Pseudo-canonical systems with rational Weyl functions: Explicit formulas and applications (Q1268416)

Differential systems called pseudo-canonical are considered on the half-axis \(x\geq 0\): \[ {d\over dx}u(x,\lambda) =i\bigl(\lambda j+V(x) \bigr) u(x, \lambda), \] with \[ j=\left[ \begin{matrix} I_m & 0\\ 0 & -I_m\end{matrix} \right], \quad V=\left[ \begin{matrix} 0 & v\\ v^* & 0\end{matrix} \right], \] \(u\) and \(U\) are \(2m\times 2m\) matrix functions, \(v\) is an \(m\times m\) locally summable matrix of a special form. The corresponding operator has both selfadjoint and skew selfadjoint parts. The basic role belongs to the introduced Weyl function. Solutions to the direct and inverse spectral problems are derived. State space methods from mathematical system theory are used. Applications to the corresponding matrix nonlinear Schrödinger equation and to the modified Korteweg-de Vries equation are given.