The interest in the Camassa-Holm equation inspired the search for various generalizations of this equation with interesting properties and applications. In this letter we deal with such a twocomponent integrable system of coupled equations. First we derive the system in the context of shallow water theory. Then we show that while small initial data develop into global solutions, for some initial data wave breaking occurs. We also discuss the solitary wave solutions. Finally, we present an explicit construction for the peakon solutions in the short wave limit of system. PACS numbers: 05.45.Yv, 47.35.Bb, 47.35.Fg, 47.35.Jk In recent years the Camassa-Holm (CH) equation [1](ω being an arbitrary constant) has caught a great deal of attention. It is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed [1-4], as well as water waves moving over an underlying shear flow [5]. The CH equation also arises in the study of a certain non-Newtonian fluids [6] and also models finite length, small amplitude radial deformation waves in cylindrical hyperelastic rods [7]. The CH equation has a bi-Hamiltonian structure [8] (and an infinite number of conservation laws), it is completely integrable (see [1] for the Lax pair formulation and [9] for the direct/inverse scattering approach), and its solitary wave solutions are solitons [1, 10-13] with stable profiles [11,12]. The equation attracted a lot of attention in recent years due to two remarkable features. The first is the presence of solutions in the form of peaked solitary waves or 'peakons' [1,10,14] for ω = 0: the peakon u(x, t) = ce −|x−ct| travelling at finite speed c = 0 is smooth except at its crest, where it is continuous, but has a jump discontinuity in its first derivative. The peakons replicate a characteristic of the travelling waves of greatest height -exact travelling solutions of the governing equations for water waves with a peak at their crest [16][17][18] whose capture by simpler approximate shallow water models has eluded researchers until recently [19]. A further remarkable property of the CH equation is the presence of breaking waves (i.e. the equation has smooth solutions which develop singularities in finite time in the form of breaking waves [1, 20, 21] -the solution remains bounded while its slope becomes unbounded in finite time [19]) as well as that of smooth solutions defined for all times [15]. These two phenomena have always fascinated the fluid mechanics community: 'Although breaking and peaking, as well as criteria for the occurrence of each, are without doubt contained in the equation of the exact potential theory, it is intriguing to know what kind of simpler mathematical equation could include all these phenomena' [19]. The short wave limit of the CH equation is the Hunter-Saxton (HS) equationobtained from (1) by taking m = −u xx . It describes the propagation of waves in a massive director field of a nematic liquid crystal [22], with the orientation of the molecule...
An Inverse Scattering Method is developed for the Camassa-Holm equation. As an illustration of our approach the solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data. The main difference with respect to the standard Inverse Scattering Transform lies in the fact that we have a weighted spectral problem. We therefore have to develop different asymptotic expansions.MSC: 35P25, 35Q15, 35Q35, 35Q51, 35Q53
In this contribution we describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of mass conservation, the simplest bottom and surface conditions and the constant vorticity condition. The approximate model equations are generated by introduction of suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order. The so obtained equations can be related to three different integrable systems: a two component generalization of the Camassa-Holm equation, the Zakharov-Ito system and the Kaup-Boussinesq system.The significance of the results is the inclusion of vorticity, an important feature of water waves that has been given increasing attention during the last decade. The presented investigation shows how -up to a certain order -the model equations relate to the shear flow upon which the wave resides. In particular, it shows exactly how the constant vorticity affects the equations.
Abstract.The reductions of the integrable N -wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analyzed. The Zakharov-Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one soliton solutions of the corresponding N -wave equations and their reductions are studied. We show that to each soliton solution one can relate a (semi-)simple subalgebra of g. We illustrate our results by 4-wave equations related to so(5) which find applications in Stockes-anti-Stockes wave generation.
We develop the Inverse Scattering Transform (IST) method for the Degasperis-Procesi equation. The spectral problem is an sl(3) Zakharov-Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST such as the construction of fundamental analytic solutions, the formulation of a Riemann-Hilbert problem, and the implementation of the dressing method are presented.
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