On asymptotically equivalent shallow water wave equations

The integrable third-order Korteweg–de Vries (KdV) equation emerges uniquely at linear order in the asymptotic expansion for unidirectional shallow water waves. However, at quadratic order, this asymptotic expansion produces an entire family of shallow water wave equations that are asymptotically equivalent to each other, under a group of nonlinear, non-local, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama–Helmholtz (KH) transformations are used to present connections between shallow water waves, the integrable fifth-order KdV equation, and a generalization of the Camassa–Holm (CH) equation that contains an additional integrable case. The dispersion relation of the full water wave problem and any equation in this family agree to fifth order. The travelling wave solutions of the CH equation are shown to agree to fifth order with the exact solution.