  We study the ion acoustic solitary waves in the four-component complex plasma
consisting of cold inertial ions, positrons, cold and hot (two-temperature)
electrons, where the electrons and positrons are the Cairns-Tsallis
distribution and have different nonthermal and nonextensive parameters. Base on
the plasma hydrodynamic equations and Sagdeev pseudo-potential theory, we
derive the conditions of the solitary waves to exist in the plasma, such as the
Sagdeev pseudo-potential, the normalized electrostatic potential, the lower and
upper limits of Mach number, and the compressive/rarefactive solitary wave. We
show that, according to the present study, the solitary wave solutions exist
only if the positrons are a Maxwellian distribution and the two-temperature
electrons have the same nonextensive and nonthermal parameters. Numerical
analyses are made for the conditions of solitary waves depending on the
nonextensive and nonthermal parameters in the Cairns-Tsallis distribution.
