Speaker 2 :  Matteo Sommacal  (Northumbria University)
    Date & Time : May 27 (Fri.), 2022 / 10:50-11:30PM (Tokyo (JST))
                  (--> May 27 (Fri.), 2022 / 2:50-3:30PM (London (BST)) )
                  (--> May 27 (Fri.), 2022 / 3:50-4:30PM (Rome (CEST)) )
    Title : Integrability, instabilities, and the onset of rogue waves
    Abstract : 
         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, leading to the construction of the 
       so-called stability spectra, which, for multi-component systems with 
       more than two components, in general differs from the continuous spectra 
       of the spatial Lax operator. 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 illustrated using the example 
       of the plane wave solutions for 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, 
       in the case of plane waves, their study makes use of some basic ideas 
       from algebraic-geometry. Indeed, it turns out that, for a Lax Pair that 
       is polynomial in the spectral parameter, the problem of classifying the 
       stability spectra is transformed into a problem of classification of 
       certain complex curves. The method is general enough to be applicable to 
       a large class of integrable systems and in principle to all typologies 
       of their solutions: additionally to the system of two coupled nonlinear 
       Schrödinger equations, it has already been successfully applied to the 
       study of the plane wave stability for the system modelling the resonant 
       interaction of three waves, and for a novel long wave-short wave system, 
       which contains both the Yajima-Oikawa and Newell models as special cases. 
       Moreover, when this method is applied to continuous wave solutions, the 
       corresponding spectra can be used to predict the values of the spectral 
       parameter leading to rational soliton (rogue wave) solutions and the 
       instability regimes allowing for their formation.
         This is a joint work with Marcos Caso-Huerta (Northumbria University), 
       Antonio Degasperis (Roma "La Sapienza"), Priscila Leal da Silva 
       (Loughborough University) and Sara Lombardo (Loughborough University).