## 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.

### MSC:

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

### Keywords:

acoustic scattering; Harry Dym flows; Camassa-Holm flows
Full Text:

### References:

 [1] Barcilon, V., Explicit solution of the inverse problem for a vibrating string, J. Math. Anal. Appl., 93, 222-234 (1983) · Zbl 0524.73061 [2] Barcilon, V., Inverse eigenvalue problems, Inverse Problems. Inverse Problems, Lecture of Notes in Mathematics, 1225 (1986), Springer-Verlag: Springer-Verlag New York, Berlin, p. 1-51 [3] Barcilon, V., Sufficient conditions for the solution of the inverse problem for a vibrating beam, Inverse Problems, 3, 181-193 (1987) · Zbl 0629.73040 [4] Burridge, R., The Gel’fand-Levitan, Marchenko, and Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of impulse-response problems, Wave Motion, 2, 305-323 (1980) · Zbl 0444.45010 [5] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521 [6] Camassa, R.; Holm, D. D.; Hyman, J. M., A new integrable shallow water equation, Adv. Appl. Mech., 31, 1-33 (1994) · Zbl 0808.76011 [9] Fuchsteiner, B., The Lie algebra structure of nonlinear evolution equations and infinite dimensional abelian symmetry groups, Progr. Theor. Phys., 65, 861 (1981) [10] Gel’fand, I. M.; Levitan, B. M., On the determination of a differential equation from its spectral function, Amer. Soc. Transl., 1, 259-309 (1955) · Zbl 0044.09301 [11] Jodeit, M.; Levitan, B. M., The isospectrality problem for the classical Sturm-Liouville equation, Adv. Differential Equations, 2, 297-318 (1997) · Zbl 1023.34502 [12] Krein, M. G., Determination of the density of an inhomogeneous symmetric string from its frequency spectrum, Dokl. Akad. Nauk. SSSR, 76, 345-348 (1951) [13] Krein, M. G., On inverse problems for an inhomogeneous string, Dokl. Akad. Nauk. SSSR, 82, 669-672 (1952) [14] Kruskal, M., Nonlinear wave equations, Dynamical Systems, Theory and Applications. Dynamical Systems, Theory and Applications, Lecture Notes in Physics, 38 (1975), Springer-Verlag: Springer-Verlag Heidelberg, p. 310-354 [15] Pöschel, J.; Trubowitz, E., Inverse Spectral Theory (1987), Academic Press: Academic Press New York [16] Symes, W. W., Impedance profile via the first transport equation, J. Math. Anal. Appl., 94, 435-453 (1983) · Zbl 0529.73029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.