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Zbl 0907.35009
Constantin, Adrian
On the inverse spectral problem for the Camassa-Holm equation. (English)
[J] J. Funct. Anal. 155, No.2, 352-363 (1998). ISSN 0022-1236

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 $y'= {1\over 4} y+\lambda my$.\par 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 $0<\lambda_1 \le \lambda_2 <\dots$ of single or double eigenvalues tending to infinity is the real analyticity of $m$. The case of a purely simple spectrum is typical of $0>m\in C^1(\bbfR)$.

MSC 2000:: *35B10 Periodic solutions of PDE
35Q35 Other equations arising in fluid mechanics
34L20 Asymptotic distribution of eigenvalues for OD operators
35G25 Initial value problems for nonlinear higher-order PDE

Keywords: periodic spectrum

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Zentralblatt MATH Berlin [Germany]