IMA/MCIM Industrial Problems Seminar

Gap solitons are localized nonlinear coherent states of light 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 c/2. It is of basic scientific interest and technological interest (for possible use in optical memory) to learn how to trap gap solitons.

We introduce a family of periodic structures with localized defects. These support linear defect modes which are shown to persist into the nonlinear regime. We investigate the capture of a gap soliton by these defects, analytically and numerically. The mechanism of capture is shown to be resonant energy transfer from a soliton to a nonlinear defect mode. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap soliton capture, reflection and transmission can be obtained. The dynamics of capture is also investigated by dynamical systems methods applied to finite dimensional reduced models.

Energy captured by a multimoded defect asymptotically settles into the nonlinear ground state defect mode. We analyze this asymptotic selection of the ground state, which occurs in many physical models. For example, in addition to the above context, this phenomena arises as well in models describing the effective dynamics of a large number of weakly interacting bosons.

References:

1. Goodman, Holmes, and Weinstein, Nonlinear propagation of light in one-dimensional periodic structure, J. Nonlinear Science, 11 (2001).

2. Goodman, Slusher, and Weinstein, Stopping light on a defect, to appear in J. Opt. Soc. Am., B (2002).

3. Goodman, Holmes, and Weinstein, Interaction of sine-Gordon kinks with defect - Phase transport in a 2-mode model, Physica D, 161 (2002).

4. Soffer and Weinstein, Resonances and radiation damping in Hamiltonian nonlinear PDEs, Inventiones Math., 136 (1999).

5. Soffer and Weinstein, Selection of the ground state for nonlinear Schrodinger equations, preprint.