A family of new integrable couplings with two arbitrary functions of TC hierarchy

Seeking new integrable couplings of the known integrable hierarchies of evolution equations is a quite new interesting aspect and is of great significance in soliton theory. In this paper, a loop algebra P and its basis are first presented. Second, a new isospectral problem with four potentials in the loop algebra P is considered to construct integrable couplings of the well-known TC hierarchy by the zero-curvature equation such that a family of new integrable couplings including two arbitrary functions are obtained. In particular, when we set two arbitrary functions to be zero, a special integrable couplings of the generalized Korteweg–de Vries equation is also given.