# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. (Chinese; English) Zbl 0765.58011

Following Tu’s setting [G. 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

${\varphi }_{x}=U\varphi ,\phantom{\rule{1.em}{0ex}}U=\left(\genfrac{}{}{0pt}{}{{\alpha }_{1}\lambda +q\left(x,t,\lambda \right)}{{\alpha }_{3}}\genfrac{}{}{0pt}{}{r\left(x,t,\lambda \right)}{{\alpha }_{2}\lambda +s\left(x,t,\lambda \right)}\right),\phantom{\rule{1.em}{0ex}}{\alpha }_{1}\ne {\alpha }_{2},\phantom{\rule{4pt}{0ex}}{\alpha }_{3}\ne 0·\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 $q={\alpha }_{4}s$, ${\alpha }_{4}\ne 1$ 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 systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q58 Other completely integrable PDE (MSC2000) 35G10 Initial value problems for linear higher-order PDE 35K25 Higher order parabolic equations, general
##### Keywords:
integrability; generalized Hamiltonian equation; reduction