Exact quasi-periodic solutions to the sine(sinh)-Gordon equations: The method for computation and analysis

The sine(sinh)-Gordon hierarchy of integrable Hamiltonian systems is described in detail, and all dynamic variables are expressed in terms of the $\wp$-functions that uniformize the associated spectral curve. Quasi-periodic solutions to the sine(sinh)-Gordon equations are obtained in terms of the function $\wp_{1,2g-1}$, reality conditions are revised, and a method of computation and analysis is presented. The proposed method is designed to analyze solutions by means of the Hamiltonian technique, which is illustrated in genera one and two.