×

zbMATH — the first resource for mathematics

A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. (Chinese, English) Zbl 0765.58011
Chin. Ann. Math., Ser. A 13, No. 1, 115-123 (1992); translation in Chin. J. Contemp. Math. 13, No. 1, 79-89 (1992).
Following Tu’s setting [G.-Z. Tu, Nonlinear physics, Proc. Int. Conf., Shanghai/China 1989, 2–11 (1990; Zbl 0728.35122)], a new hierarchy of nonlinear evolution equations is obtained, which is associated with the linear spectral problem \[ \varphi_ x=U\varphi,\quad U=\left({\alpha_ 1\lambda+q(x,t,\lambda)\atop\alpha_ 3} {r(x,t,\lambda)\atop\alpha_ 2\lambda+s(x,t,\lambda)}\right),\quad \alpha_ 1\neq\alpha_ 2,\;\alpha_ 3\neq 0.\tag{1} \] From the trace identity it follows that these equations are not only Lax integrable, but also Liouville integrable, i.e. there exists an infinite number of conservation integrals in involution with each other and functionally independent. Furthermore, the paper deals with a kind of reduction as \(q=\alpha_ 4s\), \(\alpha_ 4\neq 1\) in (1).
If the author could solve the corresponding inverse scattering problem and find the scattering coordinates, i.e. action-angle variables, the discussion about complete integrability would be more rigorous and more complete.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
PDF BibTeX XML Cite