Roberto Camassa

A New Integrable Shallow Water Equation

Publisher Summary This chapter discusses about a new integrable shallow water equation. Completely integrable nonlinear partial differential equations arise at various levels of approximation in shallow water theory. Such equations possess soliton solutions-coherent (spatially localized) structures that interact nonlinearly among themselves and then re-emerge, retaining their identity and showing particle-like scattering behavior. This chapter discusses a newly discovered, completely integrable dispersive shallow-water equation found by Camassa and Holm in 1993. This equation is obtained by using a small-wave-amplitude asymptotic expansion directly in the Hamiltonian for the vertically averaged incompressible Eulers equations, after substituting a solution ansatz of columnar fluid motion and restricting to an invariant manifold for unidirectional motion of waves at the free surface under the influence of gravity. Section II of the chapter derives the one-dimensional Green–Naghdi equations. Section III uses Hamiltonian methods to newly discovered equation for unidirectional waves. Section IV analyzes the behavior of the solutions of the equation and shows that certain initial conditions develop a vertical slope in finite time. It is also shown that there exist stable multisoliton solutions. Section V demonstrates the existence of an infinite number of conservation laws for the equation that follow from its bi-Hamiltonian property.

Fully nonlinear internal waves in a two-fluid system

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin-Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.

The geometry of peaked solitons and billiard solutions of a class of integrable PDE's

The purpose of this Letter is to investigate the geometry of new classes of soliton-like solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [10] for a shallow water equation. We put this equation in the framework of complex integrable Hamiltonian systems on Riemann surfaces and draw some consequences from this setting. Amongst these consequences, one obtains new solutions such as quasiperiodic solutions,n-solitons, solitons with quasiperiodic background, billiard, andn-peakon solutions and complex angle representations for them. Also, explicit formulas for phase shifts of interacting soliton solutions are obtained using the method of asymptotic reduction of the corresponding angle representations. The method we use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow onN-dimensional quadrics. Other topics, planned for a forthcoming paper, are outlined.

Weakly nonlinear internal waves in a two-fluid system

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin-Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

Abstract: An extension of the algebraic-geometric method for nonlinear integrable PDEs is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDEs, which are associated to nonlinear subvarieties of hyperelliptic Jacobians.The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDEs that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDEs to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions.The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations.Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location.

On the realm of validity of strongly nonlinear asymptotic approximations for internal waves

Analytical and numerical results from recently developed strongly nonlinear asymptotic models are compared and validated with experimental observations of internal gravity waves and results from the numerical integrations of Euler equations for solitary waves at the interface of two-fluid systems. The focus of this investigation is on regimes where large amplitudes are attained, where the classical weakly nonlinear theories prove inadequate. Two asymptotically different regimes are examined in detail: shallow fluids, in which the typical wavelengths of the interface displacement are long with respect to the depths of both fluids, and deep fluids, where the wavelengths are comparable to, or less than, the depth of one of the two fluids. With the aim of illustrating the breakdown of the asymptotic assumptions, the transition from a shallow to a deep regime is examined through numerical computation of Euler systems solutions and by comparisons with solution to models.

Long-time effects of bottom topography in shallow water

Abstract We present and discuss new shallow water equations that provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity. We consider the regime where the Froude number is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained at first order in an asymptotic expansion of the solutions of the Euler equations for a shallow fluid by using the small parameter δ 2 . The leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects. Both sets of equations conserve energy and circulation, convect potential vorticity and have a Hamiltonian formulation. The corresponding energy and enstrophy are quadratic integrals with which we can bound the cumulative influence of the nonhydrostatic effects.

Stability of Forced Steady Solitary Waves

This paper explores the basic mechanism underlying the remarkable phenomenon that a forcing excitation stationary in character and sustained at near resonance in a shallow channel of uniform water depth generates a non-stationary response in the form of a sequential upstream emission of solitary waves. Adopting the forced Korteweg-de Vries (fKdV) model and using two of its steady forced solitary wave solutions as primary flows, the stability of these two transcritical steady motions is investigated, and their bifurcation diagrams relating these solutions to other stationary solutions determined, with the forcing held fixed. The corresponding forcing functions are characterized by a velocity parameter for one, and an amplitude parameter for the other of the steadily moving excitations. The linear stability analysis is first pursued for small arbitrary perturbations of the primary flow, leading to a singular, non-self-adjoint eigenvalue problem, which is solved by applying techniques of matched asymptotic expansions with suitable multiscales for singular perturbations, about the isolated bifurcation points of the parametric space pertaining to the stationary perturbations. The eigenvalues and eigenfunctions are then obtained for the full range of the parameters by numerical continuation of the eigenvalues branching off from the stationary-perturbation solutions that were determined by the local analysis. A highly accurate numerical scheme is developed as required for this purpose. The linear stability analysis identifies three categories of evolution of infinitesimal disturbances superimposed to the steady state; they occur in three different parametric regimes. The first, called periodical bifurcating regime, is characterized by complex eigenvalues, with a real part much smaller than the imaginary part, signifying that small departures from the steady state will oscillate with an amplitude growing at a slow exponential rate. In the second regime, called the aperiodical bifurcating regime, the eigenvalues are purely real, implying that small departures from the steady state grow exponentially. For the third regime, linear stability theory is unable to find any eigenvalue (including zero) to exist. In this last case, however, a nonlinear analysis based on the functional hamiltonian formulation is possible, with the hamiltonian conserved for forcings of constant velocity, and the steady state is shown to be stable. For this reason, this regime will be called the stable supercritical regime. Finally, extensive numerical simulations using various finite difference schemes are carried out to find how the solution evolves once the instability of the solution manifests, with results fully confirming the predictions obtained analytically for the various regimes. The numerical simulations show that the instability in the periodical bifurcating regime, for the type of forcings considered, causes the steady solutions to evolve into the phenomenon of periodical production of upstream-advancing solitary waves.

On the geometry of an atmospheric slow manifold

Abstract We examine the hyperbolic structure and the invariant manifolds of a model proposed by Lorenz to introduce the concept of an atmospheric slow manifold within the framework of dynamical system theory. We address the question of the long time asymptotic behaviour of the system using the (global) geometric point of view. It is shown that the model can be reduced to the classical example of a pendulum coupled to a harmonic oscillator. The dynamical regimes of interest for the slow manifold hypothesis correspond to regions of phase space near the saddle-center fixed point of this model which were not previously explored. These phase space regions are analysed using a combination of Melnikov-type methods and ideas from singular perturbation theory. By using the reversible symmetries of the model, an extension of the Melnikov theory is derived. This extension allows us to find homoclinic orbits and determine their approximation by simply computing the zeros of a certain function, which is constructed in terms of the usual Melnikov function. Countable infinities of global homoclinic bifurcations and existence of chaotic dynamics can be shown to exist by using the new tool.

On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Abstract An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each “particle” in this method travels along a characteristic curve of the shallow water equation. The resulting system of nonlinear ordinary differential equations can have solutions that blow up in finite time. Conditions for global existence are isolated and convergence of the method is proved in the limit of zero spatial step size and infinite number of particles. A fast summation algorithm is introduced to evaluate integrals in the particle method so as to reduce computational cost from O(N 2) to O(N), where N is the number of particles. Accuracy tests based on exact solutions and invariants of motion assess the global properties of the method. Finally, results on the study of the nonlinear equation posed in the quarter (space-time) plane are presented. The minimum number of boundary conditions required for solution uniqueness and the complete integrability are discussed in this case, while a modified particle scheme illustrates the evolution of solutions with numerical examples.