  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
