A New Approach for Generating the TX Hierarchy as well as Its Integrable Couplings

Abstract

Tu Guizhang and Xu Baozhi once introduced an isospectral problem by a loop algebra with degree being $\lambda$, for which an integrable hierarchy of evolution equations (called the TX hierarchy) was derived under the frame of zero curvature equations. In the paper, we present a loop algebra whose degrees are $2\lambda$ and $\mathrm{2}\lambda +1$ to simply represent the above isospectral matrix and easily derive the TX hierarchy. Specially, through enlarging the loop algebra with 3 dimensions to 6 dimensions, we generate a new integrable coupling of the TX hierarchy and its corresponding Hamiltonian structure.