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PT symmetric optics: Mathematical and experimental challenges

In quantum theory a Hamiltonian describing physical system is mathematically represented by
a self-adjoint linear operator to ensure that the energy levels are all real and that the time evolution
of the system is unitary (conservation of the total probability). A few years ago, it was realized
that certain complex non-hermitian Hamiltonians obeying space-time reflection (PT) symmetry could exhibit an entirely real spectrum.听 This observation has led to intense research activities in the general area of non-hermitian quantum theory in an attempt to extend the framework of quantum mechanics into the complex domain. Most notably are non-hermitian quantum anharmonic oscillator type models, quantum field theories, and Anderson-like models, to mention a few. While much theoretical progress has been reported in the literature, none of the above mentioned research directions has led to a viable experimental proposal for which non hermitian quantum effects could be observed in laboratory experiments. Quite recently, it was suggested that the concepts of PT symmetry and non-hermiticity could be implemented, realized and investigated within the framework of classical optics. This proposal has, in turn, stimulated much research activities in the general area of parity-time or PT symmetric optics and led to the first experimental observation (in any physical system)
of parity-time symmetry-breaking.

The aim of this talk is to review recent progress in the emerging field of wave propagation in linear
and nonlinear optical PT symmetric periodic media. Here, the wave dynamics is governed by either
a linear or nonlinear Schr枚dinger type equation in the presence of an external parity-time symmetric potential. The unique properties of waves propagating in such optical media are discussed
and a number of open problems and future directions are highlighted.

Gino Biondini; Department of Mathematics; University of Buffalo, New York

How many solitons are there in the Zabusky-Kruskal experiment, and why?

In their seminal 1965 work, Zabusky and Kruskal (ZK) performed numerical simulations
of the Korteweg-de Vries (KdV) equation with small dispersion and cosine initial data.听 As is well known, the breakup of the initial pulse generated eight solitary waves interacting elastically, which they called solitons.听 Two years later, Kruskal and others went on to invent the inverse scattering transform (IST) to solve the initial value problem for the KdV equation, giving birth to the modern theory of integrable systems.听 Fifty years later, however, a precise analytical description of the ZK experiment
was surprisingly still missing.听 In this talk I will describe recent theoretical and experimental work aimed
at closing this gap.听 Specifically, I will show how a careful use of the WKB method on the scattering problem for the KdV equation allows one to completely characterize the small-dispersion limit with single-lump, periodic initial conditions, obtaining in particular explicit expressions for the number
of solitons emerging in any given situation, as well as their characteristic amplitudes and speed.听
These results are corroborated by recent experiments in water waves which fully reproduce, for the first time, the ZK simulations, including the soliton recurrence.听 This is joint work with Guo Deng
and Stefano Trillo.
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