On the isospectral problem of the dispersionless Camassa-Holm equation

Abstract

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa--Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa--Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.

arXiv comment26 pages

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Cited by in corpus (17)

• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• The inverse spectral problem for indefinite strings
• The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data
• A liouville property with application to asymptotic stability for the camassa-holm equation
• Direct and inverse spectral theory of singular left-definite Sturm-Liouville operators
• Quadratic operator pencils associated with the conservative Camassa-Holm flow
• A view of the peakon world through the lens of approximation theory
• The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation
• A Lagrangian view on complete integrability of the two-component Camassa-Holm system
• The Camassa--Holm Equation and The String Density Problem
• Unique solvability of a coupling problem for entire functions
• The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
• Integrable Motion of Curves, Spin Equation and Camassa-Holm Equation
• Trace formulas and inverse spectral theory for generalized indefinite strings
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions
• The conservative Camassa-Holm flow with step-like irregular initial data
• Orbital stability of two-component peakons