Camasa-Holm方程(英文版)

Chapter 1 Physical Backgrounds and Complete Integrability of the Camassa-Holm Equation 1.1 Physical backgrounds of the Camassa-Holm equation In the shallow water theory， a fairly wide range of wave equation and equa-tions describing weakly nonlinear effects can be regarded as the Korteweg-de Vries (KdV) equation under the assumptions that wavelengths approximate and amplitudes are small and finite. For example: (1) The motion of magnetic fluid wave of cold plasma; (2) Vibration of anharmonic lattice; (3) Ion-asoustic wave of plasma; (4) Longitudinal dispersion wave in elastic rod; (5) The pressure wave motion of the mixture of liquid and gas; (6) The rotation of fluid at the bottom of a tube; (7) The thermal excitation radiation of a phonon wave packet of the non-linear lattice at low temperature. Different degrees of approximation can yield different nonlinear partial diffe-rential equations that are completely integrable in the shallow wave theory. These equations have soliton solutions， i.e.， they tend to be flexibly given constants at infinity in space. They don’t disappear after colliding with each other and have no or only slight changes in wave form and velocity， and they show behaviors similar to particle scattering. In this section， we will use the Hamiltonian method to derive a new type of shallow water equation， i.e.， the Camassa-Holm equation (denoted as (CH) equation ([1， 19])) (1.1.1) where u is the velocity of fluid in the direction x， or equivalently， the height from the horizontal bottom to the free surface of the water wave， ω is a con-stant associated with the critical shallow wave velocity， and the subscripts ex-press partial derivatives. The equation contains the higher-order term (right branch) in small-amplitude expansion of the incompressible Euler equation with undirected motion on the free-surface water waves under the influence of gravity. The Benjamin-Bona-Mahoney (BBM) equation can be derived by removing these terms as follows: or the Korteweg-de Vries (KdV) equation (1.1.1) as an expanded form of the BBM equation. And the first derivative of the soliton solution c exp(|x-ct|) of the limit form is discontinuous at the crest when ω ≈ 0， which is called the peakon solution. These peakons affect the solutions of the initial value problem of Equation (1.1.1) as ω =0. The way in which the smooth initial conditions evolve into a series of peakons during the change is that the first derivative appears discontinuously by changing each inflection point with a negative slope to have a vertical tangent line. It should be noted that we can get multiple soliton solutions by simply overlapping single peakon solutions in a completely integrable finite dimensional Hamiltonian system and calculating their amplitude variations and the positions of the peaks. The CH equation has a double Hamiltonian structure， which can be expressed in two different Hamiltonian forms. The ratio of two compatible Hamiltonian operators is a recursive operator， resulting in an infinite number of conserved quantities. The property of the double Hamiltonian is used to rewrite our equa-tion as a compatibility condition of linear isospectral problems. So the initial value problem can be solved by the inverse scattering transform (IST). The Undirected Model Consider the inviscous incompressible fluid equa-tion with the same density: the Euler equation. Here u represents the horizontal speed in the x direction， and w represents the speed value in the z axis (vertical direction). The fluid is moved affected by gravity g， which is on a free surface z = ζ(x， t) and the horizontal infinite domain of the flat bottom z = .h0. We substitute the solutions formed asymptotically by the shallow water wave (see [1])， i.e.， u = u(x， t) and w =-(z + h0)ux into the conserved quantity (power + potential) of the Euler equation. Then we can obtain the energy by integrating z as follows: where η = ζ +h0 is the height from the water wave to the bottom. In addition， we substitute the same solution into the Euler equation and integrate in the vertical coordinate to obtain the Green-Naghdi (GN) equation ([2]). The GN equation is conserved about the above energy HGN. In fact， they can be expressed in the form of Hamiltonian ([3]) as follows: (1.1.2) where the momentum density m is defined as m = δHGN/δu. The GN equation does not allow for a thin domain expansion over the small parameter ε which rep-resents the proportion between the depth and the wavelength. In this expansion， the momentum of movement of HGN in the vertical direction (~η3ux2 ) is O(ε2). We make further small amplitude assumptions about shallow-wave theory， which is in the form of η = h0 + O(α)， and with balance quantity α = O(ε2). Conversely， the Hamiltonian HGN keeps the higher order non-essential terms (like ζ3) in such expansion. From the GN equation， we make further asymp