Constraints and Soliton Solutions for the KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equation can be generated by its recursion operator.We generalize the result to a special form of Lax pair, from which a method to constrain the integrable system to a lower-dimensional or fewer variable integrable system is proposed. A direct result is that the $n$-soliton solutions of the KdV hierarchy can be completely depicted by a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.

💡 Research Summary

The paper presents a novel framework for reducing the dimensionality of two fundamental integrable hierarchies—the Korteweg‑de Vries (KdV) hierarchy and the Ablowitz‑Kaup‑Newell‑Segur (AKNS) hierarchy—by exploiting a special form of Lax pair together with algebraic constraints.

For the KdV hierarchy the authors start from the well‑known recursion operator (R=\partial_x^2+4u+2u_x\partial_x^{-1}) that generates the infinite sequence of commuting flows and finite‑gap solutions. They introduce a Lax pair ((L,M)) where (L=\partial_x^2+u) and (M) is built directly from the recursion operator (e.g. (M=4\partial_x^3+6u\partial_x+3u_x)). The crucial step is to impose an auxiliary constraint (F(u,\psi)=0) that ties the potential (u(x,t)) to a specially chosen eigenfunction (\psi). By assuming a finite‑sum exponential ansatz for (\psi), \