On the rotating nonlinear Klein-Gordon equation: non-relativistic limit and numerical methods

Norbert J. Mauser (1) , Yong Zhang (2) , Xiaofei Zhao (3)

1 WPI - Wolfgang Pauli Institute
2 TJU - Tianjin University
3 MINGUS - Multi-scale numerical geometric schemes

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". Firstly, we formally show that in the non-relativistic limit RKG converges to coupled rotating nonlinear Schrödinger equations (RNLS), which is used to describe the particle-antiparticle pair dynamics. Investigations of the vortex state of RNLS are carried out. Secondly, we propose three different numerical methods to solve RKG from relativistic regimes to non-relativistic 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. While in non-relativistic regimes, to overcome the fast temporal oscillations, we adopt the rotating La-grangian coordinates and introduce two efficient multiscale methods with uniform accuracy, i.e., the multi-revolution composition method and the exponential integrator. Various numerical results confirm (uniform) accuracy of our methods. Simulations of vortices dynamics are presented.

Mots clés: Rotating nonlinear Klein-Gordon equation; cosmological models; non-relativistic limit; rotating two-component BEC; quantized vortex; numerical methods

Domaines: Physique Numérique [physics.comp-ph]; Analyse numérique [math.NA]; Cosmologie et astrophysique extra-galactique [astro-ph.CO]

Norbert J. Mauser, Yong Zhang, Xiaofei Zhao. On the rotating nonlinear Klein-Gordon equation: non-relativistic limit and numerical methods. 2018. ⟨hal-01956352⟩