Spectral method for solving nonlinear wave equations

In this paper, we consider the following nonlinear wave equations: $$\begin{array}{l} \frac{{\partial ^2 \phi }}{{\partial t^2 }} - \frac{{\partial ^2 \phi }}{{\partial x^2 }} + \mu ^2 \phi + v^2 \chi ^2 \phi + f\left( {|\phi |^2 } \right)\phi = 0, \\ \frac{{\partial ^2 \phi }}{{\partial t^2 }} - \frac{{\partial ^2 \chi }}{{\partial x^2 }} + \alpha ^2 \chi + v^2 \chi |\phi |^2 + g(\chi ) = 0 \\ \\ \end{array}$$ with the periodic-initial conditions: $$\begin{array}{l} \phi (x---\pi ,t) = \phi (x + \pi ,t), \chi (x---\pi ,t) = \chi (x + \pi ,t), \\ \phi (x,0) = \hat \phi _0 (x), \phi _t (x,0) = \hat \phi _1 (x), \\ \chi (x,0) = \hat \chi _0 (x), \chi _t (x,0) = \hat \chi _1 (x), \\ - \infty< x< \infty , 0 \le t \le T. \\ \end{array}$$ We put forward the computational method for solving equations (1.1)–(1.4). The method is spectral method for space variable, but is difference method for time variable. We make estimation of error, and prove convergence and stability for this method.