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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
Full Text: DOI

References:

[1] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 7, 1661-1664 (1993) · Zbl 0972.35521
[2] A. Constantin, A general-weighted Sturm-Liouville problem, Annali Sc. Norm. Sup. Pisa; A. Constantin, A general-weighted Sturm-Liouville problem, Annali Sc. Norm. Sup. Pisa · Zbl 0913.34022
[3] A. Constantin, H. P. Kean, A shallow water equation on the circle, Comm. Pure. Appl. Math.; A. Constantin, H. P. Kean, A shallow water equation on the circle, Comm. Pure. Appl. Math. · Zbl 0940.35177
[4] McKean, H. P., Integrable Systems and Algebraic Curves. Integrable Systems and Algebraic Curves, Lecture Notes in Mathematics, 755 (1979), Springer-Verlag: Springer-Verlag Berlin/New York, p. 83-200 · Zbl 0449.35080
[5] McKean, H. P.; Van Moerbeke, P., The spectrum of Hill’s equation, Invent. Math., 30, 217-274 (1975) · Zbl 0319.34024
[6] McKean, H. P.; Trubowitz, E., Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math., 29, 143-226 (1976) · Zbl 0339.34024
[7] Trubowitz, E., The inverse problem for periodic potentials, Comm. Pure Appl. Math., 30, 321-337 (1977) · Zbl 0403.34022
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