Abstract
In this chapter inverse problems to determine coefficients of the hierarchical equation and the coupled system from measurements of solitary waves are studied. The data of the problems include amplitudes and additional points on the graphs of the waves. The main purpose of the theoretical study is to establish the minimal amount of information sufficient for the unique reconstruction of the coefficients. Uniqueness theorems are proved for inverse problems with different data sets. In addition, stability for the two-wave inverse problem in the hierarchical equation is studied. The proofs use mean value theorems and the method of vanishing polynomial coefficients. In the last part of the chapter methods of numerical solution of the inverse problems for solitary waves are discussed. In particular, least squares approach and application of series expansion (incl. linearisation) are considered. Numerical examples are provided.
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© 2011 Springer-Verlag Berlin Heidelberg
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Janno, J., Engelbrecht, J. (2011). Inverse Problems for Solitary Waves. In: Microstructured Materials: Inverse Problems. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21584-1_7
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DOI: https://doi.org/10.1007/978-3-642-21584-1_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21583-4
Online ISBN: 978-3-642-21584-1
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