Lattice Boltzmann model for twodimensional generalized sineGordon equation 

Keywords:Lattice Boltzmann method, SineGordon equation, ChapmanEnskog expansion, Soliton. 
Abstract: 
The nonlinear sineGordon equation arises in various problems in science and engineering. In
this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of twodimensional generalized sineGordon equation, including damped and undamped sineGordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying
the ChapmanEnskog expansion, the governing equation is recovered correctly from the
lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained.
The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and
efficient. Numerical solutions for cases involving the most known from
the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good longtime numerical behavior for the generalized sineGordon equation. Moreover, the model can also be applied to other twodimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and KleinGordon equation. 



