We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons. PACS numbers: 5.45. Yv, 11.10.Ef, 11.10.Lm, Solitons Water wave theory first introduced solitons as solutions of unidirectional nonlinear wave equations, obtained via asymptotic expansions around simple wave motion of the Euler equations for shallow water in a particular Galilean frame [1]. Later developments identified some of these water wave equations as completely integrable Hamiltonian systems solvable by the inverse scattering transform (IST) method, see, e.g., [2]. We shall discuss the following 1+1 quadratically nonlinear equation in this class for unidirectional water waves with fluid velocity u(x, t),Here m = u − α 2 u xx is a momentum variable, partial derivatives are denoted by subscripts, the constants α 2 and γ/c 0 are squares of length scales, and c 0 = √ gh is the linear wave speed for undisturbed water at rest at spatial infinity, where u and m are taken to vanish. (Any constant value u = u 0 is also a solution.) Equation (1) was first derived by using asymptotic expansions directly in the Hamiltonian for Euler's equations in the shallow water regime and was thereby shown to be biHamiltonian and, thus, IST-integrable in [3]. Before [3], classes of integrable equations similar to (1) were known to be derivable from the theory of hereditary symmetries, [4]. However, these were not derived physically as water wave equations and their solution properties were not studied before [3]. See [5] for an insightful discussion of how the integrable equation (1) relates to the theory of hereditary symmetries.The interplay between the local and nonlocal linear dispersion in this equation is evident in its phase velocity relation, ω/k = (c 0 −γ k 2 ) /(1+α 2 k 2 ), for waves with frequency ω and wave number k linearized around u = 0. At low wave numbers, the constant dispersion parameters α 2 and γ perform rather similar functions. At high wave numbers, however, the parameter α 2 properly keeps the phase speed of the wave from becoming unbounded. The phase speed lies in the band ω/k ∈ (− γ/α 2 , c 0 ). Longer linear waves are the faster provided γ + c 0 α 2 ≥ 0. Equation (1) is not Galilean invariant. Upon shifting the velocity variable by u 0 and moving into a Galilean frame ξ = x − ct with velocity c, so that u(x, t) = u(ξ, t) + c + u 0 , this eq...

We derive the Camassa-Holm equation (CH) as a shallow water wave equation with surface tension in an asymptotic expansion that extends one order beyond the Korteweg-de Vries equation (KdV). We show that CH is asymptotically equivalent to KdV5 (the ÿfth-order integrable equation in the KdV hierarchy) by using the nonlinear/non-local transformations introduced in Kodama (Phys. Lett. A 107 (1985a) 245; Phys. Lett. A 112 (1985b) 193; Phys. Lett. A 123 (1987) 276). We also classify its travelling wave solutions as a function of Bond number by using phase plane analysis. Finally, we discuss the experimental observability of the CH solutions.

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.

The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the amplitude and form of the different kinds of peaks is analytically calculated.

We show that in the neighborhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area preserving map, there is generically a bifurcation that creates a "twistless" torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created, and eventually collides with the saddle-center bifurcation that creates the period three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the nondegeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.