Scattering of a Gross-Pitaevskii breather off a barrier: the Inverse Scattering Transform made tangible
A key observable signature of integrability---of the existence of infinitely many "higher" conservation laws---in a system supporting solitons is the fact that a collision between solitons does not change their shape or size. But then, if solitons meet on top of a strong integrability-breaking barrier, one would expect the solitons to undergo some process consistent with energy conservation but not with higher conservation laws, such as the larger soliton cannibalizing the smaller one. However, here we show that when a strongly-coupled "breather" of the integrable nonlinear Schrodinger equation is scattered off a strong barrier, the solitons constituting the breather separate but survive the collision: as we launch a breather with a fixed impact speed at barriers of lower and lower height, at first all constituent solitons are fully reflected, then, at a critical barrier height, the smallest soliton gets to be fully transmitted, while the other ones are still fully reflected. This persists as the barrier is lowered some more until, at another critical height, the second smallest soliton begins to be fully transmitted as well, etc., resulting in a staircase-like transmission plot, with _quantized_ plateaus. We show how this effect makes tangible the _inverse scattering transform_: the powerful, but otherwise physically opaque mathematical formalism for solving completely integrable partial differential equations.
Supported by the NSF, ONR, and NSF-BSF (US-Israel).
In collaboration with V. Dunjko