On the dissipative evolution equations associated with the Zakharov-Shabat system with a quadratic spectral parameter

Lee, Jyh-Hao

doi:10.1090/S0002-9947-1989-0943304-6

On the dissipative evolution equations associated with the Zakharov-Shabat system with a quadratic spectral parameter
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by Jyh-Hao Lee

Trans. Amer. Math. Soc. 316 (1989), 327-336

DOI: https://doi.org/10.1090/S0002-9947-1989-0943304-6

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Abstract:

In this paper we derive some results for the Zakharov-Shabat system of the form $dm/dx = {z^2}[J,m] + (zQ + P)m$; $J$ is diagonal and skew-Hermitian $[8,10,12]$. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of $m$ by ${L^2}$-norm of the potential $(Q,P) [4]$. By this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data $(Q(x,0), P(x,0)) = ({Q_0},{P_0})$ is of Schwartz class.

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