Plasma is 99% of the cosmic matter between stars and galaxies and some planets make up the most of its material. Plasma is found in other forms. In the industry, it is used in television screens, in neon lamps, and in special research, as well as in welding, and many industrial fields. Dusty and complex plasma have captured the attention of plasma physicists especially in what concerns envisage of technological applications, so we take the plasma system and we solve it by numerical analytical methods. In the book we research the plasma and talk about it in general, describe its types, characteristics, the difference between plasma blood and plasma physics and the difference between plasma and gas. The book also discusses the soliton waves, their properties, the mathematical description of plasma and some ways to solve them. Finally, the book mentions applications for soliton solution in the plasma ionic waves by numerical schemes to obtain soliton solutions for systems of partialdifferential equations.