A novel geometric method is applied to the problem of describing traveling wave solutions of the generalized Korteweg–de Vries (gKdV) equation in the form \( u_t + u_{xxx} + a(u)u_x = 0, \) where a(u) is a smooth function characterizing the nonlinearity. Using the traveling wave ansatz, the gKdV equation reduces to an ordinary differential equation (ODE), which we analyze via the \(\mathcal {C}^\infty \) -structure-based method, a geometric framework involving sequences of involutive distributions and Pfaffian equations. Starting with the symmetry \(\partial _z\) , we construct a \(\mathcal {C}^\infty \) -structure for the ODE and apply the stepwise integration algorithm to obtain an implicit general solution. Then, we derive explicit solutions for specific forms of a(u), including the modified KdV and Schamel–KdV equations, as well as power-law nonlinearities.
