Integrability and continuous wave instabilities: an algebraic-geometric approach

Dr Matteo Sommacal, Northumbria University

Recently, a direct construction of the eigenmodes of the linearization of 1+1, multicomponent, nonlinear, partial differential equations of integrable type has been introduced. This construction employs only the associated Lax pair, with no reference to spectral data and boundary conditions. In particular, this technique allows to study the instabilities of continuous wave solutions in the parameter space of their amplitudes and wave numbers, as well as to compute and potentially classify the so-called stability spectra. In the context of modulation instability, it provides also a necessary condition in the parameters for the onset of rational solitons.

The theory will be presented using the example of a system of two coupled nonlinear Schrödinger equations in the defocusing, focusing and mixed regimes. The derivation of the stability spectra is completely algorithmic and make use of elementary algebraic-geometry. It turns out indeed that, for a Lax Pair that is polynomial in the spectral parameter, the problem of classifying the spectra is transformed into a problem of classification of certain algebraic varieties. The method is general enough to be applicable to a large class of integrable systems.

If time will allow, different applications of this theory will be illustrated, including the resonant interaction of three waves.

This work has been carried out in collaboration with Professor Antonio Degasperis (Roma "La Sapienza") and Professor Sara Lombardo (Loughborough University).