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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

ϕ x =Uϕ,U=α 1 λ+q(x,t,λ) α 3 r(x,t,λ) α 2 λ+s(x,t,λ),α 1 α 2 ,α 3 0·(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=α 4 s, α 4 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:
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
35Q58Other completely integrable PDE (MSC2000)
35G10Initial value problems for linear higher-order PDE
35K25Higher order parabolic equations, general