Darboux transformations associated with Boiti-Tu eigenvalue problem (Q1122708)

\textit{M. Boiti} and \textit{G. Z. Tu} [A simple approach to the Hamilton structure of soliton equation, Nuovo Cimento 75B, 145 (1983)] introduced the following eigenvalue problem (Boiti-Tu eigenvalue problem): \[ \quad (1)\quad \phi_ x=U_{\phi},\quad U=-i\lambda \sigma_ 3+u\sigma_ 1+\lambda^{-1}(i\leq \sigma_ 3-v\sigma_ 2), \] where \(\sigma_ 1=\left( \begin{matrix} 0\quad 1\\ 1\quad 0\end{matrix} \right)\), \(\sigma_ 2=\left( \begin{matrix} 0\\ i\end{matrix} \begin{matrix} -i\\ 0\end{matrix} \right)\), \(\sigma_ 3=\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} 0\\ -1\end{matrix} \right)\) are Pauli matrices. \textit{M. Boiti}, \textit{Jerome J. P. Leon} and \textit{E. Pempinelli} [J. Math. Phys. 25, 1725-1734 (1984; Zbl 0557.35099)] introduced the auxiliary eigenvalue problem \[ (2)\quad \phi_ t=V_{\phi},\quad V=\sum^{n}_{j=0}V_ j \lambda^{n- j}+\sum^{p}_{\ell =0}W_{\ell} \lambda^{\ell -p-1}. \] (1), (2) are called evolution equations associated with the Boiti-Tu eigenvalue problem. The authors give a uniform method to find new solutions from a given system of solutions of the same system of equations. It is called Darboux transformation. For finding a new solution of every system of equations of the hierarchy, it is sufficient to solve some linear problems. The Darboux transformation has many merits.