A new super integrable hierarchy and a generalized super-AKNS hierarchy

TL;DR

This work develops a non-isospectral integrable framework on the loop algebra of the Lie superalgebra $\\mathfrak{osp}(1,6)$ using a Lax pair $\\phi_x=U\\phi$, $\\phi_t=V\\phi$ with compatibility $U_t - V_x + [U,V]_s =0$ and $\\lambda_t \\neq 0$. Employing the supertrace identity, it constructs a $(1+1)$-dimensional super integrable hierarchy with Hamiltonians $H_{n+1}^1=\\frac{2}{n+1}(a_{n+2}+b_{n+2}+c_{n+2})$ and flows $u_t=J_1P_{1,n+1}$, with reductions to both the AKNS and generalized super-AKNS hierarchies under appropriate spectral choices. The paper then derives a $(2+1)$-dimensional generalization by restricting to a reduced pair $U_{sAKNS},V_{sAKNS}\\in\\widetilde{\\mathfrak{osp}}(1,2)$, producing a hierarchy $u_t=J_2(P_{2,n+1}+\\tilde{u}_x)$ with $P_{2,n+1}=\\frac{2}{n+1}\\frac{\\delta}{\\delta u}a_{n+2}$ and $H^2_{n+1}=\\frac{2}{n+1}a_{n+2}$, i.e., a $(2+1)$-D generalization of the super-AKNS hierarchy. Together, these results provide a unified framework for constructing and Hamiltonian-structuring super-integrable hierarchies on $\\mathfrak{osp}(1,6)$ and its $(2+1)$-D extension, with explicit reductions to known super AKNS-type systems.

Abstract

In this paper, we investigate a non-isospectral problem on the loop algebra of the Lie superalgebra osp(1,6), and construct an super integrable system using the supertrace identity. The resulting super integrable system can be reduced to the super-AKNS hierarchy under certain conditions. By reconsidering a new (2 + 1)-dimensional non-isospectral problem with spectral matrices satisfying these conditions, we obtain a (2+1)-dimensional generalization of the superAKNS hierarchy.