Singular Asymptotics for Nonlinear Dispersive Waves

Peter Miller
University of Michigan, Ann Arbor

A common feature of nonlinear wave propagation in many physical systems is an instability causing long wavepackets to break into pulses now called solitons (e.g. the Benjamin-Feir instability of water-waves and the modulational instability of fiber optics). In some contexts (e.g. in dispersion-shifted fibers), the solitons are small compared to the size of a typical wavepacket. Thus, wavepackets decay by emitting an enormous number of solitons. We want to make whatever predictions we can about the asymptotic behavior of this complicated process in the limit when the scale ratio goes to zero.

Effectiveness of numerical computation limited in such problems by stiffness, {i.e.}, relevant features occur on widely separated scales. Approaches based on formal expansions sometimes lend insight, but equally often yield apparently nonsensical model equations for which the initial-value problem is ill-posed.

This presentation will describe these phenomena in some detail, and will show how the asymptotic behavior can be analyzed precisely for
integrable nonlinear wave equations.

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