GRB 211211A: a Neutron Star$-$White Dwarf Merger?

The gamma-ray burst GRB 211211A and its associated kilonova-like emission were reported recently. A significant difference between this association event and GRB 170817A/AT 2017gfo is that GRB 211211A has a very long duration. In this paper, we show that this association event may arise from a neutron star$-$white dwarf (NS$-$WD) merger if a magnetar leaves finally in the central engine. Within the NS$-$WD merger, the main burst of GRB 211211A could be produced by magnetic bubble eruptions from toroidal magnetic field amplification of the pre-merger NS. This toroidal field amplification can be induced by the runaway accretion from the WD debris disc if the disc is in low initial entropy and efficient wind. While the extended emission of GRB 211211A is likely involved with magnetic propelling. The observed energetics and duration of the prompt emission of GRB 211211A can be fulfilled in comparison with those of accretion in hydrodynamical thermonuclear simulation, as long as the WD has a mass $\gtrsim1M_{\odot}$. Moreover, if the X-ray plateau in GRB afterglows is due to the magnetar spin-down radiation, GRB optical afterglows and kilonova-like emission can be well jointly modeled combining the standard forward shock with the radioactive decay power of $^{56}{\rm Ni}$ adding a rotational power input from the post-merger magnetar.