The field of nonlinear waves is an active mathematical research area with a
long tradition and storied history. Starting with the observation of John Scott
Russell in 1834 of the great “wave of translation,” now called a soliton, nonlinear
dispersive wave equations have proven over the years to be fundamental for mod-
eling nonlinear wave phenomena in such diverse fields as fluid and gas dynamics,
nonlinear optics, low-temperature physics, biology, and more. The discipline can
be succinctly described as a study of waves usually resulting from a precise bal-
ance between nonlinearity and dispersion. While dispersion attenuates in physical
problems, it does so weakly and typically over long spatial and temporal scales.
Coupled with an intricate process of wave mixing due to nonlinearity, the analy-
sis and computation of nonlinear dispersive waves becomes challenging and com-
plicated. Furthermore, computing solitons and describing their elastic collisions
requires mathematical methods that are quite distinct from those used to analyze
and solve linear partial differential equations.
Major progress in the study of soliton equations came with the discovery of the
“Inverse Scattering Transform” (IST) by Gardner, Greene, Kruskal, and Miura in
1967. As the name suggests, the IST combines techniques from inverse problems
and scattering theory to solve a special class of nonlinear dispersive equations,
called completely integrable systems, of which the Korteweg-de Vries (KdV) and
nonlinear Schr¨ odinger (NLS) equations are prototypical examples. The IST method
can be viewed as a nonlinear analog of the Fourier transform. Not only does the
IST allow one to solve the initial-value problem for nonlinear integrable equations
for fairly general initial conditions, it has also provided unprecedented insight into
how nonlinear and dispersive effects influence one another. Research in integrable
systems has developed into a rich discipline at the crossroads between analysis,
geometry, algebra, and mathematical physics. Recently discovered connections with
the study of Riemann–Hilbert problems and Riemann surfaces are examples of such
Unfortunately, not every nonlinear dispersive system can be solved with the
IST. Notably, the problem of modeling free surface irrotational waves in incom-
pressible and inviscid fluids is beyond the scope of the IST. As a “parent” model
for several of the most important problems in integrable systems though, one might
question how phenomena seen in integrable systems manifest themselves in this
more complicated, yet physically more realistic setting. Likewise, within the last
several years, techniques from integrable systems have found their way into research
of the free surface problem. Despite such efforts to solve a problem with a nearly
200 year history, the benefits and limitations of these novel lines of attack are still
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