Abstract
Otis C. Wright
On the Exact Solution for the Geometric Optics Approximation
of the Defocusing Nonlinear Schr\"odinger Hierarchy
Otis C. Wright, M. G. Forest, K. T.-R. McLaughlin
It is well-known for the inviscid Burgers' equation that the precise
space-time location of the onset of the ``shock" front can be
characterized
analytically in terms of the initial data. Interestingly, this explicit
characterization
of the onset of the ``shock'' can be generalized to certain quasi-linear
systems of pde's. This talk examines an exact solution method for
systems of quasi-linear p.d.e.'s that arise
in the semi-classical limit of the integrable defocusing nonlinear
Schr\"odinger equation.
The classical hodograph transform provides an implicit solution of the
geometric optics
approximation equations (a quasilinear first-order system)
of the defocusing nonlinear Schr\"odinger equation. The implicit
solution involves certain \emph{auxiliary} functions which encode the
initial data.
These auxiliary functions are, in turn, solutions of certain
second-order linear p.d.e.'s,
which can be solved using a
classical method of Riemann. The relevant \emph{Riemann-Green}
functions which
appear in the
solution can be found \emph{explicitly}.
In this talk, the construction of the relevant Riemann-Green functions
for generic
smooth monotone initial data (consistent with the geometrical optics
approximation)
is discussed for members of the defocusing NLS integrable hierarchy.
Return to
Return to conference page