A twisted integrable hierarchy with \(\mathbb{D}_2\) symmetry (Q2872316)

A loop algebra approach to the Gerdjikov-Mikhailov-Valchev (GMV) equation is provided to exploit the associated twisted integrable structure and a new twisted integrable hierarchy is discovered. Using the twisted loop algebra structure, a transparent treatment of the associated scattering and inverse scattering theory is obtained and the initial value problem for the GMV equation is solved.NEWLINENEWLINEThe paper is organized as follows. In Section 2, the twisted \(\frac{\mathrm{U}(3)}{\mathrm{U}(1)\times \mathrm{U}(2)}\) hierarchy via a non-split factorization of the loop algebra is defined and explicit formulas of a decisive coefficient, for the GMV equation, in the Lax pair are computed. Section 3 discusses the GMV equation and its relation with the twisted \(\frac{\mathrm{U}(3)}{\mathrm{U}(1)\times \mathrm{U}(2)}\) hierarchy. Section 4 and 5 are devoted to the scattering and inverse scattering theory of the twisted \(\frac{\mathrm{U}(3)}{\mathrm{U}(1)\times \mathrm{U}(2)}\) hierarchy. The Cauchy problems of twisted \(\frac{\mathrm{U}(3)}{\mathrm{U}(1)\times \mathrm{U}(2)}\) flows and the GMV equation are solved in Section 6.