American Institute of Mathematical Sciences

Advanced
September  2007, 19(3): 469-481. doi: 10.3934/dcds.2007.19.469

Solitons from the Lagrangian perspective

Adrian Constantin 1,

1. 

School of Mathematics, Trinity College Dublin, Dublin 2

Received  March 2007 Revised  June 2007 Published  July 2007

The soliton solution of the Korteweg-de Vries equation provides a good approximation to the shape of a solitary wave solution to the governing equations for water waves. However, the corresponding velocity field below the soliton is not an accurate approximation. We propose an approach that provides us with a better approximation. By describing the particle paths below the free surface, we show that the qualitative features of the entire flow in a solitary water wave is captured by our approximation of the velocity field.
Keywords: soliton, particle trajectory., solitary wave.
Mathematics Subject Classification: Primary: 35Q51, 76B2.
Citation: Adrian Constantin. Solitons from the Lagrangian perspective. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 469-481. doi: 10.3934/dcds.2007.19.469
[1]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051
[2]

Mathias Nikolai Arnesen. Existence of solitary-wave solutions to nonlocal equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3483-3510. doi: 10.3934/dcds.2016.36.3483
[3]

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

Min Chen, Nghiem V. Nguyen, Shu-Ming Sun. Solitary-wave solutions to Boussinesq systems with large surface tension. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1153-1184. doi: 10.3934/dcds.2010.26.1153
[5]

Jibin Li, Yi Zhang. Exact solitary wave and quasi-periodic wave solutions for four fifth-order nonlinear wave equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 623-631. doi: 10.3934/dcdsb.2010.13.623
[6]

Jibin Li. Family of nonlinear wave equations which yield loop solutions and solitary wave solutions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 897-907. doi: 10.3934/dcds.2009.24.897
[7]

Y. A. Li, P. J. Olver. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 419-432. doi: 10.3934/dcds.1997.3.419
[8]

Rui Liu. Several new types of solitary wave solutions for the generalized Camassa-Holm-Degasperis-Procesi equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 77-90. doi: 10.3934/cpaa.2010.9.77
[9]

Xiaowan Li, Zengji Du, Shuguan Ji. Existence results of solitary wave solutions for a delayed Camassa-Holm-KP equation. Communications on Pure & Applied Analysis, 2019, 18 (6) : 2961-2981. doi: 10.3934/cpaa.2019132
[10]

Justin Holmer, Maciej Zworski. Slow soliton interaction with delta impurities. Journal of Modern Dynamics, 2007, 1 (4) : 689-718. doi: 10.3934/jmd.2007.1.689
[11]

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 & Continuous Dynamical Systems - A, 1998, 4 (1) : 159-191. doi: 10.3934/dcds.1998.4.159
[12]

Joey Y. Huang. Trajectory of a moving curveball in viscid flow. Conference Publications, 2001, 2001 (Special) : 191-198. doi: 10.3934/proc.2001.2001.191
[13]

Rodrigo Samprogna, Cláudia B. Gentile Moussa, Tomás Caraballo, Karina Schiabel. Trajectory and global attractors for generalized processes. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3995-4020. doi: 10.3934/dcdsb.2019047
[14]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579
[15]

Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049
[16]

Jerry Bona, Hongqiu Chen. Solitary waves in nonlinear dispersive systems. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 313-378. doi: 10.3934/dcdsb.2002.2.313
[17]

Orlando Lopes. A linearized instability result for solitary waves. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 115-119. doi: 10.3934/dcds.2002.8.115
[18]

Monique Chyba, Thomas Haberkorn, Ryan N. Smith, George Wilkens. A geometric analysis of trajectory design for underwater vehicles. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 233-262. doi: 10.3934/dcdsb.2009.11.233
[19]

Dale McDonald. Sensitivity based trajectory following control damping methods. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 127-143. doi: 10.3934/naco.2013.3.127
[20]

V. V. Chepyzhov, A. Miranville. Trajectory and global attractors of dissipative hyperbolic equations with memory. Communications on Pure & Applied Analysis, 2005, 4 (1) : 115-142. doi: 10.3934/cpaa.2005.4.115

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences