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

Abstract

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 ÿfth-order integrable equation in the KdV hierarchy) by using the nonlinear/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.

A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith)

“…The two-body dynamics is integrable for any b (see Figure 1), and for all values of b > 1 the peakons appear to be numerically stable and dominate the initial value problem [24]. Furthermore, with the addition of linear dispersion (u x and u xxx terms) it has been shown that not only the Camassa-Holm equation [20] but also the whole b-family (1.6) (apart from b = −1) belong to a class of asymptotically equivalent shallow water wave equations [21].…”

Blow-Up Phenomena and Global Existence to a Weakly Dissipative Shallow Water Equation

“…Dullin etal. [25] showed that Eq. (1.4) can be obtained from the shallow water elevation equation by an appropriate Kodama transformation.…”

A Note on the Degasperis-Procesi Equation

“…was derived recently as a shallow water approximation to the Euler equation [14]. The equation (1.1) presents some similarities to the Camassa-Holm equation [1,17] u t − u txx + 3uu x = 2u x u xx + uu xxx , x ∈ R, t ≥ 0.…”