Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Classification of two types of weak solutions to the Casimir equation for the Ito system


Authors: John Haussermann and Robert A. Van Gorder
Journal: Quart. Appl. Math. 72 (2014), 471-490
MSC (2010): Primary 35Q53, 37K10, 35D30, 34E05
DOI: https://doi.org/10.1090/S0033-569X-2014-01347-X
Published electronically: February 26, 2014
MathSciNet review: 3237560
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Abstract: The existence and non-uniqueness of two classes of weak solutions to the Casimir equation for the Ito system is discussed. In particular, for (i) all possible travelling wave solutions and (ii) one vital class of self-similar solutions, all possible families of local power series solutions are found. We are then able to extend both types of solutions to the entire real line, obtaining separate classes of weak solutions to the Casimir equation. Such results constitute rare globally valid analytic solutions to a class of nonlinear wave equations. Closed-form asymptotic approximations are also given in each case, and these agree nicely with the numerical solutions available in the literature.


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Additional Information

John Haussermann
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816

Robert A. Van Gorder
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: rav@knights.ucf.edu

DOI: https://doi.org/10.1090/S0033-569X-2014-01347-X
Keywords: Casimir equation; Ito system; extended KdV equation; weak solutions; asymptotic series
Received by editor(s): June 22, 2012
Published electronically: February 26, 2014
Article copyright: © Copyright 2014 Brown University


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