--->   CHANGE IN TIME AND PLACE   <---

               SPECIAL DIFFERENTIAL GEOMETRY/PDE SEMINAR

                            Tuesday, July 29

                      --->   Padelford C36   <---
                         --->   4:00 pm   <---

                               Luis Vega
                  (Universidad del Pais Vasco, Spain)


      SMOOTHING EFFECTS AND LOCAL EXISTENCE THEORY FOR GENERALIZED
                    NONLINEAR SCHROEDINGER EQUATIONS


Consider the initial value problem (IVP) for nonlinear Schroedinger
equations of the form
                                _    _                   n
(1)      d u = i Lu + P(u, d u, u, d u),   t in R, x in R ,
          t                 x       x

         u(x,0) = u (x),
                   0

where u=u(x,t) is a complex-valued function, L is a non-degenerate,
constant-coefficient, second-order operator
              2         2
    L = sum  d   - sum d      for some k in {1,...,n},
        j<=k  xj   j>k  xj

    du = (d  u, ..., d  u),
           x1         xn

and P: C^{2n+2} -> C is a polynomial, having no constant or linear
terms, i.e.
                                          a
    P(z) = P(z ,...,z    ) =    sum    A z ,       l >= 2.
              1      2n+2    l<=|a|<=d  a

In this paper we study the local solvability of the IVP (1).  More
precisely, we shall establish that (1) is locally well posed in
appropriate Sobolev spaces.

When a special form of the nonlinear term P is assumed, the standard
energy estimates provide the desired result.  In these particular cases,
the dispersive part of the equation modeled by the operator L does not
play any role in the proof of this local result. However we shall see
that thanks to the dispersive properties of L no specific structure in P
will be necessary for the existence of a local theory.
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For more information about this seminar, visit the DG/PDE Seminar Web
page at https://www.math.washington.edu/~lee/Seminars/dg.html, or from
the Seminars, Colloquia, and Conferences section of the Math Department
home page.
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| John M. Lee, Professor of Mathematics       University of Washington |
| lee@math.washington.edu                     Mathematics Department   |
| https://www.math.washington.edu/~lee         Box 354350               |
| Phone 206-543-1735, Fax 206-543-0397        Seattle, WA 98195-4350   |
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