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 Ly = m -1 (-y + 1 4 y) acting in the space L 2 (R, m(x)dx), where m = u xx -u > 0 is a 1-periodic positive function and u is the solution of the Camassa-Holm equation u tu xxt + 3uu x = 2u x u xx + uu xxx.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.Consider the well known Camassa-Holm equation [2]:(This equation describes the motion of solitary waves on shallow water, u(x, t) being the fluid velocity in the x-direction.We add some backgraund of the Camassa-Holm equation.Firstly, note that the paper [17] presents an alternative derivation of the equation, different from the original approach devised by Camassa-Holm.Secondly, we remark that the Camassa-Holm equation was derived as a model in elasticity in the paper [13].Finally, the Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group on the circle (see [37] for a formal derivation and see [11,12] for a detailed discussion of the geometric viewpoint and its physical implications).Thus, there is a purely geometric interpretation of the Camassa-Holm equation as a geodesic flow on the diffeomorphism group of the circle [37,11,12].It is known (see [7]) that m ≡ u xxu retains its signature under the shallow water flow if it is of one sign to start with.In looking for spatially periodic solutions of Eq.(1) a key point is to understand the associated spectral problem: