Abstract
•We present application in mathematical physics models.•Higher order first and order PDE equations.•Solitary wave solutions.•The hydrodynamic mathematical methods.•Modulation instability analysis.
Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations. In our research, we ascertain abundant new closed form traveling wave solution of the nonlinear integro-differential equations via Ito equation, integro-differential Sawada–Kotera equation, first integro-differential KP hierarchy equation and second integro-differential KP hierarchy equation by two variable (G′/G, 1/G)-expansion method with the help of computer package like Mathematica. Some shape of solutions like, bell profile solution, anti-king profile solution, soliton profile solution, periodic profile solution etc. are obtain in this investigation. Trigonometric function solution, hyperbolic function solution and rational function solution are established by using our eminent method and comparing with our results to all of the well-known results which are given in the literature. By means of free parameters, plentiful solitary solutions are derived from the exact traveling wave solutions. The method can be easier and more applicable to investigate such type of nonlinear evolution models.