Abstract
Resorting to the characteristic polynomial of Lax matrix for a Harry–Dym-type hierarchy, we define a trigonal curve, on which appropriate vector-valued Baker–Akhiezer function and meromorphic function are introduced. With the help of the theory of trigonal curve and three kinds of Abelian differentials, we obtain the explicit Riemann theta function representations of the meromorphic function, from which we obtain the quasi-periodic solutions for the entire Harry–Dym-type hierarchy.
Highlights
The Harry–Dym equation ut = (u− 2 ) xxx, Citation: Feng, Q.; Wu, L
It was shown that the Harry–Dym equation admits many properties typical for soliton equations, such as inverse scattering transform, bi-Hamiltonian structure, and an infinite number of conservation laws and symmetries
The principal aim of the present paper is to study the algebro-geometric constructions and quasi-periodic solutions [25,26,27,28,29] of the Harry–Dym-type hierarchy, with the aid of the theory of trigonal curve [30,31]
Summary
The quasi-periodic and involutive solutions of Harry–Dym equation were discussed in [8,9,10,11]. [22,23] derived a hierarchy of Harry–Dym-type equations and discussed their parametric solutions through the method of nonlinearization. The principal aim of the present paper is to study the algebro-geometric constructions and quasi-periodic solutions [25,26,27,28,29] of the Harry–Dym-type hierarchy, with the aid of the theory of trigonal curve [30,31].
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