A finite-dimensional involutive system is presented, and the Wadati-Konno- Ichikawa (WKI) hierarchy of nonlinear evolution equations and their commutator representations are discussed in this article. By this finite-dimensional involutive system, it is proven that under the so-called Bargmann constraint between the potentials and the eigenfunctions, the eigenvalue problem (called the WKI eigenvalue problem) studied by Wadati, Konno, and Ichikawa [J. Phys. Sot. Jpn. 47, 1698 (1979)] is nonlinearized as a completely integrable Hamiltonian system in the Liouville sense. Moreover, the parametric representation of the solution of each equation in the WKI hierarchy is obtained by making use of the solution of two compatible systems.
Qiao, Zhijun. 1995. “A Completely Integrable System and Parametric Representation of Solutions of the Wadati–Konno–Ichikawa Hierarchy.” Journal of Mathematical Physics 36 (7): 3535–40. https://doi.org/10.1063/1.530979.
Journal of Mathematical Physics