The Period Function of the Generalized Sine-Gordon Equation and the Sinh-Poisson Equation

In this paper, we consider the generalized sine-Gordon equation ${\psi }_{tx}=(1+a{\partial }_x^2)\sin \psi$ and the sinh-Poisson equation $u_{xx}+u_{yy}+\sigma \sinh u=0$, where a is a real parameter, and $\sigma$ is a positive parameter. Under different conditions, e.g., $a=0$, $a\ne 0$, and $\sigma >0$, the periods of the periodic wave solutions for the above two equations are discussed. By the transformation of variables, the generalized sine-Gordon equation and sinh-Poisson equations are reduced to planar dynamical systems whose first integral includes trigonometric terms and exponential terms, respectively. We successfully handle the trigonometric terms and exponential terms in the study of the monotonicity of the period function of periodic solutions.