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
Full-text PDF Free Access
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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.
References
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- Masaaki Ito, Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A 91 (1982), no. 7, 335–338. MR 670869, DOI https://doi.org/10.1016/0375-9601%2882%2990426-1
- Robert A. Van Gorder, Solutions to a novel Casimir equation for the Ito system, Commun. Theor. Phys. (Beijing) 56 (2011), no. 5, 801–804. MR 2954306, DOI https://doi.org/10.1088/0253-6102/56/5/02
- Robert A. Van Gorder, Computing the region of convergence for power series in many real variables: a ratio-like test, Appl. Math. Comput. 218 (2011), no. 5, 2310–2317. MR 2831505, DOI https://doi.org/10.1016/j.amc.2011.07.052
- M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover publications 1964.
- Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, Int. J. Math. Math. Sci. 58 (2003), 3699–3715. MR 2031140, DOI https://doi.org/10.1155/S0161171203303291
References
- Peter J. Olver and Philip Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3) 53 (1996), no. 2, 1900–1906. MR 1401317 (97c:35172), DOI https://doi.org/10.1103/PhysRevE.53.1900
- Masaaki Ito, Symmetries and conservation laws of a coupled nonlinear wave equation, Phys. Lett. A 91 (1982), no. 7, 335–338. MR 670869 (84c:58036), DOI https://doi.org/10.1016/0375-9601%2882%2990426-1
- Robert A. Van Gorder, Solutions to a novel Casimir equation for the Ito system, Commun. Theor. Phys. (Beijing) 56 (2011), no. 5, 801–804. MR 2954306, DOI https://doi.org/10.1088/0253-6102/56/5/02
- Robert A. Van Gorder, Computing the region of convergence for power series in many real variables: a ratio-like test, Appl. Math. Comput. 218 (2011), no. 5, 2310–2317. MR 2831505, DOI https://doi.org/10.1016/j.amc.2011.07.052
- M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover publications 1964.
- Diego Dominici, Nested derivatives: a simple method for computing series expansions of inverse functions, Int. J. Math. Math. Sci. 58 (2003), 3699–3715. MR 2031140 (2005f:41079), DOI https://doi.org/10.1155/S0161171203303291
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Additional Information
John Haussermann
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816
MR Author ID:
961861
Robert A. Van Gorder
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email:
rav@knights.ucf.edu
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
