Nonlocal integrable equations: blow-up phenomena and beyond

I will carry out this fellowship at the University of Oslo, under the supervision of Professor Kenneth Karlsen at the Department of Mathematics. The project is devoted to study existence and blow-up properties of the solutions of initial value problems (IVPs) for nonlocal nonlinear integrable PDEs. Nonlocal integrable equations were introduced by M. Ablowitz and Z. Musslimani in 2013 and have become a hot topic in the theory of integrable systems. By applying inverse scattering transform (IST) method it was shown that soliton solutions of nonlocal equations blow-up in isolated points in x,t plane. Therefore the natural question is to study the existence and blow-up properties of the general IVPs for such equations. To this end we will use the mixture of the IST method and PDE techniques. The former will be used for predicting the presence and the form of the blow-up of the solution, while the PDE approaches will be used for describing the blow-up of the solution of the general IVP and for finding a suitable concept of a weak solution, which exists even after the collapse. Also by the IST method it was shown that apart from zeros, which correspond to solitons, the associated to the initial data spectral functions have the new type of singularities. These singularities correspond to the winding of the argument of the spectral function and play a significant role in long-time asymptotic behavior of the solutions ? they are responsible for a transition between qualitatively different asymptotic zones in x,t plane. It is a challenging open problem to describe analytically the nonlinear effects corresponding to these new singularities and in the framework of the project we are going to investigate this phenomena numerically. The fellowship will allow me to expand my skills in PDE methods, which is different, but related to my present expertise field. Also it will enable me to become an established interdisciplinary researcher, who could pursue top academic positions.