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.
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