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Zbl 0765.58011
Ma, Wenxiu
A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. (Chinese, English)
[J] Chin. Ann. Math., Ser. A 13, No. 1, 115-123 (1992); translation in Chin. J. Contemp. Math. 13, No. 1, 79-89 (1992). ISSN 1000-8314

Following Tu's setting [{\it 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 $$\varphi\sb x=U\varphi,\quad U=\left({\alpha\sb 1\lambda+q(x,t,\lambda)\atop\alpha\sb 3} {r(x,t,\lambda)\atop\alpha\sb 2\lambda+s(x,t,\lambda)}\right),\quad \alpha\sb 1\ne\alpha\sb 2,\ \alpha\sb 3\ne 0.\tag 1 $$ 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 $q=\alpha\sb 4s$, $\alpha\sb 4\ne 1$ in (1).\par 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.
[Dawei Zhuang (Guangzhon)]

MSC 2000:: *37K10 Completely integrable systems etc.
35Q58 Other completely integrable PDE
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general

Keywords: integrability; generalized Hamiltonian equation; reduction

Citations: Zbl 0728.35122

Cited in: Zbl 1014.37043

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