Dullin, Holger R.; Gottwald, Georg A.; Holm, Darryl D. Camassa-Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. (English) Zbl 1032.76518 Fluid Dyn. Res. 33, No. 1-2, 73-95 (2003). 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. Cited in 1 ReviewCited in 149 Documents MSC: 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations Keywords:Camassa-Holm equation; shallow water wave equation with surface tension; asymptotic expansion PDFBibTeX XMLCite \textit{H. R. Dullin} et al., Fluid Dyn. Res. 33, No. 1--2, 73--95 (2003; Zbl 1032.76518) Full Text: DOI