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Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. (English) Zbl 1032.76518
Summary: 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 Y. Kodama [Phys. Lett. A 107, No. 6, 245–249 (1985); Phys. Lett. A 112, No. 5, 193–196 (1985); Phys. Lett. A 123, 276–282 (1987)]. 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.

MSC:
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q53KdV-like (Korteweg-de Vries) equations
35Q51Soliton-like equations