Chaotic Scattering and then-Bounce Resonance in Solitary-Wave Interactions

Abstract

We present a new and complete analysis of the n-bounce resonance and chaotic scattering in solitary wave collisions. In these phenomena, the speed at which a wave exits a collision depends in a complicated fractal way on its input speed. We present a new asymptotic analysis of collectivecoordinate ODEs, reduced models that reproduce the dynamics of these systems. We reduce the ODEs to discrete-time iterated separatrix maps and obtain new quantitative results unraveling the fractal structure of the scattering behavior. These phenomena have been observed repeatedly in many solitary-wave systems over 25 years. In dissipative systems such as electrical signal propagation in nerve fibers or reaction-diffusion systems, two interacting waves generally merge into a single larger wave. In completely integrable, or soliton, equations, by contrast, interacting solitary waves emerge from a collision intact and with their original speeds, but a slight shift in their position, which is well-understood through the theory of inverse scattering.Collisions in dispersive wave systems that are neither dissipative nor completely integrable may produce a much wider range of behaviors. We focus on one, the 2-bounce, or, more generally n-bounce phenomenon. Two counterpropagating waves with sufficient relative initial speed (or one wave incident on a localized defect) will pass by or reflect off each other with little interaction, while for most initial speeds below some critical velocity v c they will become trapped, forming a localized bound state. At certain velocities below v c , the waves become trapped, begin to move apart, and come together a second time before finally moving apart for good-the socalled 2-bounce solutions. In addition to the 2-bounce resonant solutions, one often finds 3-, 4-, or, more generally, n-bounce solutions. Figure 1a shows a 2-bounce resonant solution to (1), and figure 1b shows the sensitive dependence of the final speed on the initial speed, with the number of 'bounces' indicated by color. The initial conditions leading to these behaviors are interleaved in a manner often described as fractal. This was first seen * Electronic address: goodman@njit.edu † Electronic address: rhaberma@smu.edu . Figure 1c shows a 2-bounce resonant solution of the model ODEs for system (1) (discussed below), and figure 1d shows that the ODE model reproduces the fractal interaction structure of the PDE, if not the exact structure. We analyze these phenomena through systematic asymptotics applied to 'collective coordinate' models, low-dimensional model systems of ordinary differential equations (ODEs) derived from a variational principle that reproduce the dynamics in numerical simulations. We construct, using Melnikov-integrals and formal matching procedures, approximate n-bounce resonant solutions to the ODEs and derive an iterated map arXiv:nlin/0702048v1 [nlin.PS]

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