Campbell et al. ' have extracted information about kink-antikink interactions from a series of numerical simulations of the P and modified sine-Gordon systems; including the double sine Gordon. Sasaki has investigat- ed the kink-kink collision in the case of the sine-Gordon system" in terms of a position shift parameter in an effort to explain the physical significance of the second virial coefficient obtained by TIM. Thus it is desirable to develop a framework within which it is possible to treat the kink-kink collisions as an interparticle interaction and then investigate the proper- ties of the ensuing interaction potential and its dynamics. It is the purpose of this paper to introduce such a scheme using as example kink-kink co11isions in the double-sine- Gordon (DSG) system. ' The DSG kink has found diverse applications in condensed matter physics: ele- mentary excitations in He, ' ' domain walls in one- II. COLLECTIVE VARIABLES IN KK SCATTERING We start with the field Lagrangian 2 2 a ae 2 Bt 2 Bx '2 V(4) Io (2. 1) 1 — cos — + g(1 — cos4) . (2.2) 2 dimensional magnetic chains' and in ferroelectrics, ' and very recently as misfit dislocations in the reconstructed (111) surface of gold. ' It has also been used to model self-induced transparency in nonlinear optics' and as an extended object in particle physics. ' Another interesting aspect of the DSG kink is that it possesses internal dynamical structure ' ' which adds to the richness and complexity of the collision problem and provides a chal- lenging test to the proposed approach. In the present approach we extend our recently developed Hamiltonian formalism which treats the DSG kink as a particle with one internal degree of free- dom to the analysis of the DSG kink-kink (KK) collision. We have used computer simulations as a posteriori check on the dynamical details resulting from the formalism. In Sec. II we will present the outline of the DSG collec- tive variable formalism. In Sec. III we derive the La- grangian and equations of motion and consider the asymptotic motion of the DSG kinks with special con- sideration to the coupling between the internal and exter- nal degrees of freedom. In Sec. IV we present the results of the kink-kink scattering and a comparison with the re- sults obtained from computer simulations. The con- clusion is presented in Sec. V and in the Appendix we dis- cuss the importance of using the Lorentz-boosted DSG kink solution in the kink-kink ansatz. 38 4817 1988 The American Physical Society