Numerical solution of Boussinesq systems of KdV-KdV type: II. Evolution of radiating solitary waves

Nonlinearity

Volumn 21, Issue 12, 2008, Pages 2825-2848

  Bona, J L ^a   Dougalis, V A ^b,c   Mitsotakis, D E ^b,c  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b UNIVERSITY OF ATHENS   (Greece)

^c INSTITUTE OF COMPUTER SCIENCE   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 58149337147     PISSN: 09517715     EISSN: 13616544     Source Type: Journal    
DOI: 10.1088/0951-7715/21/12/006     Document Type: Article

References (32)

1. Alazman A A, Albert J P, Bona J L, Chen M and Wu J 2006 Comparisons between the BBM equation and a Boussinesq system Adv. Diff. Eqns 11 121-66
2. Antonopoulos D C, Dougalis V A and Mitsotakis D E 2008 Numerical solution of Boussinesq systems of the Bona-Smith type preprint to be submitted
3. Amick C J and Toland J F 1992 Solitary waves with surface tension: I. Trajectories homoclinic to periodic orbits in four dimensions Arch. Ration. Mech. Anal. 118 37-69
4. Bakholdin I and Il'ichev A 1996 Radiation and modulational instability described by fifth-order Korteweg-de Vries equation Contemp. Math. 200 1-15
5. Beale J T 1991 Exact solitary water waves with capillary ripples at infinity Commun. Pure Appl. Math. 44 211-47
6. Benilov E S, Grimshaw R and Kuznetsova E P 1993 The generation of radiating waves in a singularly-perturbed Korteweg-de Vries equation Physica D 69 270-8
7. Bona J L 1981 Convergence of periodic wavetrains in the limit of large wavelength Appl. Sci. Res. 37 21-30
8. Bona J L and Chen M 1998 A Boussinesq system for two-way propagation of nonlinear dispersive waves Physica D 116 191-224
9. Bona J L, Chen M and Saut J-C 2002 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and linear theory J. Nonlinear Sci. 12 283-318
10. Bona J L, Chen M and Saut J-C 2004 Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory Nonlinearity 17 925-52
11. Bona J L, Colin T and Lannes D 2005 Long wave approximations for water waves Arch. Ration. Mech. Anal. 178 373-410
12. Bona J L, Dougalis V A, Karakashian O A and McKinney W R 1995 Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation Phil. Trans. R. Soc. Lond. A 351 107-64
13. Bona J L, Dougalis V A and Mitsotakis D E 2007 Numerical solution of KdV-KdV systems of Boussinesq equations: I. The numerical scheme and existence of generalized solitary waves Math. Comput. Simulation 74 214-28
14. Bona J L, McKinney W R and Restrepo J M 2000 Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation J. Nonlinear Sci. 10 603-38
15. Boyd J P 1991 Weakly non-local solitons for capillary-gravity waves: fifth-degree Korteweg-de Vries equation Physica D 48 129-46
16. Champneys A R and Lord G J 1997 Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem Physica D 102 101-24
17. Champneys A R, Vanden-Broeck J-M and Lord G J 2002 Do true elevation gravity-capillary solitary waves exist? A numerical investigation J. Fluid. Mech. 454 403-17
18. Chen H 2006 Long-period limit of nonlinear dispersive waves: the BBM-equation Diff. Integr. Eqns 19 463-80
19. Dias F and Iooss G 2003 Water-waves as a spatial dynamical system Handbook for Mathematical Fluid Dynamics ed S Friedlander and D Serre (Amsterdam: Elsevier) pp 443-99
20. Dougalis V A, Duran A, Lopez-Marcos M A and Mitsotakis D E 2007 A numerical study of the stability of solitary waves of the Bona-Smith system J. Nonlinear. Sci. 17 569-607
21. Mitsotakis D E 2007 Theory and numerical analysis of nonlinear dispersive wave equations: Boussinesq systems in one and two dimensions PhD Thesis University of Athens (in Greek)
22. Grimshaw R and Iooss G 2003 Solitary waves of a coupled Korteweg-de Vries system Math. Comput. Simulation 62 31-40
23. Grimshaw R and Joshi N 1995 Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation SIAM J. Appl. Math. 55 124-35
24. Hunter J K and Scheurle J 1988 Existence of perturbed solitary wave solutions to a model equation for water waves Physica D 32 253-68
25. Hunter J K and Vanden-Broeck J-M 1983 Solitary and periodic gravity-capillary waves of finite amplitude J. Fluid Mech. 134 253-68
26. Lombardi E 2000 Oscillatory integrals and phenomena beyond all algebraic orders, with applications to homoclinic orbits in reversible systems Lecture Notes in Mathematics vol 1741 (Berlin: Springer)
27. Michallet H and Dias F 1999 Numerical study of generalized interfacial waves Phys. Fluids 11 1502-11
28. Pelloni B and Dougalis V A 2001 Numerical modelling of two-way propagation of non-linear dispersive waves Math. Comput. Simulation 55 595-606
29. Pomeau Y, Ramani A and Grammaticos B 1988 Structural stability of the Korteweg-de Vries solitons under a singular perturbation Physica D 31 127-34
30. Sun S M 1991 Existence of a generalized solitary wave with positive Bond number smaller than J. Math. Anal. Appl. 156 471-504
31. Vanden-Broeck J-M 1991 Elevation solitary waves with surface tension Phys. Fluids A 11 2659-63
32. Yang T-S and Akylas T R 1996 Weakly nonlocal gravity-capillary solitary waves Phys. Fluids 8 1506-14