On asymptotically equivalent shallow water wave equations

Abstract

The integrable 3rd-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, nonlocal, normal-form transformations introduced by Kodama in combination with the application of the Helmholtz-operator. These Kodama-Helmholtz transformations are used to present connections between shallow water waves, the integrable 5th-order Korteweg-de Vries 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 5th order. The travelling wave solutions of the CH equation are shown to agree to 5th order with the exact solution.

Well-posedness, blow-up phenomena, and global solutions for the b-equation

“…[21], [22]. For the case b ¼ À1, the corresponding Kodama transformation is singular and the asymptotic ordering is violated, cf.…”

“…For the case b ¼ À1, the corresponding Kodama transformation is singular and the asymptotic ordering is violated, cf. [21], [22]. The solutions of the b-equation (1.2) with c 0 ¼ G ¼ 0 were studied numerically for various values of b in [27], [28], where b was taken as a bifurcation parameter.…”

“…The Degasperis-Procesi equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the Camassa-Holm shallow water equation ([21], [22]). An inverse scattering approach for computing n-peakon solutions to the Degasperis-Procesi equation was presented in [38].…”

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].…”

A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions

“…In the context of nonlinear fifth-order KdV type equations, studies are flourishing because these equations are able to describe the real features in a variety of scientific applications and engineering areas and would have much practical/physical meaning [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Fifthorder KdV (KdV5) type equations take the form u t = u xxxxx + f (x, t, u x , u xx , u xxx ),…”