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Nonlinear Scattering in the Lamb System

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Recent Progress in Operator Theory and Its Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 220))

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Abstract

The goal of this paper is to survey recent results on scattering in nonlinear conservative Lamb’ s systems. A Lamb’ s system is a wave equation coupled with an equation of motion of a particle of mass \(m\geq 0.\)We describe the long time asymptotics in a global energy norm for all finite energy solutions with \(m=0[6]\) and \(m> 0[7].\)Under certain conditions on the potential, each solution in an appropriate functional space decays, in a global energy norm asc \(t\rightarrow\pm \infty,\)towards the sum of a stationary state and an outgoing wave. The outgoing waves correspond to the ’in’ and ’out’ asymptotic states. For \(m > 0,\)we define nonlinear wave operators corresponding to the ones introduced in [6] and obtain a necessary condition for the existence of the asymptotic states. For m = 0 we state a conjecture for the asymptotic completeness and verify this for some particular potentials.

Mathematics Subject Classification (2000).Primary 37K05, 35A05, 35A30.

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Authors and Affiliations

  1. Universidad Autónoma de Guerrero Unidad Académica de Matemáticas, Av. Lázaro Cárdenas S/N, Ciudad Universitaria Chilpancingo de los Bravo, Guerrero, 39087, Mexico

    Marco A. Taneco-Hernández

  2. Departamento de Ciencias Básicas, UAM-A, Av. San Pablo 180, Col. Reinosa Tamaulipas, México, D. F., 02200, Mexico

    Marco A. Taneco-Hernández

Authors
  1. Marco A. Taneco-Hernández
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Correspondence to Marco A. Taneco-Hernández .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute, McBryde Hall 524, Blacksburg, 24061, Virginia, USA

    Joseph A. Ball

  2. Dept. Mathematics, University of Iowa, MacLean Hall 14, Iowa City, 52242-1419, Iowa, USA

    Raúl E. Curto

  3. , CINVESTAV, IPN - Instituto Politécnico Nacional, Av. IPN 2508, Mexico, D.F., 07360, Distrito Federal, Mexico

    Sergei M. Grudsky

  4. Dept. Mathematics, University of California, San Diego, Gilman Dr. 9500, La Jolla, 92093-0112, California, USA

    J. William Helton

  5. , Departamento di Matemática, Centro de Investigación en Matemáticas, Guanajuato, Gto., 36000, Guanajuato, Mexico

    Raúl Quiroga-Barranco

  6. Departamento de Matemática, CINVESTAV-IPN, Apdo. Postal, México, 07300, Distrito Federal, Mexico

    Nikolai L. Vasilevski

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© 2012 Springer Basel AG

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Taneco-Hernández, M.A. (2012). Nonlinear Scattering in the Lamb System. In: Ball, J., Curto, R., Grudsky, S., Helton, J., Quiroga-Barranco, R., Vasilevski, N. (eds) Recent Progress in Operator Theory and Its Applications. Operator Theory: Advances and Applications(), vol 220. Springer, Basel. https://doi.org/10.1007/978-3-0348-0346-5_20

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