The generalised Zakharov-Shabat system and the gauge group action (Q2865517)

In this paper, which extends the results of the paper by \textit{G. Grahovski} and \textit{M. Condon} [``On the Caudrey-Beals-Coifman system and the gauge group action'', J. Nonlinear Math. Phys. 15 (Suppl. 3), 197--208 (2008; \url{doi:10.2991/jnmp.2008.15.s3.20})], the author studies the generalized Zakharov-Shabat system with complex-valued and non-regular Cartan elements.NEWLINENEWLINELet \({\mathbf g}\) be a Lie algebra. In the first part of the paper the author sketches the construction of the fundamental analytic solutions of the operator \(L\) defined by NEWLINE\[NEWLINE L\Psi(t,x,\lambda)=\left (i{d\over dx}+Q(t,x)-\lambda J\right )\psi(t,x,\lambda), \eqno{(1)} NEWLINE\]NEWLINE which arises as the Lax operator in multi-component nonlinear Schrödinger equations, considering separately the cases of real and complex Cartan elements. In this latter case, the algebra \({\mathbf g}\) is semisimple. Here, NEWLINE\[NEWLINEQ(t,x)=\sum_{\alpha\in\Delta_+}(q_{\alpha}(t,x)E_{\alpha}+q_{-\alpha}(t,x)E_{-\alpha})\in {\mathbf g}_J,\qquad\;\, J=\sum_{j=1}^{\text{rank}({\mathbf g})}a_jH_j\in {\mathbf h}, NEWLINE\]NEWLINE \({\mathbf h}\) is a Cartan subalgebra of \({\mathbf g}\), \({\mathbf g}_J\) is the subalgebra of \({\mathbf g}\) consisting of the elements commuting with \(J\), which is non-regular (which means that \(\alpha(J)\neq 0\) for any root \(\alpha\) of \({\mathbf g}\)), \(\{E_{\alpha},H_j\}\) forms the Cartan-Weil basis in \({\mathbf g}\), \(\Delta_+\) is the set of positive roots of \({\mathbf g}\). Finally, the potential terms are assumed to vanish fast enough as \(|x|\to +\infty\).NEWLINENEWLINENext, the author describes the gauge-equivalent multi-component Heisenberg ferromagnetic to the multi-component nonlinear Schrödinger equation. Finally, the results are illustrated by an example of the multi-component nonlinear Schrödinger equations, related to the \(so(5, {\mathbf C})\) Lie algebra, where the corresponding Heisenberg ferromagnetic-type models are determined.