Exact expression for the propagating front velocity in nonlinear discrete systems under nonreciprocal coupling

Abstract

Nonlinear waves are a robust phenomenon observed in complex systems ranging from mechanics to ecology. Fronts into the stable state, nonlinear waves appearing in bistable and multistable systems, are fundamental due to their robustness against perturbations and capacity to propagate one state over another. Controlling and understanding these waves is then crucial to make use of their properties. Their velocity is one of the most important features, which can be analytically computed only for specific dynamical systems and under restricted conditions on the parameters, and it becomes more elusive in the presence of spatial discreteness and nonreciprocal coupling. A key difficulty in developing expressions for the front velocity is the lack of a front rigid shape exploiting translational invariance, a property that is broken in discrete systems with a finite number of elements. This work reveals that fronts in discrete systems can be treated as rigid objects when analyzing their whole trajectory by collecting the system state at each time step instead of just observing the instantaneous, current state. Then, a relationship between the front velocity and its reconstructed rigid shape is found. Applying this method to a generic model for nonreciprocally coupled bistable systems reveals that the derived velocity formula provides insight into fronts’ long-observed properties, such as the oscillatory trajectory for the front position and the pinning–depinning transition, and agrees with the approximative and parameterized methods described in the literature. Furthermore, it reveals an explicit linear relationship between the velocity and the nonreciprocal coupling constant. Numerical simulations show perfect agreement with the theory.