Traveling wave solutions for wave equations with two exponential nonlinearities

We use a simple method that leads to the integrals involved in obtaining the traveling wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained while when that term is nonzero all the basic traveling wave solutions of Liouville, Tzitzeica and their variants, as well as sine/sinh-Gordon equations with important applications in the phenomenology of nonlinear physics and dynamical systems are found through a detailed study of the corresponding elliptic equations