A Two-Dimensional Version of the Camassa-Holm Equation

Kruse, H.‐P.; Scheurle, Jürgen; Du, Wenju

doi:10.1142/9789812794543_0017

Cited by 15 publications

(21 citation statements)

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“…This equation coincides with the dispersionless case of the CH equation for shallow water waves in one and two dimensions, discussed in Camassa and Holm [1993]; Kruse, Scheurle and Du [2001]. It also coincides with the ATME equation (the averaged template matching equation) in two dimensions.…”

Section: The Epdiff Equationsupporting

confidence: 76%

The Breadth of Symplectic and Poisson Geometry

Marsden¹,

Raţiu²

2007

Progress in Mathematics

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This paper concerns the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of R n or of an n-dimensional manifold M ). The paper focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives; that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.The paper shows that the EPDiff equations are generated by a smooth vector field on the diffeomorphism group for sufficiently smooth solutions. This is analogous to known results for incompressible fluids-both the Euler equations and the LAE equations-and it shows that for sufficiently smooth solutions, the equations are well-posed for short time. Numerical evidence suggests that, as time progresses, these smooth solutions break up into singular solutions which, at least in one dimension, exhibit soliton behavior.These non-smooth, or measure-valued, solutions are higher dimensional generalizations of the peakon solutions of the CH equation in one dimension. One of the main purposes of the paper is to show that many of the properties of these measure-valued solutions can be understood through the fact that their solution Ansatz is a momentum map. Some additional geometry is also pointed out, for example, that this momentum map is one part of a dual pair.

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Section: The Epdiff Equationsupporting

confidence: 76%

The Breadth of Symplectic and Poisson Geometry

Marsden¹,

Raţiu²

2007

Progress in Mathematics

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“…8 The second term proportional to α 2 approximates (twice) the vertically averaged kinetic energy due to vertical motion. For more details of the latter approximation for the two-dimensional CH shallow water equation, see Kruse, Schreule, and Du [36].…”

Section: Zero-dispersion Shallow Water Waves In Two Dimensionsmentioning

confidence: 99%

Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs

Holm¹,

Staley²

2003

SIAM J. Appl. Dyn. Syst.

249

220

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Abstract. We investigate the following family of evolutionary 1+1 PDEs that describes the balance between convection and stretching for small viscosity in the dynamics of one-dimensional nonlinear waves in fluids:mt + umx Here u = g * m denotes u(x) = ∞ −∞ g(x − y)m(y) dy. This convolution (or filtering) relates velocity u to momentum density m by integration against the kernel g(x). We shall choose g(x) to be an even function so that u and m have the same parity under spatial reflection. When ν = 0, this equation is both reversible in time and parity invariant. We shall study the effects of the balance parameter b and the kernel g(x) on the solitary wave structures and investigate their interactions analytically for ν = 0 and numerically for small or zero viscosity.This family of equations admits the classic Burgers "ramps and cliffs" solutions, which are stable for −1 < b < 1 with small viscosity. These pulson solutions obey a finite-dimensional dynamical system for the time-dependent speeds pi(t) and positions qi(t). We study the pulson and peakon interactions analytically, and we determine their fate numerically under adding viscosity.Finally, as outlook, we propose an n-dimensional vector version of this evolutionary equation with convection and stretching, namely,for a defining relation, u = G * m. These solutions show quasi-one-dimensional behavior for n, k = 2, 1 that we find numerically to be stable for b = 2. The corresponding superposed solutions of * Received by the editors July 10, 2002; accepted for publication (in revised form) by M. Golubitsky January 30, 2003; published electronically August 23, 2003. This work was performed by an employee of the U.S. Government or under U.S. Government contract. The U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes. Copyright is owned by SIAM to the extent not limited by these rights.http Pi(s, t) G x − Q i(s, t) ds, u ∈ R n .These are momentum surfaces (or filaments for k = 1), defined on surfaces (or curves) x = Q i(s, t), i = 1, 2, . . . , N. For b = 2, the Pi(s, t), Qi(s, t) ∈ R n satisfy canonical Hamiltonian equations for geodesic motion on the space of n-vector valued k-surfaces with cometric G.

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“…In particular, it applies in a variety of fluid situations ranging from solitons to turbulence. The EPDiff equation coincides with the dispersionless case of the Camassa-Holm (CH) equation for shallow water in one-dimension (1-D) and two-dimensions (2-D), discussed in [4,30]. Applying viscosity to the incompressible, three-dimensional (3-D) analog of this equation produces the Navier-Stokes-α model for the averaged fluid equations (see, e.g., [8]).…”

Section: Introductionmentioning

confidence: 76%

Integration of the EPDiff equation by particle methods

2012

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Abstract. The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.

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A Two-Dimensional Version of the Camassa-Holm Equation

Cited by 15 publications

References 0 publications

The Breadth of Symplectic and Poisson Geometry

The Breadth of Symplectic and Poisson Geometry

Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs

Integration of the EPDiff equation by particle methods

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