演講公告 90年1月12日 (星期五)

摘要

Abstract
In this section, we review some results ofJ.H. Lee . Let $J =\pmatrix -i&0\\ 0&i\endpmatrix, \ Q = \pmatrix 0&q\\ r&0\endpmatrix, r = \va q^*, \va = \pm 1, \ P = Q(adJ)^{-1}Q =\pmatrix qr/2i&0\\ 0&-qr/2i\endpmatrix$. Consider the following system
$$dM/dx = z^2[J, M] + (zQ+P)M, Imz^2 \ne 0, \tag 1$$
$M(\cdot, z)$ is bounded and absolutely continuous and $M(x, z) \to I$ as $x \to -\infty $. By the constraints $P = Q(adJ)^{-1}Q$; we have $M(x$, $z) \to I$ as $\mid z\mid \to\infty $. Hence we may pose the inverse problem as the previous case. For $Q = \pmatrix 0&q \\ \va q^* & 0 \endpmatrix , P = Q(adJ)^{-1}Q$, then an associated evolution equation is of the form :
$$q_t = (i/2)q_{xx} + (\va/2)q^2q_x + q|q|^4, \ \va = \pm 1. \tag 2$$
Let $u = q \exp (\int ^x_{-\infty} \va iqq^*)$, then $u$ satisfies the derivative nonlinear Schr\"odinger equation (DNLS)
$$u_t = iu_{xx} + \va(u^2u*)_x. \tag 3$$
In the case $\va = -1$, we get the soliton solutions of DNLS by a system of linear algebra. Then we modify slightly the argument of Zakharov-Shabat for the solitons of nonlinear Schr\" odinger equation (NLS). It can be shown that for large time $t$, the $N$-soliton solution breakdowon into individual one-solitons. See J.H. Lee and C.T. Lin . The global existence (in time $t$) of Schwartz class Solution of Eq. (2) was obtained for the generic initial data by J.H. Lee via $L^2$-estimate. When $\va = -1, $ Eq (2) is solvable for all $t$ because of the symmetric condition $r = -q^*$. $N$-soliton solutions are also obtained via Hirota's bilinearization method.
{\bf Related Developments}
It is impossible in a short space to recount and survey all the related
results. For the analytic results, we refer to the surrey paper of R. Beals, P. Deift and X. Zhou and references their in. Recently there are many interesting books in soliton theory in the Springer Series in Nonlinear Dynamics. For examples:
What is integrability
Ed. by V.E. Zakharov;
Nonlinear Process in Physics
Ed. by A.S. Foka, D.J. Kaup, A.C. Newell and V.E. Zakharov;
Important Dovelepments in soliton theory
Ed. by A.S. Fokas and V.E. Zakharov,etc