Class of localized structures in nonlinear lattices

Dirac solitons and topological edge states in the β-Fermi-Pasta-Ulam-Tsingou dimer lattice

We consider a dimer lattice of the Fermi-Pasta-Ulam-Tsingou (FPUT) type, where alternating linear couplings have a controllably small difference and the cubic nonlinearity (β-FPUT) is the same for all interaction pairs. We use a weakly nonlinear formal reduction within the lattice band gap to obtain a continuum, nonlinear Dirac-type system. We derive the Dirac soliton profiles and the model's conservation laws analytically. We then examine the cases of the semi-infinite and the finite domains and illustrate how the soliton solutions of the bulk problem can be glued to the boundaries for different types of boundary conditions. We thus explain the existence of various kinds of nonlinear edge states in the system, of which only one leads to the standard topological edge states observed in the linear limit. We finally examine the stability of bulk and edge states and verify them through direct numerical simulations, in which we observe a solitonlike wave setting into motion due to the instability.

Gap compactonlike solutions of coupled Kortweg-de Vries equations with linear and nonlinear dispersions

We show the existence of a type of excitation, which we term as "gap compactonlike," within the gap of the linear spectrum of a system of coupled Kortweg-de Vries equations with linear and nonlinear dispersions. Since the solutions lie in the gap region of the spectra, they avoid resonance with the linear oscillatory wave and, therefore, do not decay into radiations. These types of solutions are important in energy localization and transport in polymers and biopolymers, optical systems, etc.

Collective proton transport with weak proton-proton coupling

A mechanism based on the peculiar dynamical properties of the hydrogen bond is suggested to describe fast proton transfers in hydrogen-bonded chains with realistic (weak) intersite proton-proton interactions. For this purpose, a two-sublattice model is suggested and studied analytically that admits an exact soliton solution. The solitons, which are topological objects (kinks and antikinks) on the one hand, but dynamical entities (like Boussinesq solitons) on the other hand, are shown to have a sufficiently large width to allow free propagation along the chain.

Kink dynamics in one-dimensional coupled map lattices

We examine the problem of the dynamics of interfaces in a one-dimensional space-time discrete dynamical system. Two different regimes are studied: the non-propagating and the propagating one. In the first case, after proving the existence of such solutions, we show how they can be described using Taylor expansions. The second situation deals with the assumption of a travelling wave to follow the kink propagation. Then a comparison with the corresponding continuous model is proposed. We find that these methods are useful in simple dynamical situations but their application to complex dynamical behaviour is not yet understood. (c) 1995 American Institute of Physics.