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 nonlinearityis characterized by beta/ETXwith small parameter epsilon is an element of(0,1]and interaction parameter beta is an element of (0,+ infinity). 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 areO(alpha(2)tau + h(m))up to the long time at T-is an element of= T/alpha(2), where alpha = is an element of for beta >= 1 and alpha = is an element of/beta for 0 < is an element of < 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(is an element of(2))and O(is an element of(2)/beta(2))wavelength in time. Numerical examples are provided to illustrate the accuracy and discrete energy property of the proposed method.