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We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et. al. [1], we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finitedimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.

Gap solitons are localized nonlinear coherent states which have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed v, 0 ≤ v ≤ c, they have been observed in experiments at speeds of approximately 50% of c. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as nonlinear defect modes. Using mathematical analysis and numerical simulations we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap soliton capture, reflection and transmission can be obtained by simple conservation of energy and resonant energy transfer principles.

Summary. We consider the nonlinear propagation of light in an optical fiber waveguide as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is assumed to have an index of refraction which varies periodically along its length. The wavelength of light is selected to be in resonance with the periodic structure (Bragg resonance). The AMLE system considered incorporates the effects of non-instantaneous response of the medium to the electromagnetic field (chromatic or material dispersion), the periodic structure (photonic band dispersion) and nonlinearity. We present a detailed discussion of the role of these effects individually and in concert. We derive the nonlinear coupled mode equations (NLCME) which govern the envelope of the coupled backward and forward components of the electromagnetic field. We prove the validity of the NLCME description and give explicit estimates for the deviation of the approximation given by NLCME from the exact dynamics, governed by AMLE. NLCME is known to have gap soliton states. A consequence of our results is the existence of very long-lived gap soliton states of AMLE. We present numerical simulations which validate as well as illustrate the limits of the theory. Finally, we verify that the assumptions of our model apply to the parameter regimes explored in recent physical experiments in which gap solitons were observed.

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]