Abstract
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 P̃ and its basis are first presented. Second, a new isospectral problem with four potentials in the loop algebra P̃ 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.
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