Quasi-periodic solutions to the sine-Gordon equation are expressed in terms of $\wp_{ 1,2g-1}$-function, and reality conditions are completely specified. This new result leads to computation and graphical representation of non-linear wave solutions. A way of obtaining the finite-gap solutions by means of algebro-geometric integration is exposed in detail, as well as the construction of the sine-Gordon hierarchy on coadjoint orbits in the loop algebra of $\mathfrak{ su}(2)$.
Categories: nlin.SI math-ph math.MP