Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves

We derive the Camassa–Holm equation (CH) as a shallow water wave equation with surface tension in an asymptotic expansion that extends one order beyond the Korteweg–de Vries equation (KdV). We show that CH is asymptotically equivalent to KdV5 (the fifth-order integrable equation in the KdV hierarchy) by using the non-linear/non-local transformations introduced in Kodama ( Phys. Lett. A 107 (1985a) 245 ; Phys. Lett. A 112 (1985b) 193 ; Phys. Lett. A 123 (1987) 276 ). We also classify its travelling wave solutions as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.