American Institute of Mathematical Sciences

Advanced
February  2003, 3(1): 115-139. doi: 10.3934/dcdsb.2003.3.115

Characteristics and the initial value problem of a completely integrable shallow water equation

Roberto Camassa 1,

1. 

Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States

Received  March 2002 Revised  September 2002 Published  November 2002

The initial value problem for a completely integrable shallow water wave equation is analyzed through its formulation in terms of characteristics. The resulting integro-differential equations give rise to finite dimensional projections onto a class of distributional solutions of the equation, equivalent to taking the Riemann sum approximation of the integrals. These finite dimensional projections are then explicitly solved via a Gram-Schmidt orthogonalization procedure. A particle method based on these reductions is implemented to solve the wave equation numerically.
Keywords: lattice dynamics., Hamiltonian structures, Completely integrable systems.
Mathematics Subject Classification: 37K1.
Citation: Roberto Camassa. Characteristics and the initial value problem of a completely integrable shallow water equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 115-139. doi: 10.3934/dcdsb.2003.3.115
[1]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
[2]

Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
[3]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61
[4]

Rafael De La Llave, Victoria Sadovskaya. On the regularity of integrable conformal structures invariant under Anosov systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 377-385. doi: 10.3934/dcds.2005.12.377
[5]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
[6]

Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587
[7]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67
[8]

William D. Kalies, Konstantin Mischaikow, Robert C.A.M. Vandervorst. Lattice structures for attractors I. Journal of Computational Dynamics, 2014, 1 (2) : 307-338. doi: 10.3934/jcd.2014.1.307
[9]

Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829
[10]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
[11]

Yangyou Pan, Yuzhen Bai, Xiang Zhang. Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1761-1774. doi: 10.3934/dcdss.2019116
[12]

Aristophanes Dimakis, Folkert Müller-Hoissen. Bidifferential graded algebras and integrable systems. Conference Publications, 2009, 2009 (Special) : 208-219. doi: 10.3934/proc.2009.2009.208
[13]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873
[14]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227
[15]

Colin Rogers, Tommaso Ruggeri. q-Gaussian integrable Hamiltonian reductions in anisentropic gasdynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2297-2312. doi: 10.3934/dcdsb.2014.19.2297
[16]

Vladimir Dragović, Milena Radnović. Pseudo-integrable billiards and arithmetic dynamics. Journal of Modern Dynamics, 2014, 8 (1) : 109-132. doi: 10.3934/jmd.2014.8.109
[17]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
[18]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[19]

Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889
[20]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (28)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences