Modulation of the Camassa-Holm equation and reciprocal transformations

Abstract

We derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.

A Two-component Generalization of the Camassa-Holm Equation and its Solutions

“…When κ = 0, the Camassa-Holm equation (1.1) has a peculiar property that its soliton solutions become piecewise smooth and have corners at their crests, such solutions are weak solutions of (1.1) and are called "peakons". Since the works of Camassa and Holm, this equation has become a well known example of integrable systems and has been studied from various point views in, for example, [1,3,9,10,14,15,18,20,26,27,28] and references therein.…”

Initial value problem of the Whitham equations for the Camassa–Holm equation

“…for equation (1). Since the solution of (2) will never develop a shock when a ≤ b, we will be interested only in the case a > b.…”

“…The Camassa-Holm equation u t + (3u + 2ν)u x − 2 (u xxt + 2u x u xx + uu xxx ) = 0 , u(x, 0; ) = u 0 (x) (1) describes waves in shallow water when surface tension is present [2]. Here, ν is a constant parameter.…”

“…The single phase Whitham equations have been found in [1] and they can be written in the Riemann invariant form u ix + λ i (u 1 , u 2 , u 3 )u ix = 0 for − ν < u 3 < u 2 < u 1 ,…”

“…Although both the Camassa-Holm equation (1) and the KdV equation (6) are dispersive approximations to the Burgers equation (2) or (7), there are significant differences in the limiting dynamics. The biggest difference is that the Whitham equations (3) for the former equation are non-strictly hyperbolic (cf.…”