Self-dual configurations in a generalized Abelian Chern-Simons-Higgs model with explicit breaking of the Lorentz covariance

We have studied the existence of self-dual solitonic solutions in a generalization of the Abelian Chern-Simons-Higgs model. Such a generalization introduces two different nonnegative functions, $\omega_1(|\phi|)$ and $\omega(|\phi|)$, which split the kinetic term of the Higgs field - $|D_\mu\phi|^2 \rightarrow\omega_1 (|\phi|)|D_0\phi|^2-\omega(|\phi|) |D_k\phi|^2$ - breaking explicitly the Lorentz covariance. We have shown that a clean implementation of the Bogomolnyi procedure only can be implemented whether $\omega(|\phi|) \propto \beta |\phi|^{2\beta-2}$ with $\beta\geq 1$. The self-dual or Bogomolnyi equations produce an infinity number of soliton solutions by choosing conveniently the generalizing function $\omega_1(|\phi|)$ which must be able to provide a finite magnetic field. Also, we have shown that by properly choosing the generalizing functions it is possible to reproduce the Bogomolnyi equations of the Abelian Maxwell-Higgs and Chern-Simons-Higgs models. Finally, some new self-dual $|\phi|^6$-vortex solutions have been analyzed both from theoretical and numerical point of view.