Solitary Waves and N-Particle Algorithms for a Class of Euler–Poincaré Equations

Roberto Camassa, Dongyang Kuang, Long Lee

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Abstract

We study a class of partial differential equations (PDEs) in the family of the so-called Euler–Poincaré differential systems, with the aim of developing a foundation for numerical algorithms of their solutions. This requires particular attention to the mathematical properties of this system when the associated class of elliptic operators possesses nonsmooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particle system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when nonhomogeneous “constant stream” boundary conditions are assumed. We show how this simple change at the boundary deeply affects the nature of the evolution, from hyperbolic-like to dispersive with a nontrivial dispersion relation, and examine the potentially regularizing properties of singular kernels offered by this deformation. From the particle algorithm viewpoint, kernel singularities affect the existence and uniqueness of solutions to the corresponding ordinary differential equations systems. We illustrate this with the case when the operator kernel assumes a conical shape over the spatial variables, and examine in detail two-particle dynamics under the resulting lack of Lipschitz continuity. Curiously, we find that for the conically shaped kernels the motion of the related two-dimensional waves can become completely integrable under appropriate initial data. This reduction projects the two-dimensional system to the one-dimensional completely integrable Shallow-Water equation [1], while retaining the full dependence on two spatial dimensions for the single channel solutions. Finally, by comparing with an operator-splitting pseudospectral method we illustrate the performance of the particle algorithms with respect to their Eulerian counterpart for this class of nonsmooth kernels.

LanguageEnglish (US)
Pages502-546
Number of pages45
JournalStudies in Applied Mathematics
Volume137
Issue number4
DOIs
StatePublished - Nov 1 2016

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Euler equations
Solitary Waves
Solitons
Poincaré
Euler
kernel
Boundary conditions
Kernel Operator
Singular Kernel
Operator Splitting
Pseudospectral Method
Lipschitz Continuity
Integrable Equation
Splitting Method
Shallow Water Equations
Particle System
Two-dimensional Systems
Free Space
Dispersion Relation
Casting

ASJC Scopus subject areas

  • Applied Mathematics

Cite this

Solitary Waves and N-Particle Algorithms for a Class of Euler–Poincaré Equations. / Camassa, Roberto; Kuang, Dongyang; Lee, Long.

In: Studies in Applied Mathematics, Vol. 137, No. 4, 01.11.2016, p. 502-546.

Research output: Contribution to journalArticle

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