An efficient finite difference/Hermite-Galerkin spectral method for time-fractional coupled sine-Gordon equations on multidimensional unbounded domains and its application in numerical simulations of vector solitons

Abstract This study is devoted to the numerical simulation of vector solitons described by the time-fractional coupled sine–Gordon equations in the sense of Caputo fractional derivative, where the problem is defined on the multidimensional unbounded domains R d ( d = 2 , 3 ) . For this purpose, we employ the Hermite–Galerkin spectral method with scaling factor for the spatial approximation to avoid the errors introduced by the domain truncation, and we apply the finite difference method based on the Crank–Nicolson method for the temporal discretization. Comprehensive numerical studies are carried out to verify the accuracy and the stability of our method, which shows that the method is convergent with ( 3 − max { α 1 , α 2 } ) -order accuracy in time and spectral accuracy in space. Here, α i ( 1 α i 2 , i = 1 , 2 ) are the orders of the Caputo fractional derivative. In addition, the effect of the Caputo fractional derivative on the evolutions of the vector solitons is numerically studied. Finally, several numerical simulations for both two- and three-dimensional cases of the problem are performed to illustrate the robustness of the method as well as to investigate the collisions of circular and elliptical ring vector solitons.