Mixed hierarchy of soliton equations

The mixed hierarchy of soliton equations in (1+1) dimensions is introduced. It contains nonholonomic deformations of soliton equations such the KdV6 equation and the Kupershmidt deformations of soliton equations as special members. Based on the commutator representation method, a recipe for constructing zero curvature representations of mixed hierarchy is proposed. As applications, we obtain the mixed hierarchies and their zero curvature representations for the Korteweg–de Vries hierarchy, the Ablowitz–Kaup–Newell–Segur hierarchy, the modified Korteweg–de Vries hierarchy, the Toda lattice hierarchy, and the Volterra lattice hierarchy.

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