Solitons in the Camassa-Holm shallow water equation. (English) Zbl 0961.76512

Cooper, Fred; Shepard, Harvey

Phys. Lett., A 194, No. 4, 246-250 (1994).

Summary: We study the class of Camassa-Holm shallow water equations derived from the Lagrangian \[ L= \int[\frac 12(\varphi_{xxxx}-\varphi_x)\varphi_t-\frac 12(\varphi_{x})^3-1/2\varphi_x(\varphi_{xx})^2-\frac 12\kappa\varphi^2_x]dx, \] using a variational approach. This class contains “peakons” for \(\kappa=0\), which are solitons whose peaks have a discontinuous first derivative. We derive approximate solitary wave solutions to this class of equations using trial variational functions of the form \(u(x, t)=\varphi_x=A(t)\exp[-\beta (t)|x-q(t)|^2]\) in a time-dependent variational calculation. For the case \(\kappa=0\) we obtain the exact answer. For \(\kappa\neq 0\) we obtain the optimal variational solution. For the variational solution having fixed conserved momentum \(P=\int \frac 12(u^{2}+u^2_x) dx\), the soliton’s scaled amplitude, \(A/P^{1/2}\), and velocity, \(q/P^{1/2}\), depend only on the variable \(z=\kappa/\sqrt{P}\). We prove that these scaling relations are true for the exact soliton solutions to the Camassa-Holm equation.

MSC:

76B25	Solitary waves for incompressible inviscid fluids
35Q53	KdV equations (Korteweg-de Vries equations)
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M30	Variational methods applied to problems in fluid mechanics

References:

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