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

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.