New soliton wave structures of fractional Gilson-Pickering equation using tanh-coth method and their applications

•A physical model of fractional Gilson-Pickering equation is considered.•A set of some new solitary wave solutions are constructed using highly performance and efficient the tanh-coth method.•Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given.•The graphical illustrations of two, three-dimensional, and contour graphs have been depicted. The present article discovers the new solitary wave solutions of the well–known Gilson-Pickering model by considering the fractional derivative. The solitary wave solutions are established by considering the proficient technique named as a tanh-coth method with aid of the computational program. The equation is reformulated to a fractional order derivative by using the conformable derivative operator. As a result, a set of new soliton solutions of fractional Gilson-Pickering are efficaciously assembled. Moreover, different types of soliton solutions in the form of 3D-plots, contour plots, and 2D-plots by considering the several different values of parameters are presented. The attained outcomes are generally new for the considered model equation, and the results demonstrate that the used method is competent, direct, and concise which can be used in more complex phenomena.