  Chiral anomaly and the novel quantum phenomena it induces have been widely
studied for Dirac and Weyl fermions. In most typical cases, the Lorentz
covariance is assumed and thus the linear dispersion relations are maintained.
However, in realistic materials, such as Dirac and Weyl semimetals, the
non-linear dispersion relations appear naturally. We develop a kinetic
framework to study the chiral anomaly for Weyl fermions with non-linear
dispersions by using the methods of Wigner function and semi-classical
equations of motion. In this framework, the chiral anomaly is sourced by Berry
monopoles in momentum space and could be enhanced or suppressed due to the
windings around the Berry monopoles. Our results can help understand the chiral
anomaly-induced transport phenomena in non-relativistic systems.
