Tom Trogdon, CIMS

Riemann--Hilbert problems and the inverse scattering transform: From asymptotics to computation

Riemann--Hilbert problems and the inverse scattering transform: From asymptotics to computation

The inverse scattering transform (IST) is used to solve the
Cauchy problem for integrable nonlinear partial differential
equations on the line. Matrix Riemann--Hilbert problems
(RHPs) are a key component in the IST. Historically, RHPs
have made the IST amenable to rigorous asymptotic analysis with
the Deift--Zhou method of nonlinear steepest descent. More
recently, techniques for oscillatory singular integral equations
have been employed to solve RHPs numerically and compute the
IST. Importantly, nonlinear dispersive evolution equations
can be solved numerically without any need for time-stepping.
Errors are seen to be uniformly small for arbitrarily
large times. Combining this approach with the so-called
dressing method allows for the computation of a wide class of
non-decaying solutions.