In this article, a system of finite-dimensional involutive functions is presented and proven to be integrable in the Liouville sense. By using the nonlinearization method, the C. Neumann system associated with the modified Korteweg-de Vries (mKdV) hierarchy is obtained. Thus, the C. Neumann system is shown to be completely integrable via a gauge transformation between it and an integrable Hamiltonian system. Finally, the solution of a stationary mKdV equation and the involutive solutions of the mKdV hierarchy are secured. As two examples, the involutive solutions are given for the mKdV equation: u,+ ;uXXX- $u2u,=0 and the 5th mKdV equation v,- ~v,,,~~+~u~v,,~+~vv,v,,+ &~-$v4uX=0.
Qiao, Zhijun, "An involutive system and integrable C. Neumann system associated with the modified Korteweg-de Vries hierarchy" (1994). Mathematical and Statistical Sciences Faculty Publications and Presentations. 9.
Journal of Mathematical Physics