Stationary Modes and Localized Metastable States in a Triangular Lattice of Active Particles

Abstract

The dynamics of a triangular lattice consisting of active particles is studied. Particles with nonlinear friction interact via nonlinear forces of Morse potential. Nonlinear friction slows down fast particles and accelerates slow ones. Each particle interacts mainly with the nearest neighbors due to the choice of the cutoff radius. Stationary modes (attractors) and metastable states of the lattice are studied by methods of numerical simulation. It is shown that the main attractor of the system under consideration is the so-called translational mode-the state with equal and unidirectional velocities of all particles. For some parameter values translational modes with defects in the form of vacancies and interstitial particles are possible. Metastable localized states are presented by the plane soliton-like waves (M-solitons) with inherent velocity and density maxima. The lifetime of such states depends on the lattice parameters and the wavefront width. All metastable states transform into the translational mode after a transient process.

Discrete Variants of the $$\phi ^4$$ Model: Exceptional Discretizations, Conservation Laws and Related Topics

“…The static kink solution (5) with v = 0 to (3) is also a static solution to its first integral (62). Now we square both sides of (61) and of the shift (φn − φn−1)/h = F (φn, φn−1) and find their difference…”

“…Localized nonlinear excitations such as topological solitons and breathers play an important role in many areas of physics and very often they are considered in discrete media. For example, in solid state physics they are used to describe domain walls, dislocations, and crowdions in crystals [1,2,3,4,5], in macroscopic models of coupled pendulums [6], in granular crystals [7], in the arrays of electric circuits [8] and micromechanical cantilevers [9], among many others. In all these applications the mobility of solitary waves is an important issue, especially given that a typical discretization breaks the translational invariance of the continuum model and thus renders the discrete case far less amenable to genuine traveling dynamics.…”