New optical solitons of double Sine-Gordon equation using exact solutions methods

•To investigate the Sine-Gordon equation by the rational Exp–function, tan, extended tan, sech and sine–cosine methods.•To get exact solutions with distinct physical structures such as kink, bright-dark, and bright soliton solutions.•To examine the validity of all the obtained exact solutions through the use of the symbolic computation system.•Applications of rational Exp–function, tan, extended tan, sech and sine–cosine methods in various fields of physical sciences and engineering.