Description
This mini-course provides a systematic introduction to explicit formulae, which have recently found a broad range of applications in the analysis of nonlocal completely integrable PDEs. Central examples for this approach via explicit formulae arise for the Benjamin-Ono equation (BO), Calogero-Moser(-Sutherland) derivative NLS (CM-DNLS), the cubic Szegö equation, and the Half-Wave Maps equation (HWM). A unifying feature of these completely integrable nonlocal PDEs is a Lax pair structure on Hardy spaces. The first part of this course will highlight the operator-theoretic analysis, posed on the torus as well as the real-line case. In the second part of the mini-course, we discuss some fundamental applications covering scaling-critical global well-posedness, finite-time blowup, and soliton resolution.