Acoustic scattering and the extended Korteweg-de Vries hierarchy. (English) Zbl 0919.35118

Summary: The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transformation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa-Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals. \(\copyright\) Academic Press.


35Q53 KdV equations (Korteweg-de Vries equations)
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
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