GLOBAL CONSERVATIVE SOLUTIONS OF THE NONLOCAL NLS EQUATION BEYOND BLOW-UP

We consider the Cauchy problem for the integrable nonlocal nonlinear Schro center dot dinger (NNLS) equation i partial differential tq(x,t)+/- partial differential x2q(x,t)+/- 2 sigma q2(x,t)q(-x, t) = 0 with initial data q(x, 0) is an element of H1,1(R). It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak H1,1 local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.