The set of coupled difference equations describing the dynamics of a monatomic chain is reduced to that of uncoupled anharmonic oscillators through the use of the translational invariance of the equations. Poincare's perturbation scheme is then used to obtain a space- and time-wise periodic wave solution together with a nonlinear dispersion relation between frequency, wavenumber and the amplitude. For all wavelengths, the frequency of vibration is observed to increase beyond the corresponding value for the linear case by terms proportional to the square of the amplitude multiplied by the coefficients of the cubic as well as quartic interactions. This solution of the lattice equations suggests a class of solutions of the generalized Korteweg-de Vries (K. de V.) equation which is related to the long wave limit of the nonlinear lattice by means of a semicharacteristic variable stretching transformation. Numerical results for the dispersion relations are presented for the semi-empirical Morse, Born-Meyer and Lennard-Jones potentials. For these physical examples, the contributions of the cubic and quartic terms of the interaction potential are found to be of the same order.