Luis Vega

Higher-order nonlinear dispersive equations

We study nonlinear dispersive equations of the form atU + 2j+U P(uAx,..., u)-O x,t EX, E = where P(.) is a polynomial having no constant or linear terms. It is shown that the associated initial value problem is locally well posed in weighted Sobolev spaces. The method of proof combines several sharp estimates for solutions of the associated linear problem and a change of dependent variable which allows us to consider data of arbitrary size. INTRODUCTION In this paper we consider the initial value problem (1.1) { 8tu~a~j+ u+PO, a 2Ju) O x, t E R, j E Z+, u(x,O) = uo(X), where At = a/Ot, Ax = 0/Ox, u = u(x, t) is a real(or complex-) valued function, and P :R2j+ I R (or P :C2j+ IC) is a polynomial having no constant or linear terms; i.e., e, (1.2) P(z)= E aaz a witheo > 2 andz=(z , z2j+l) The class described in (1.1) generalizes several models arising in both mathematics and physics. In particular, it contains the KdV hierarchy [14], higherorder models in water waves problems and in elastic media (see [12] and references therein), and the equations discussed in [3, ?7]. Our purpose is to study local well-posedness of the IVP (1.1). Here the difficulties appear from the fact that, in general, techniques such as standard energy estimates, space-time (LPLq-) estimates, Galerkins method, and so on cannot be applied. Received by the editors October 19, 1992 and, in revised form, December 7, 1992. 1991 Mathematics Subject Classification. Primary 35Q30; Secondary 35G25, 35D99.

The Vortex Filament Equation as a Pseudorandom Generator

In this paper, we consider the evolution of the so-called vortex filament equation (VFE),

Spectral stability of Schroedinger operators with subordinated complex potentials

\mathbf{X}_t = \mathbf{X}_s\wedge\mathbf{X}_{ss},

On the regularity of solutions to the

taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based.