Abstract
A new generalized coupled Korteweg–de Vries (KdV) hierarchy is presented starting from a 4×4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Most importantly, a new generalized coupled KdV equation is produced. Moreover, a Darboux transformation for the generalized coupled KdV equation is established with the aid of the gauge transformation between the corresponding 4×4 matrix spectral problems, by which some explicit solutions of the generalized coupled KdV equation are obtained. As a reduction, a Darboux transformation of the KdV equation and its explicit solutions are given.
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