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

2006/01/01 by Helge Holden, Xavier Raynaud · 98 citations
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Nonlinear Photonic Systems #Algebraic structures and combinatorial models #Camassa–Holm equation #Sequence (biology) #Mathematics #Initial value problem #Scheme (mathematics) #Mathematical physics #Physics #Mathematical analysis #Pure mathematics #Integrable system

paper · doi:10.3934/dcds.2006.14.505

published in Discrete and Continuous Dynamical Systems 14(3), 505-523 (American Institute of Mathematical Sciences)

openalex publication_date 2006/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The Camassa--Holm equationu_t-uxxt+3uu_x-2u_xuxx-uuxxx=0 enjoys special solutionsof the form u(x,t)=Σi=1np_i(t)e^-|x-q_i(t)|, denotedmultipeakons, that interact in a way similar to that of solitons.We show that given initial data u|_t=0=u₀ in H¹(R)such that u-uxx is a positive Radon measure, one canconstruct a sequence of multipeakons that converges inLloc∞(R, Hloc1(R)) to the unique global solution of theCamassa--Holm equation. The approach also provides a convergent,energy preserving nondissipative numerical method which isillustrated on several examples.

Citations

• A New Integrable Shallow Water Equation
• Camassa–Holm, Korteweg–de Vries and related models for water waves
• Characteristics and the initial value problem of a completely integrable shallow water equation
• Global Weak Solutions for a Shallow Water Equation
• On the Cauchy problem for the Camassa–Holm equation
• Integral and integrable algorithms for a nonlinear shallow-water wave equation
• Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation
• Convergence of a Finite Difference Scheme for the Camassa–Holm Equation
• On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation

Cited by

• Periodic conservative solutions of the Camassa–Holm equation
• Convergent difference schemes for the Hunter–Saxton equation
• Camassa–Holm equation [wikipedia]