Ground states of coupled nonlinear oscillator systems

Abstract

Abstract The dynamics of coupled nonlinear oscillator systems is often described by the classical discrete nonlinear Schrodinger equation (DNLSE). In its simplest version, the DNLSE is made up of two terms—a nearest-neighbor hopping term and an on-site cubic nonlinear term. Each of the terms is preceded by a coefficient that can take on either a positive or a negative sign. Each of the DNLSE versions is derived from a corresponding equivalent Hamiltonian. The result is a small family of four versions of the DNLSE Hamiltonian, each with its own associated ground state, all indeed scattered in myriad of scientific publications. Here we present a comprehensive picture for the ground states of DNLSE systems, summarize existing results and provide new insights. First we classify the four DNLSE Hamiltonians into two pairs according to the sign of the nonlinear term—a “positive/negative Hamiltonian pair” if the sign of the nonlinear term is positive/negative respectively. Ground states of the positive Hamiltonian pair are discrete plane waves in either a ferromagnetic-like or an antiferromagnetic-like configuration, depending on the sign of the hopping term. Ground states of the negative Hamiltonian pair are either unstaggered or staggered site-centered discrete breathers. The instantaneous state of a DNLSE system is described by a set of one-parameter complex functions each with its own amplitude and phase. We show that except for the sign of the phase, a ground state associated with a positive/negative Hamiltonian is the maximum energy state associated with the sign-reversed (negative/positive) Hamiltonian. Next we discuss some properties of the ground states associated with the positive-Hamiltonian pair—entropy, temperature, correlations and stability. We extend our ground state stability discussion to include excited plane waves. We propose to engineer a specific perturbation that preserves both density and energy—the two conserved quantities of a DNLSE system—and to test plane wave s stability based on entropy change. We show that under such conserved-quantities-preserved perturbation, all excited plane waves are entropy-unstable. For site-centered discrete breathers—the ground states of the negative-Hamiltonian pair—we have divided system nonlinearity into two ranges and wrote very good analytic approximations for the breathers in each range. Lastly, in a dedicated section, we very briefly discuss the specific implementation of the DNLSE in the fields of magnetism, optics, and ultracold atoms, emphasizing ground states. For example, following a 2002 article, we show that the dynamics of a 1d optically-trapped ultracold bosonic atoms, in a rather wide range of system densities and system nonlinearities, can be described by a particular version of the here-discussed classical DNLSEs.