| 李志豪
教授 ( 中央研究院 )
N×N Zakharov-Shabat with a Polynomial Spectral Parameter I, II
摘要 |
Abstract
Here we generalize Beals-Coifman's results for the first order system $dM/dx =z[J,
M]+QM$. Let $J = diag(id_1, id_2, ..., id_N), d_1 < d_2 < \cdot \cdot \cdot <
d_N, i = \sqrt{-1}. \; \sum = \{z: Im(z^n) = 0\}$. Let $q_1, q_2, ..., q_n$ be $M_N({\Bbb
C})$-valued functions and $q_j, dq_j/dx \in L^1$ for $j = 1, 2, ..., n$. We consider the
following system:
$$\aligned
& dM/dx = z^n[J, M] + (z^{n-1}q_1 + z^{n-2}q_2 + \cdot \cdot \cdot +q_n)M, \; z \notin
\sum,\\
& M(\cdot, z) \text{ is bounded and absolutely continuous, } M(x, z)
\to I \text{as } x \to -\infty.
\endaligned
$$
We define a satisfactory scattering transform. We also solve the inverse problem.
In order to pose the invere problem properly, the following constraints on the potentials
are necessary: The diagonal parts of $q_j$ must be expressible by the off-diagonal parts
of the preceding potentials. These constraints described below enable us to have $M$
normalized at $z = \infty$, which is crucial in our argument. Our procedure is similar to
that of Beals-Coifman, but with the following new feature: for the forward problem we have
to control $M$ at $z = \infty $. A transform of the spectral equation enables us to
control the $z$-dependence in the potentials. Following the set-up of Beals-Coifman, we
may solve $M$ via Fredholm integral equation, or we may apply the wedge product technique
to solve the columns of $M$ Volterra equations. We refer the readers to the papers
Beals-Coifman, J. H. Lee .
In the case $n=2$ and $N = 2$, we solve a nonlinear evolution equation which can be
transformed explicitly into the derivative nonlinear Schr\"odinger equation
considered by Kaup-Newell. |
90年1月10日 (星期三)
14:00-16:00
新數館308室
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