  Employing first principles electronic structure calculations in conjunction
with the frozen-magnon method we calculate exchange interactions, spin-wave
dispersion, and spin-wave stiffness constants in inverse-Heusler-based spin
gapless semiconductor (SGS) compounds Mn$_2$CoAl, Ti$_2$MnAl, Cr$_2$ZnSi,
Ti$_2$CoSi and Ti$_2$VAs. We find that their magnetic behavior is similar to
the half-metallic ferromagnetic full-Heusler alloys, i.e., the intersublattice
exchange interactions play an essential role in the formation of the magnetic
ground state and in determining the Curie temperature, $T_\mathrm{c}$. All
compounds, except Ti$_2$CoSi possess a ferrimagnetic ground state. Due to the
finite energy gap in one spin channel, the exchange interactions decay sharply
with the distance, and hence magnetism of these SGSs can be described
considering only nearest and next-nearest neighbor exchange interactions. The
calculated spin-wave dispersion curves are typical for ferrimagnets and
ferromagnets. The spin-wave stiffness constants turn out to be larger than
those of the elementary 3$d$-ferromagnets. Calculated exchange parameters are
used as input to determine the temperature dependence of the magnetization and
$T_\mathrm{c}$ of the SGSs. We find that the $T_\mathrm{c}$ of all compounds is
much above the room temperature. The calculated magnetization curve for
Mn$_2$CoAl as well as the Curie temperature are in very good agreement with
available experimental data. The present study is expected to pave the way for
a deeper understanding of the magnetic properties of the inverse-Heusler-based
SGSs and enhance the interest in these materials for application in spintronic
and magnetoelectronic devices.
