Dynamics of coupled gap solitons in diatomic lattices with cubic and quartic nonlinearities Bambi Hu, 1,2 Guoxiang Huang, 1,3 and Manuel G. Velarde 4 1 Centre for Nonlinear Studies and Department of Physics, Hong Kong Baptist University, Hong Kong, China 2 Department of Physics, University of Houston, Houston, Texas 77204 3 Department of Physics and Laboratory for Quantum Optics, East China Normal University, Shanghai 200062, China 4 Instituto Pluridisciplinar, Universidad Complutense, Paseo Juan XXIII, No. 1, Madrid 28040, Spain ~Received 29 November 1999! The dynamics of coupled gap solitons in diatomic lattices with cubic and quartic nonlinearities is considered analytically based on an extended quasidiscreteness approach. For various mass differences ~and thus different gap widths of the phonon spectrum!, the coupled gap solitons are shown to display very rich dynamical behavior and their properties are strongly dependent on the force-constant ratio K 3 2 /( K 2 K 4 ), where K j ( j 51,2,3) are the force constants for the quadratic, cubic, and quartic parts of the intersite interaction poten- tial, respectively. Several previous theoretical approaches for studying gap soliton dynamics in diatomic lat- tices are recovered in our scheme, and the relations between these methods are elucidated in a systematic way. PACS number~s!: 63.20.Pw, 63.20.Ry I. INTRODUCTION Anharmonicity in lattices is responsible for many impor- tant phenomena, such as transfer of energy, thermal conduc- tivity, structural phase transitions, and the associated soft mode and central peak phenomena, etc. The study of nonlin- ear lattice dynamics and related lattice solitons has been greatly influenced by the pioneering work of Fermi, Pasta, and Ulam @1#. Most of the early work in this area focused on monatomic lattices. In recent years, much attention has been paid to nonlinear dynamics in diatomic lattices. The particu- lar interest in studying the band gap and related nonlinear excitations @2–14# has been greatly stimulated by the discov- ery of optical gap solitons in periodic dielectric materials @15#. For a diatomic lattice, the phonon spectrum consists of two branches ~acoustic and optical!, induced by mass or force-constant differences. Due to the interplay between dis- creteness and nonlinearity, types of nonlinear localized exci- tations that have no direct analog in continuum models are possible. In particular, gap solitons may appear with their vibration frequencies in the phonon band gap. Since gap soli- tons occur in perfect lattices with discrete translational sym- metry, the terms ‘‘anharmonic gap mode’’ and ‘‘intrinsic gap mode’’ have been used also @4,12#. It is possible that gap solitons may be created experimentally in diatomic lattices. References @16–18# reported observation of gap solitons in damped and parametrically driven one-dimensional ~1D! di- atomic pendulum lattices. The mechanism for the appearance of gap solitons in non- linear diatomic lattices can be briefly explained. Assume that there is an excited lattice wave with its vibration frequency falling within the phonon band gap. In the linear limit, such a lattice wave is strongly reflected ~Bragg reflection!. Only exponentially growing and decaying solutions for lattice dis- placements are possible and, for a finite system, an exponen- tially decaying solution results, leading to very low transmis- sivity. The situation is changed when the amplitude of the lattice wave is high enough. In this circumstance the nonlin- earity of the system begins to play its role. If the nonlinearity has an appropriate sign, the exponentially growing and de- caying solutions to the left and right can be connected in the large-amplitude region to form a self-consistent nonlinear localized solution that is finite everywhere. Such a solution is just the lattice gap soliton mentioned above. There exist three different analytical approaches for the gap soliton dynamics in nonlinear diatomic lattices. The first one was provided by Kivshar and Flytzanis @3#. The starting point is that, in the case of small mass difference ~thus a narrower phonon band gap!, because of nonlinearity there exists a strong coupling between the optical lower cutoff mode and the acoustical upper cutoff mode at the boundary of the Brillouin zone ~BZ!. Under the rotating-wave approxi- mation, they derived coupled nonlinear envelope equations for the two cutoff modes for the diatomic lattice with non- linear on-site potential. Some interesting coupled soliton so- lutions were obtained. Later, this approach was used to study the coupled gap solitons in a diatomic lattice with nonlinear intersite potentials @19–21#. Such coupled-mode theory is similar to the corresponding theory for optical gap solitons in shallow nonlinear gratings @22#, valid only for a narrow band gap, and the coupled-mode equations obtained are essentially the same as the coupled-mode equations obtained in Ref. @22#. The second theory was given by Konotop @11# based on an envelope function approach. In his approach Konotop also considered the small-band-gap case. However, instead of the coupled envelope equations he obtained a nonlinear Schro ¨ - dinger ~NLS! equation. The solitons obtained can propagate with the group velocity of the carrier wave at wave vector q 5p / d of the corresponding monatomic lattice with the lat- tice constant d 0 5d /2, where d is the lattice constant of the diatomic lattice @11#. The solitons obtained by this approach display tails ~companion modes! at the rear of the solitonic pulses. The third method is based on a quasidiscreteness ap- proach ~QDA!@5,13#. The amplitude equation derived in this approach is also a NLS equation but it is valid for the whole BZ of the phonon spectrum. Using the results from the QDA, PHYSICAL REVIEW E AUGUST 2000 VOLUME 62, NUMBER 2 PRE 62 1063-651X/2000/62~2!/2827~13!/$15.00 2827 ©2000 The American Physical Society