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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 x u xx +uu xxx

is to understand the associated spectral problem y ' =1 4y+λmy.

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 λ 2n -λ 2n-1 for a sequence 0<λ 1 λ 2 < 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>mC 1 ().

MSC:
35B10Periodic solutions of PDE
35Q35PDEs in connection with fluid mechanics
34L20Asymptotic distribution of eigenvalues for OD operators
35G25Initial value problems for nonlinear higher-order PDE