A ${\mathcal{C}}^{\mathrm{\infty }}$-structure-based approach to traveling wave solutions of the gKdV equation

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 ${\mathcal{C}}^{\mathrm{\infty }}$-structure-based method, a geometric framework involving sequences of involutive distributions and Pfaffian equations. Starting with the symmetry ${\mathrm{\partial }}_z$, we construct a ${\mathcal{C}}^{\mathrm{\infty }}$-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.