The Fermi–Pasta–Ulam (FPU) model, which was proposed ago to examine thermalization in nonmetallic solids and develop “experimental” techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems—including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose–Einstein condensates (BECs) in optical lattices. BECs, in particular, due to their widely ranging and easily manipulated dynamical apparatuses—with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials—provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.
The Fermi–Pasta–Ulam (FPU) model was formulated in 1954 in an attempt to explain heat conduction in non-metallic lattices and develop “experimental” (computational) methods for research on nonlinear dynamical systems.1 Further studies of this problem later led to the first analytical description of solitons (using the Korteweg–de Vries equation, which is a continuum approximation of the discrete FPU system), which have since become one of the fundamental paradigms of nonlinear science. These nonlinear waves occur ubiquitously in rather diverse physical situations ranging from water waves to plasmas, optical fibers, superconductors (long Josephson junctions), quantum field theories, and more. Over the past several years, the study of solitons and coherent structures in Bose–Einstein condensates(BECs) has come to the forefront of experimental and theoretical efforts in soft condensed matter physics, drawing the attention of atomic and nonlinear physicists alike. Observed experimentally for the first time in 1995 in vapors of sodium and rubidium,2,3 a BEC—a macroscopic cloud of coherent quantum matter—is attained when atoms, confined in magnetic traps, are optically and evaporatively cooled to a fraction of a microkelvin. The macroscopic behavior of BECs near zero temperature is modeled very well by the Gross–Pitaevskii equation (a time-dependent nonlinear Schrödinger equation with an external potential), which admits a wide range of coherent structure solutions. Especially attractive is that experimentalists can now engineer a wide variety of external trapping potentials (of either magnetic or optical origin) confining the condensate. As a key example, we focus on BECs loaded into deep, spatially periodic optical potentials, effectively splitting the condensate into a chain of linearly-interacting, intrinsically nonlinear droplets, the dynamics of which is accurately characterized by nonlinear lattice models. This paper highlights some of the quasidiscrete nonlinear dynamical structures in BECs reminiscent of the discoveries that originated from the FPU model.
We thank David Campbell for the invitation to write this article and Predrag Cvitanović and Norm Zabusky for useful discussions during the course of this work. P.G.K. gratefully acknowledges support from NSF-DMS-0204585, from the Eppley Foundation for Research and from a NSF-CAREER award. The work of B.A.M. was supported in a part by Grant No. 8006/03 from the Israel Science Foundation. R.C.G. gratefully acknowledges support from an SDSU Grant-In-Aid. M.A.P. acknowledges support provided by a VIGRE grant awarded to the School of Mathematics at Georgia Tech.
II. THE FPU PROBLEM
III. BOSE–EINSTEIN CONDENSATION
A. BECs in optical lattices and superlattices
B. Lattice dynamics
IV. SOLITON–SOLITON TAIL-MEDIATED INTERACTIONS AND THE TODA LATTICE
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