- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 809824, 10 pages
Wave-Breaking Criterion for the Generalized Weakly Dissipative Periodic Two-Component Hunter-Saxton System
Department of Mathematics, Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China
Received 22 May 2013; Accepted 22 July 2013
Academic Editor: Michael Meylan
Copyright © 2013 Jianmei Zhang and Lixin Tian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- J. K. Hunter and R. Saxton, “Dynamics of director fields,” SIAM Journal on Applied Mathematics, vol. 51, no. 6, pp. 1498–1521, 1991.
- Z. Yin, “On the structure of solutions to the periodic Hunter-Saxton equation,” SIAM Journal on Mathematical Analysis, vol. 36, no. 1, pp. 272–283, 2004.
- H.-H. Dai and M. Pavlov, “Transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation,” Journal of the Physical Society of Japan, vol. 67, no. 11, pp. 3655–3657, 1998.
- J. K. Hunter and Y. X. Zheng, “On a completely integrable nonlinear hyperbolic variational equation,” Physica D, vol. 79, no. 2–4, pp. 361–386, 1994.
- R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.
- R. S. Johnson, “Camassa-Holm, Korteweg-de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002.
- A. Constantin and B. Kolev, “On the geometric approach to the motion of inertial mechanical systems,” Journal of Physics A, vol. 35, no. 32, pp. R51–R79, 2002.
- P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, no. 2, pp. 1900–1906, 1996.
- R. Beals, D. H. Sattinger, and J. Szmigielski, “Inverse scattering solutions of the Hunter-Saxton equation,” Applicable Analysis, vol. 78, no. 3-4, pp. 255–269, 2001.
- A. Bressan and A. Constantin, “Global solutions of the Hunter-Saxton equation,” SIAM Journal on Mathematical Analysis, vol. 94, pp. 68–92, 2010.
- A. Bressan, H. Holden, and X. Raynaud, “Lipschitz metric for the Hunter-Saxton equation,” Journal de Mathématiques Pures et Appliquées, vol. 94, no. 1, pp. 68–92, 2010.
- J. K. Hunter and Y. X. Zheng, “On a nonlinear hyperbolic variational equation. I. Global existence of weak solutions,” Archive for Rational Mechanics and Analysis, vol. 129, no. 4, pp. 305–353, 1995.
- J. Lenells, “The Hunter-Saxton equation describes the geodesic flow on a sphere,” Journal of Geometry and Physics, vol. 57, no. 10, pp. 2049–2064, 2007.
- J. B. Li and Y. S. Li, “Bifurcations of travelling wave solutions for a two-component Camassa-Holm equation,” Acta Mathematica Sinica (English Series), vol. 24, no. 8, pp. 1319–1330, 2008.
- W. Rui and Y. Long, “Integral bifurcation method together with a translation-dilation transformation for solving an integrable 2-component Camassa-Holm shallow water system,” Journal of Applied Mathematics, vol. 2012, Article ID 736765, 21 pages, 2012.
- H. Wu and M. Wunsch, “Global existence for the generalized two-component Hunter-Saxton system,” Journal of Mathematical Fluid Mechanics, vol. 14, no. 3, pp. 455–469, 2012.
- B. Moon and Y. Liu, “Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,” Journal of Differential Equations, vol. 253, no. 1, pp. 319–355, 2012.
- S. Wu and Z. Yin, “Blow-up, blow-up rate and decay of the solution of the weakly dissipative Camassa-Holm equation,” Journal of Mathematical Physics, vol. 47, no. 1, Article ID 013504, 2006.
- S. Wu, J. Escher, and Z. Yin, “Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation,” Discrete and Continuous Dynamical Systems B, vol. 12, no. 3, pp. 633–645, 2009.
- T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, vol. 448 of Lecture Notes in Mathematics, pp. 25–70, Springer, Berlin, Germany, 1975.
- G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010.
- A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.