Nonlinear chains inside walls

Abstract

The conservative dynamics of a 1D chain of units coupled with (FPU type) nonlinear interactions is considered. Stationary patterns in such chains emerge due to a balance of coupling energy between neighbouring units. Particularly interesting are the nontrivial stationary states which contain segments of positive and negative slope. This results in a zig-zag pattern, in the case of periodic boundary conditions, and in kink (or anti-kink) solutions in the case of the free boundary conditions. Imposing constraints on the chain, by way of two confining infinitely high walls, has repercussions for the stability of these stationary states. Here, such stationary states, commensurable with the available space between the two walls, are examined in detail, and their respective stability properties are determined analytically by invoking the transfer matrix method. Strikingly, stationary anti-kink solutions and periodic zig-zag states, being unstable in the absence of confining walls, become stable when confining walls are introduced. Furthermore, simulations reveal that chains with randomly generated initial conditions can seek out these patterns, thus localising energy, and persist for considerable time.