Article

**Adv. Math. 235, 469-495 (2013)**[DOI: 10.1016/j.aim.2012.12.006]

## On the isospectral problem of the dispersionless Camassa-Holm equation

### Jonathan Eckhardt and Gerald Teschl

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.

** MSC2010:** Primary 37K15, 34B40; Secondary 35Q35, 34L05

**Keywords:** *Camassa-Holm equation, isospectral problem, inverse spectral theory*

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