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.

26 pages

References in corpus (6)

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• Inverse Scattering Transform for the Camassa-Holm equation
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Cited by in corpus (8)

• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• A view of the peakon world through the lens of approximation theory
• The Camassa--Holm Equation and The String Density Problem
• The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
• Trace formulas and inverse spectral theory for generalized indefinite strings
• Orbital stability of two-component peakons
• The conservative Camassa-Holm flow with step-like irregular initial data
• A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions