With the purpose of bringing together researchers working on these different
problems, a special session called “Nonlinear Waves and Integrable Systems” was
organized at the AMS Western Sectional Meeting in Boulder, Colorado, in the
spring of 2013. The organizers were fortunate in having such a diverse group of
speakers agree to present their research and commit to write a research paper for
this volume which attempts to capture the scope of the special session. While by no
means exhaustive, this volume addresses and explains many of the major analytic
and computational techniques used across several sub-disciplines of the study of
nonlinear wave equations.
The volume begins with an article by Ben Herbst, Garrett Nieddu, and A. David
Trubatch who return to questions from the genesis of the field of integrable systems.
The authors study whether discretizations of continuous integrable systems inherit
their dynamic properties, in this case “recurrence,” from the integrability of the
Next, a series of papers on techniques related to the IST is presented. Francesco
Demontis, Cornelis van der Mee, and Federica Vitale study the scattering data of
a defocusing Zakharov–Shabat system with non-zero boundary conditions. This
problem is of particular relevance to the study of dispersive shock-waves, a subject
of much recent interest.
The next two papers address questions related to the Novikov–Veselov (NV)
equation, a two-dimensional generalization of the KdV equation. The first pa-
per, co-authored by Ryan Croke, Jennifer L. Mueller, Michael Music, Peter Perry,
Samuli Siltanen, and Andreas Stahel, is devoted to the use of the IST to study the
NV equation. The authors illustrate the use of powerful techniques from inverse
problems and scattering theory to determine for what types of initial conditions the
NV equation is well posed. In the second paper, the stability of particular solutions
to the NV equation is studied by Ryan Croke, Jennifer L. Mueller, and Andreas
One last paper concerned with integrable systems is written by Gregory Lyng
who studies the semi-classical limit of the focusing NLS equation showing the role
of Riemann–Hilbert problems in investigating integrable systems.
The subsequent series of papers addresses questions on the free surface problem
in fluids. The first paper by Jon Wilkening looks at a classic question on the
correspondence of phenomena found in the KdV equation and the free surface
problem from which the KdV equation is derived.
In the next paper, Katie Oliveras and Bernard Deconinck investigate modula-
tion instabilities in the free surface problem. Modulational instabilities were first
found in the NLS equation, which is derived from the free surface problem. Using
a formulation of the free surface problem first proposed by Ablowitz, Fokas and
Musslimani (AFM) – an approach strongly motivated by methods from integrable
systems – the authors extend the results from the NLS equation to a more physically
In the same spirit, Katie Oliveras and Vishal Vasan show how to use the AFM
formulation to determine the height of free surface waves from pressure measure-
ments along the “sea-floor”. This work has practical importance for oceanogra-
phers, and represents a significant improvement over existing methodologies.
Finally, Jon Wilkening and Vishal Vasan investigate several competing numer-
ical schemes for simulating the free surface problem. The merits and drawbacks