Improved Uniform Error Bounds on a Lawson-type Exponential Integrator Method for Long-Time Dynamics of the Nonlinear Double Sine-Gordon Equation

A Lawson-type exponential integrator combined with the Fourier pseudo-spectral method is provided for the nonlinear Double Sine-Gordon equation (DSGE), while the nonlinearity is characterized by \(\beta /\epsilon \) $\beta /ϵ$ with small parameter \(\epsilon \in (0,1]\) $ϵ\in (0,1]$ and interaction parameter \(\beta \in (0,+\infty )\) $\beta \in (0,+\infty )$ . In comparison to the Sine-Gordon equation, DSGE has many properties of solitons as well as its own unique new features. This is the first work to numerically simulate the physical phenomena arising from DSG kinks collisions. The improved uniform error bounds are proved by using the regularity compensation oscillatory (RCO) technique, which are \(O(\alpha ^2\tau +h^m)\) $O({\alpha }^2\tau +h^m)$ up to the long time at \(T_{\epsilon }=T/\alpha ^2\) $T_{ϵ}=T/{\alpha }^2$ , where \(\alpha =\epsilon \) $\alpha =ϵ$ for \(\beta \ge 1\) $\beta \ge 1$ and \(\alpha ={\epsilon }/{\beta }\) $\alpha =ϵ/\beta$ for \(0<\epsilon<\beta <1\) $0<ϵ<\beta <1$ . Based on the uniform bounds, the error estimation for the discrete energy is derived. Furthermore, the improved error bounds are extended to two oscillatory DSGEs with \(O(\epsilon ^2)\) $O({ϵ}^2)$ and \(O(\epsilon ^2/\beta ^2)\) $O({ϵ}^2/{\beta }^2)$ wavelength in time. Numerical examples are provided to illustrate the accuracy and discrete energy property of the proposed method.