In this paper we review a number of recent developments in the study of the Discrete Nonlinear Schrodinger (DNLS) equation. Results concerning ground and excited states, their construction, stability and bifurcations are presented in one and two spatial dimensions. Combinations of such steady states lead to the study of coherent structure bound states. A special case of such structures are the so-called twisted modes and their two-dimensional discrete vortex generalization. The ideas on such multi-coherent structures and their interactions are also useful in treating finite system effects through the image method. The statistical mechanics of the system is also analyzed and the partition function calculated in one spatial dimension using the transfer integral method. Finally, a number of open problems and future directions in the field are proposed.

The discrete nonlinear schrödinger equation: a survey of recent results

We prove for a class of nonlinear Schrodinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as ground state selection. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree–Fock type.

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Selection of the ground state for nonlinear schrödinger equations

We prove for a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as {\it ground state selection}. 
Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear Master equations. 
Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type.

Selection of the ground state for nonlinear Schroedinger equations

On a class of discretizations of Hamiltonian nonlinear partial differential equations

For the nonlinear Klein–Gordon type models, we describe a general method of discretization in which the static kink can be placed anywhere with respect to the lattice. These discrete models are, therefore, free of the Peierls–Nabarro potential. Previously reported models of this type are shown to belong to a wider class of models derived by means of the proposed method. A relevant physical consequence of our findings is the existence of a wide class of discrete Klein–Gordon models where slow kinks practically do not experience the action of the Peierls–Nabarro potential. Such kinks are not trapped by the lattice and they can be accelerated by even weak external fields.

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Discrete Klein–Gordon models with static kinks free of the Peierls–Nabarro potential

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Dynamics of lattice kinks

In this paper we consider two models of soliton dynamics (the sine Gordon and the \phi^4 equations) on a 1-dimensional lattice. We are interested in particular in the behavior of their kink-like solutions inside the Peierls- Nabarro barrier and its variation as a function of the discreteness parameter. We find explicitly the asymptotic states of the system for any value of the discreteness parameter and the rates of decay of the initial data to these asymptotic states. We show that genuinely periodic solutions are possible and we identify the regimes of the discreteness parameter for which they are expected to persist. We also prove that quasiperiodic solutions cannot exist. Our results are verified by numerical simulations.

Panos Kevrekidis

Papers

Dynamics of lattice kinks

Dynamics of Lattice Kinks