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2006, Volume 14, Issue 3: 505-523. Doi: 10.3934/dcds.2006.14.505
This issue Previous Article Optimal fusion of sensor data for Kalman filtering Next Article A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential

A convergent numerical scheme for the Camassa--Holm equation based on multipeakons

  • 1.

    Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim

  • 2.

    Department of Mathematical Sciences, Norwegian University of Science and Technology, NO--7491 Trondheim

Received:  February 2005
Revised:  May 2005
Published:  July 2006
Abstract / Introduction Related Papers Cited by
    Abstract
  • The Camassa--Holm equation $u_t$$-$uxxt+3u$u_x-2u_x$uxx-uuxxx=0 enjoys special solutions of the form $u(x,t)=$Σi=1n$p_i(t)e^{-|x-q_i(t)|}$, denoted multipeakons, that interact in a way similar to that of solitons. We show that given initial data $u|_{t=0}=u_0$ in $H^1$(R) such that u-uxx is a positive Radon measure, one can construct a sequence of multipeakons that converges in Lloc∞(R, Hloc1(R)) to the unique global solution of the Camassa--Holm equation. The approach also provides a convergent, energy preserving nondissipative numerical method which is illustrated on several examples.
    Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35B10, 35Q53.

    Citation:
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