Abstract <p> Based on the theory of hyperelliptic curves, the algebraic curve method is extended to construct algebro-geometric quasiperiodic solutions of nonlocal reverse space–time soliton equations. The nonlocal reverse space–time sine-Gordon equation is chosen as an example to illustrate our method. Given the Lax matrix of the nonlocal reverse space–time sine-Gordon equation, we introduce an algebraic hyperelliptic curve <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2585_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal K_n\)</EquationSource> </InlineEquation> of genus <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2585_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>, from which the Dubrovin-type equations, a meromorphic function <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2585_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi\)</EquationSource> </InlineEquation>, and a Baker–Akhiezer function <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11232_2025_2585_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\psi_{1}\)</EquationSource> </InlineEquation> are found. Using the theory of algebraic curves, the nonlocal reverse space–time sine-Gordon flows are straightened by using the Abel–Jacobi coordinates. In accordance with the asymptotic properties of the Baker–Akhiezer function, we construct explicit theta-function representations of the Baker–Akhiezer function and the meromorphic function, including that for solutions of the nonlocal reverse space–time sine-Gordon equation. </p>

Algebro-geometric quasiperiodic solutions of the nonlocal reverse space–time sine-Gordon equation

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发表日期：

2025年1月27日

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Liang Guan,
Xianguo Geng,
Xue Geng

摘要

Abstract

Based on the theory of hyperelliptic curves, the algebraic curve method is extended to construct algebro-geometric quasiperiodic solutions of nonlocal reverse space–time soliton equations. The nonlocal reverse space–time sine-Gordon equation is chosen as an example to illustrate our method. Given the Lax matrix of the nonlocal reverse space–time sine-Gordon equation, we introduce an algebraic hyperelliptic curve \(\mathcal K_n\) of genus \(n\) , from which the Dubrovin-type equations, a meromorphic function \(\phi\) , and a Baker–Akhiezer function \(\psi_{1}\) are found. Using the theory of algebraic curves, the nonlocal reverse space–time sine-Gordon flows are straightened by using the Abel–Jacobi coordinates. In accordance with the asymptotic properties of the Baker–Akhiezer function, we construct explicit theta-function representations of the Baker–Akhiezer function and the meromorphic function, including that for solutions of the nonlocal reverse space–time sine-Gordon equation.