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In this article, we investigate the exact and solitary wave solutions for the shallow water wave equations and the generalized Klein-Gordon equation using the exp − ( ) -expansion method. A wave transformation is applied to convert the problem into the form of an ordinary differential equation. By using this method, we found the explicit solitary wave solutions in terms of the hyperbolic functions, trigonometric functions, exponential functions and rational functions. The extracted solution plays a significant role in many physical phenomena such as electromagnetic waves, nonlinear lattice waves, ion sound waves in plasma, nuclear physics, shallow water waves and so on. It is noted that the method is reliable, straightforward and an effective mathematical tool for analytic treatment of nonlinear systems of partial differential equation in mathematical physics and engineering.

Solitary Wave Solutions for the Shallow Water Wave Equations and the Generalized Klein-Gordon Equation Using Exp(-ϕ(η) )-Expansion Method