A Zakharov-Shabat inverse scattering problem with a polynomial spectral dependence in the potential (Q762672)

The inverse scattering problem for the Zakharov-Shabat system \(y'+iK\sigma_ 3y=vy\), \(x\in {\mathbb{R}}\), \(y=\left( \begin{matrix} y_ 1\\ y_ 2\end{matrix} \right)\) is studied, where the 'potential' v depends on the spectral parameter \(K: v(K,x)=\sum^{n}_{p=0}K^{p/2n}v_ p(x),\) \(v_ p=\left( \begin{matrix} 0\\ u^-_ p\end{matrix}^{u^+_ p}_{0}\right)\). The \(2(n+1)\) 'scalar potentials' \(u^-_ p\) and \(u^-_ p\) are supposed to be sufficiently regular complex functions of \(x\in {\mathbb{R}}\), decreasing fast enough as \(| x| \to \infty.\) The problem is solved by reduction to the inverse scattering problem for two coupled Zakharov-Shabat systems in 4n-dimensional space \(y^{\pm '}+iK(\sigma_ 3\otimes I_{2n})y^{\pm}=(U^{\pm}\pm KQ^{\pm})y^{\pm}.\)