American Institute of Mathematical Sciences

Advanced
July  1997, 3(3): 419-432. doi: 10.3934/dcds.1997.3.419

Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons

Y. A. Li 1, and  P. J. Olver 1,

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States, United States

Received  September 1996 Revised  January 1997 Published  April 1997

We investigate how the non-analytic solitary wave solutions -- peakons and compactons -- of an integrable bi-Hamiltonian system arising in fluid mechanics, can be recovered as limits of classical solitary wave solutions forming analytic homoclinic orbits for the reduced dynamical system. This phenomenon is examined to understand the important effect of linear dispersion terms on the analyticity of such homoclinic orbits.
Mathematics Subject Classification: 34A20, 34C35, 35B65, 58F05, 76B2.
Citation: Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 419-432. doi: 10.3934/dcds.1997.3.419
[1]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-hamiltonian dynamical system ii. complex analytic behavior and convergence to non-analytic solutions. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159
[2]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215
[3]

Mathias Nikolai Arnesen. Existence of solitary-wave solutions to nonlocal equations. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3483-3510. doi: 10.3934/dcds.2016.36.3483
[4]

Jerry L. Bona, Didier Pilod. Stability of solitary-wave solutions to the Hirota-Satsuma equation. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1391-1413. doi: 10.3934/dcds.2010.27.1391
[5]

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153
[6]

Tong Yang, Seiji Ukai, Huijiang Zhao. Stationary solutions to the exterior problems for the Boltzmann equation, I. Existence. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 495-520. doi: 10.3934/dcds.2009.23.495
[7]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[8]

David Cheban. I. U. Bronshtein's conjecture for monotone nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1095-1113. doi: 10.3934/dcdsb.2019008
[9]

Stephen C. Anco, Elena Recio. Accelerating dynamical peakons and their behaviour. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6131-6148. doi: 10.3934/dcds.2019267
[10]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[11]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[12]

Feimin Huang, Yi Wang, Tong Yang. Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity. Kinetic and Related Models, 2010, 3 (4) : 685-728. doi: 10.3934/krm.2010.3.685
[13]

Anna Capietto, Walter Dambrosio, Tiantian Ma, Zaihong Wang. Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1835-1856. doi: 10.3934/dcds.2013.33.1835
[14]

Zhensu Wen, Guanrong Chen. Solitary waves, periodic peakons and compactons on foliations in a Hertz chain model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022072
[15]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[16]

Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[17]

Kai Yan. On the blow up solutions to a two-component cubic Camassa-Holm system with peakons. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4565-4576. doi: 10.3934/dcds.2020191
[18]

Jibin Li. Family of nonlinear wave equations which yield loop solutions and solitary wave solutions. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 897-907. doi: 10.3934/dcds.2009.24.897
[19]

Oleg Makarenkov, Paolo Nistri. On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes. Communications on Pure and Applied Analysis, 2008, 7 (1) : 49-61. doi: 10.3934/cpaa.2008.7.49
[20]

Jibin Li, Yi Zhang. On the traveling wave solutions for a nonlinear diffusion-convection equation: Dynamical system approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1119-1138. doi: 10.3934/dcdsb.2010.14.1119

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (120)
  • HTML views (0)
  • Cited by (61)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy