[Paper Review] Solitary waves and $N$-particle algorithms for a class of Euler-Poincaré equations

Roberto Camassa, Dongyang Kuang+1 more|arXiv (Cornell University)|Apr 18, 2014

Nonlinear Waves and Solitons18 references4 citations

TL;DR

This paper develops N-particle particle algorithms for a class of Euler-Poincaré equations with non-smooth, conically-shaped kernels, demonstrating that such systems can yield completely integrable two-particle dynamics under specific initial conditions. The study reveals that the resulting two-dimensional wave dynamics reduce to the one-dimensional completely integrable shallow water equation, while maintaining full 2D spatial dependence in single-channel solutions, and validates the particle method against pseudospectral Eulerian methods with high accuracy.

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 non-smooth kernels. By casting the system in its Lagrangian (or characteristics) form, we first formulate a particles system algorithm in free space with homogeneous Dirichlet boundary conditions for the evolving fields. We next examine the deformation of the system when non-homogeneous "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 non-trivial 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 [Camassa, R. and Holm, D. D., Phys. Rev. Lett., 71, 1961-1964, 1993], while retaining the full dependence on two spatial dimensions for the single channel solutions.

Motivation & Objective

• To establish foundational numerical algorithms for solving a class of Euler-Poincaré equations with non-smooth elliptic operators.
• To investigate the impact of non-homogeneous 'constant stream' boundary conditions on the dispersive and hyperbolic nature of the system.
• To analyze the existence and uniqueness of solutions in the N-particle ODE system when the kernel lacks Lipschitz continuity.
• To explore the integrability of two-particle dynamics under conically-shaped kernels and its implications for wave structure.
• To compare the performance of particle-based algorithms against pseudospectral Eulerian methods for non-smooth kernels.

Proposed method

• The system is reformulated in Lagrangian (characteristics) form to derive an N-particle finite-dimensional ODE system from the PDEs.
• The particle algorithm is developed under free space with homogeneous Dirichlet boundary conditions, then extended to non-homogeneous 'constant stream' boundary conditions.
• The kernel is assumed to have a conical shape, leading to non-Lipschitz continuity and affecting solution uniqueness in the ODE system.
• The two-particle dynamics are analyzed using Hamiltonian mechanics, with exact solutions derived for head-on collisions via the Hamiltonian H = c₁² + c₂².
• A Lyapunov function is constructed to analyze stability and the structure of the stable manifold in the two-particle system.
• Numerical validation is performed using a sixth-order Runge-Kutta method, comparing particle algorithm solutions with exact analytical solutions for relative distance and momentum.

Experimental results

Research questions

• RQ1 How do non-smooth, conically-shaped kernels affect the existence and uniqueness of solutions in the N-particle ODE system for Euler-Poincaré equations?
• RQ2 Can two-particle dynamics governed by such singular kernels lead to complete integrability, and if so, under what initial conditions?
• RQ3 What is the impact of non-homogeneous 'constant stream' boundary conditions on the dispersive versus hyperbolic character of the system?
• RQ4 To what extent can particle algorithms accurately simulate PDE solutions with non-smooth kernels compared to pseudospectral Eulerian methods?
• RQ5 Does the reduction of two-dimensional wave dynamics to the one-dimensional shallow water equation occur while preserving full 2D spatial dependence in single-channel solutions?

Key findings

• For conically-shaped kernels, the two-particle system becomes completely integrable under appropriate initial conditions, with exact solutions expressible in terms of hyperbolic functions.
• The exact solution for a head-on collision with γ = 4c² yields q(t) = -2 log[sech(ct)] and p(t) = ±2c / tanh(ct), demonstrating soliton-like behavior.
• The numerical solution using sixth-order Runge-Kutta achieves a 2-norm error of 2.9506e-05 for position and 8.7397e-07 for momentum compared to the exact solution.
• The stable manifold for the two-particle system is given by z = √[(G_{b-1}(0) - G_{b-1}(q)) / (4H)], which reduces to the known form for the shallow water equation when b = 3/2.
• The particle algorithm performs with high accuracy, showing that the method is viable for non-smooth kernels where standard ODE existence theorems fail.
• The system transitions from hyperbolic-like to dispersive behavior under non-homogeneous boundary conditions, with a non-trivial dispersion relation emerging from the kernel singularity.