In this talk, some interesting (we believe !) new features of the Riemann-Hilbert method of the asymptotic analysis of integrable systems will be discussed. These features have emerged during the study of the global solution of the tt* equations of Cecotti and Vafa which
the speaker has been pursuing, for some time already, together with M. Guest
and Chang-Shou Lin. The method which we use in this study is based on a combination
of the isomonodrtomy technique and Iwasawa factorization from the theory of loop groups.
This allows us to simplify significantly the asymptotic
analysis of the tt* equations and, simultaneously, to bring a new light on some aspects of
the well known relation between the Birkhoff-Grothendieck and Iwasawa factorizations.
Another important link - to a 1980 paper by I. Krichever on the nonlinear analog of the d'Alembert's
formula, will be also highlighted.

The talk represents an ongoing joint project of M. Guest and the speaker, and it is dedicated
to the memory of Igor Krichever with whom the project was started.

A recording of the talk can be found at
https://youtu.be/DEPsUV0sFwU