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On the inverse spectral problem for the Camassa-Holm equation. (English) Zbl 0907.35009
Summary: A key basis for seeking periodic solutions of the Camassa-Holm equation \[ u_t- u_{xxt} +3uu_x= 2u_xu_{xx} +uu_{xxx} \] is to understand the associated spectral problem \(y'= {1\over 4} y+\lambda my\).
The periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum. The potential can then be reconstructed from the periodic spectrum. A necessary and sufficient condition for exponential decrease of the widths \(\lambda_{2n} -\lambda_{2n-1}\) for a sequence \(0<\lambda_1 \leq \lambda_2 <\dots\) of single or double eigenvalues tending to infinity is the real analyticity of \(m\). The case of a purely simple spectrum is typical of \(0>m\in C^1(\mathbb{R})\).

MSC:
35B10 Periodic solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
35G25 Initial value problems for nonlinear higher-order PDEs
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