Existence and decay of solitary wave solutions for Korteweg–de Vries-type equations

In this paper, we investigate the existence of solitary wave solutions for a class of Korteweg–de Vries (KdV) equations, which essentially reduces to solving a class of quasilinear second-order elliptic equations. We first employ the dual approach to transform the original equation into a semilinear second-order elliptic equation. Then, using variational methods, truncation techniques, and minimax arguments, we conduct a detailed analysis of the existence, regularity, decay properties, and the mountain pass geometry of the solutions. In particular, we establish the existence, smoothness, and exponential decay of solitary wave solutions to the KdV equation.