Giant boundary layer induced by nonreciprocal coupling in discrete systems

Abstract

Nonreciprocally coupled systems present rich dynamical behavior such as unidirectional amplification, fronts, localized states, pattern formation, and chaotic dynamics. Fronts are nonlinear waves that may connect an unstable equilibrium with a stable one and can suffer a convective instability when the coupling is nonreciprocal. Namely, a state invades the other one, and due to boundary conditions, the front stops and creates a boundary layer. Unexpectedly, in nonreciprocal coupled systems, we observe arbitrarily large boundary layers in the convective regime when the condition at the fixed edge does not match the equilibrium value. We analytically determine the boundary layer size using map iterations; these results agree with numerical simulations. On the other hand, if one of the boundary conditions matches the unstable equilibrium state, the boundary layer size diverges; however, due to the computer numerical truncation, it is finite in numerical simulations. Our result shows that, in nonreciprocally coupled systems, this mismatch in the boundary condition is relevant in controlling the boundary layer size, which exhibits a logarithm scaling with the mismatch value.