Inverse Spectral Problem for the Periodic Camassa-Holm Equation

Abstract

We consider the direct/inverse spectral problem for the periodic Camassa-Holm equation. In fact, we survey the direct/inverse spectral problem for the periodic weighted operator acting in the space L 2(ℝ, m(x)dx), where m = uxx −u > 0 is a 1-periodic positive function and u is the solution of the Camassa-Holm equation ut − uxxt + 3uux = 2uxuxx + uuxxx . For the operator L we describe the complete solution of the inverse spectral problem: i) uniqueness, prove that the spectral data uniquely determines the potential, ii) characterization, give conditions for some data to be the spectral data of some potential, iii) reconstruction, give an algorithm for recovering the potential from the spectral data, iv) a priori estimates, obtain two-sided a priori estimates of u, m in terms of gap lengths.