Evolution of the scattering coefficients of the Camassa-Holm equation, for general initial data

We consider the Camassa–Holm equation for general initial data, particularly when the potential in the scattering problem of the Lax pair, m + κ, becomes negative over a finite region. We show that the direct scattering problem of the eigenvalue problem of the Lax pair for this equation may be solved by dividing the spatial infinite interval into a union of separate intervals. Inside each of these intervals, the initial potential is uniformly either positive or negative. Due to this, one can define Jost functions inside each interval, each of which will have a uniform asymptotic form. We then demonstrate that one can obtain the t-evolution of the scattering coefficients of the scattering matrix of each interval. In the process, we also demonstrate that the evolution of the zeros of m + κ can be given entirely in terms of limits of the scattering coefficients at singular points. 1