Math @ Duke

Publications [#9522] of Stephen P. Shipman
Papers Published
 Stephen P Shipman, Modulated Waves in a Semiclassical Continuum Limit of an Integrable NLS Chain,
Communications on Pure and Applied Mathematics, Vol. LIII, No. 2
[ps]
(last updated on 2000/01/18)
Abstract: A onedimensional integrable lattice system of ODEs for
complex functions Q(n,T) that exhibits dispersive phenomena
in the phase is studied. We consider wave solutions of the
local form Q(n,T) ~ q exp(i(kn+wT+c)), in which q, k, and w
modulate on long time and long space scales t=eT and x=en.
Such solutions arise from initial data of the form
Q(n,0) = q(ne)exp(ip(ne)/e), the phase derivative p' giving
the local value of the phase difference k. Formal
asymptotic analysis as e tends to 0 yields a firstorder
system of PDEs for q and p' as functions of x and t. A
certain finite subchain of the discrete system is solvable
by an inverse spectral transform. We propose formulas for
the asymptotic spectral data and use them to study the
limiting behavior of the solution in the case of initial
data Q(n)<1, which yield hyperbolic PDEs in the formal
limit. We show that the hyperbolic case is amenable to
LaxLevermore theory. The associated maximization problem
in the spectral domain is solved by means of a scalar
RiemannHilbert problem for a special class of data for all
times before breaking of the formal PDEs. Under certain
assumptions on asymptotic behaviors, the phase and amplitude
modulation of the discrete systems is shown to be governed
by the formal PDEs. Modulation equations after breaking time
are not studied. Full details of the WKB theory and
numerical results are left to another exposition.
(c) 1999 John Wiley & Sons, Inc.


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