IIMAS - FENOMEC
UNAM
Resumen:
While less well known and studied than solitary waves, undular
bores, also termed dispersive shock waves, are a nonlinear
wave form common in nature, examples being tidal bores and
tsunamis. In contrast to solitary waves, undular bores are
unsteady wave forms. They are modulated periodic wavetrains
with solitary waves at one edge and linear waves at the other
and continually expand in width. As they are unsteady, they
are more difficult to analyse than solitary waves. The main
tool used to find undular bore solutions of nonlinear wave
equations is Whitham modulation theory. Undular bores are
found as simple wave solutions of the hyperbolic modulation
equations for modulationally stable wavetrains. In practice,
however, undular bore solutions are mainly limited to
integrable equations as the Whitham modulation equations are
then guaranteed to be set in Riemann invariant form, which is
a prerequisite for determining simple wave solutions. However,
G. El has developed a method, termed dispersive shock fitting,
which allows the leading and trailing edges of undular bores
to be found for non-integrable equations of Korteweg-de Vries
(KdV) type. This seminar will outline the development of
dispersive shock fitting for nonlinear wave equations with
nonlocal Benjamin-Ono dispersion given by a nonlocal Hilbert
transform. Dispersive shock fitting for the KdV equation will
be outlined to set this in context. Examples considered will
be the standard Benjamin-Ono equation of water wave theory and
an equation with joint Benjamin-Ono and nonlinear
Schr\"odinger dispersion from quantum mechanics.
Informes: coloquiomym@gmail.com, o al 5622-3564.