In this study, exact analytical solutions of the nonlinear Tzitzéica–Dodd–Bullough (TDB) equation are obtained by employing the φ⁶-expansion method. The TDB equation is an important nonlinear evolution model that arises in several areas of applied mathematics and physics, including differential geometry, plasma physics, nonlinear field theory, and wave propagation phenomena. By introducing an appropriate traveling wave transformation, the governing nonlinear partial differential equation is reduced to an ordinary differential equation, which is subsequently solved using a systematic algebraic procedure associated with the φ⁶-expansion framework. The resulting solutions are expressed in logarithmic forms, leading to diverse classes of nonlinear wave structures such as solitary waves, periodic waves, and singular wave profiles. These analytical expressions reveal the rich dynamical behavior of the model and provide insight into the influence of system parameters on wave amplitude, localization, and propagation characteristics. Furthermore, the obtained results demonstrate that the proposed method offers an efficient and reliable mathematical tool for constructing exact solutions of nonlinear equations involving exponential nonlinearities. The derived solutions may also serve as useful benchmarks for validating numerical simulations and for investigating nonlinear wave interactions in related physical systems. Overall, this work contributes to the theoretical understanding of nonlinear wave dynamics and highlights the applicability of the φ⁶-expansion method to a broad class of nonlinear evolution equations arising in applied science and engineering contexts.