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Explicit quasi-periodic solutions of the Vakhnenko equation. (English) Zbl 1304.35045

Summary: The trigonal curve associated with the Vakhnenko equation is introduced by using the Lax matrix for the \(n\)th stationary positive flow. Based on the theory of the trigonal curve and the properties of the three kinds of Abel differentials, the Riemann theta function representation for the Baker-Akhiezer function is derived, from which the straightening out of flows in the Jacobian variety is exactly given through the Abel maps. We finally arrive at quasi-periodic solutions of the Vakhnenko equation.{
©2014 American Institute of Physics}

MSC:

35B15 Almost and pseudo-almost periodic solutions to PDEs
35G20 Nonlinear higher-order PDEs
14K25 Theta functions and abelian varieties
35G25 Initial value problems for nonlinear higher-order PDEs
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