On the Rotating Nonlinear Klein--Gordon Equation: NonRelativistic Limit and Numerical Methods
Norbert J. Mauser, Yong Zhang, Xiaofei Zhao
Abstract
We consider numerics/asymptotics for the rotating nonlinear Klein--Gordon (RKG) equation, an important PDE in relativistic quantum physics that can model a rotating galaxy in Minkowski metric and serves also as a model, e.g., for a “cosmic superfluid.” First, we formally show that in the nonrelativistic limit RKG converges to coupled rotating nonlinear Schrödinger equations (RNLS), which are used to describe the particle-antiparticle pair dynamics. Investigations of the vortex state of RNLS are carried out. Second, we propose three different numerical methods to solve RKG from relativistic regimes to nonrelativistic regimes in polar and Cartesian coordinates. In relativistic regimes, a semi-implicit finite difference Fourier spectral method is proposed in polar coordinates where both rotation terms are diagonalized simultaneously. In nonrelativistic regimes, to overcome the fast temporal oscillations, we adopt the rotating Lagrangian coordinates and introduce two efficient multiscale methods with uniform accuracy, i.e., the multirevolution composition method and the exponential integrator. Various numerical results confirm (uniform) accuracy of our methods. Simulations of vortices dynamics are presented.