A characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy
The authors study spectral properties of a family of the first-order differential operators \[ L(t)=Jd/dx+Q(x,t), \] with \[ J=\begin{pmatrix} i&0\\ 0&-i\end{pmatrix},\qquad Q(x,t)=\begin{pmatrix} 0&-iq(x,t)\\ ip(x,t)&0 \end{pmatrix}, \] and \(t\) is a deformation parameter. With every operator \(L(t)\) they connect a hierarchy of the \((n+1)\)th-order \(2\times 2\)-matrix-valued differential operators \(P_{n+1}\) and the appropriate AKNS hierarchy of nonlinear evolution equations for the functions \(q(x,t), p(x,t).\) Here, \[ P_{n+1}=-\sum_{l=0}^{n+1}(g_{n-l+1}J+iA_{n-l})L^l, \quad A_{l}=\begin{pmatrix} 0&-f_l\\ h_l&0\end{pmatrix}, \] and \(f_l, g_l, h_l\) are defined by the following recursion procedure: \[ \begin{multlined} f_l=(1/2)if_{l,x}-iqg_{l+1},\;g_{l+1,x}=pf_l+qh_l,\;h_{l+1}=-(1/2)ih_{l,x}+ ipg_{l+1},\;l=-1,0,1,\ldots,\\ f_{-1}=0,\;g_{0}=1,\;h_{-1}=0.\end{multlined} \] The AKNS hierarchy of nonlinear evolution equations for the functions \(q(x,t), p(x,t)\) is defined is follows: \[ dL/dt=[P_{n+1},L]. \] The authors define the matrix function \(Q(x,t)\) whose entries are elliptic functions with a common period lattice as the Picard-AKNS potential if for all values of the spectral parameter \(E\in{\mathbb{C}}\) the differential equation \[ L\Psi=E\Psi, \] has a fundamental system of solutions meromorphic with respect to \(x.\) They define the matrix function \(Q(x,t)\) as the algebro-geometric AKNS potential if the appropriate entries \(q(x,t), p(x,t)\) are stationary solutions to some equation of the AKNS hierarchy. The principal new result of the paper is the following theorem: A matrix-valued function \(Q(x,t)\) is an elliptic algebro-geometric AKNS potential if and only if it is a Picard-AKNS potential. This theorem is a generalization from the scalar to matrix case of previous results by the authors for the KdV hierarchy. The authors illustrate the theorem by a number of examples. In order to prove the theorem the authors have developed special technical means which are to be useful for other similar problems.
scientific article; zbMATH DE number 1230684
scientific article