  Soliton-based computing is relied on their unique properties for transporting
energy and emerging intact from head-on collisions. Magnetic domain walls are
often referred to as solitons disregarding the strict mathematical definition
requiring the above scattering property. Here we demonstrate the conditions of
elastic and inelastic scattering for spin-orbit torque-induced dynamics of
antiferromagnetic domain walls on the example of a technologically relevant
Mn$_2$Au material. We show that even domain walls with opposite winding numbers
can experience elastic scattering and we present a corresponding phase diagram
as a function of the spin-orbit field strength and duration. The elastic
collision requires minimum domain walls speed which we explain assuming an
attractive potential created by domain wall pair. On the contrary, when the
domain walls move at lower speeds, their collision is inelastic and results in
a dispersing breather. Our findings will be important for the development
soliton-based computing using antiferromagnetic spintronics and we discuss
these perspective on our suggestions of how to create NOT and XOR gates.
