Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation

Journal of Nonlinear Science

Volumn 10, Issue 6, 2000, Pages 603-638

  Bona, J L ^a   McKinney, W R ^b   Restrepo, J M ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b NORTH CAROLINA STATE UNIVERSITY   (United States)

^c UNIVERSITY OF ARIZONA   (United States)

Author keywords

BBM equation; Generalized BBM equation; Generalized RLW equation; RLW equation; Stable solitary waves; Unstable solitary waves

Indexed keywords

EID: 0005700573     PISSN: 09388974     EISSN: None     Source Type: Journal    
DOI: 10.1007/s003320010003     Document Type: Article

References (51)

1. D. H. Peregrine, "Calculations of the development of an undular bore," J. Fluid Mechanics 25 (1996), 321-330.
2. D. H. Peregrine, "Long waves on a beach," J. Fluid Mechanics 27 (1967), 815-827.
3. T. B. Benjamin, J. L. Bona, & J. J. Mahony, "Model equations for long waves in nonlinear dispersive systems," Phil. Trans. Royal Soc. London, A 227 (1972), 47-78.
4. J. L. Bona, W. G. Pritchard, & L. R. Scott, "A comparison of solutions of two model equations for long waves," in Fluid Dynamics in Astrophysics and Geophysics, Norman R. Lebovitz, ed., Lectures in Applied Mathematics #20, 1983, 235-267.
5. J. L. Bona, W. G. Pritchard, & L. R. Scott, "An evaluation of a model equation for water waves," Phil. Trans. Royal Soc. London, A 302 (1981), 457-510.
6. J. C. Eilbeck & G. R. McGuire, "Numerical study of the regularized long wave equation I: Numerical methods," J. Computational Physics 19 (1975), 63-73.
7. B.-Y. Guo & V. S. Manoranjan, "Spectral method for solving the RLW equation," J. Computational Math. 3 (1985), 228-237.
8. J. L. Bona, W. G. Pritchard, & L. R. Scott, "Numerical schemes for a model for nonlinear dispersive waves," J. Computational Physics 60 (1985), 167-186.
9. J. B. McLeod & P. J. Olver, "The connection between completely integrable partial differential equations and ordinary differential equations of Painlevé type," SIAM J. Math Analysis 14 (1983), 56-75.
10. J. Tasi, "Evolution of shockwaves in a one-dimensional lattice," J. Applied Physics 51 (1980), 5804-5815.
11. J. Tasi, "A second-order Korteweg-de Vries equation for a lattice," J. Applied Physics 51 (1980), 5816-5827.
12. T. B. Benjamin, "Lectures on Nonlinear Wave Motion," in Nonlinear Wave Motion, Alan Newell, ed., Lectures in Applied Mathematics #15, American Math. Soc., Providence, 1974, 3-47.
13. T. B. Benjamin, "A new kind of solitary wave," J. Fluid Mechanics 245 (1992), 401-411.
14. T. B. Benjamin, "Solitary and periodic waves of a new kind," Phil. Trans. Royal Soc. London, A 354 (1996), 1775-1806.
15. J.-C. Saut, "Sur quelques généralisations de l'équation de Korteweg-de Vries," J. Math. Pures Appl. 58 (1979), 21-61.
16. J.-C. Saut, "Quelques Généralisations de l'Équation de Korteweg-de Vries, II," J. Integro-Differential Eq. 33 (1979), 320-335.
17. J. P. Albert, J. L. Bona, & J. M. Restrepo, "Solitary-wave solutions of the Benjamin equation," SIAM J. Applied Math. 59 (1999), 2139-2161.
18. J. L. Bona, "Solitary waves and other phenomena associated with model equations for long waves," Fluid Dynamics Trans. 10 (1980), 77-111.
19. J. L. Bona, "On solitary waves and their role in the evolution of long waves," in Applications of Nonlinear Analysis in the Physical Sciences, H. Amann, N. Bazley, K. Kirchgässner, ed., Pitman Press, London, 1981, 183-205.
20. L. Abdelouhab, J. L. Bona, M. Felland, & J.-C. Saut, "Nonlocal models for nonlinear dispersive waves," Physica D 40 (1989), 360-392.
21. M. J. Ablowitz & H. Segur, Solitons and the Inverse Scattering Transform, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1981.
22. J. L. Bona, W. G. Pritchard, & L. R. Scott, "Solitary-wave interaction," Physics of Fluids 23 (1980), 438-441.
23. J. L. Bona & A. Soyeur, "On the stability of solitary-wave solutions for model equations for long waves," J. Nonlinear Sci. 4 (1994), 449-470.
24. M. I. Weinstein, "Existence and dynamic stability of solitary-wave solutions of equations arising in long-wave propagation," Comm. Partial Differential Eq. 12 (1987), 1133-1173.
25. T. B. Benjamin, J. L. Bona, & D. K. Bose, "Solitary-wave solutions of nonliner problems," Phil. Trans. Royal Soc. London, A 340 (1990), 195-244.
26. J. P. Albert, J. L. Bona, & J.-C. Saut, "Model equations for stratified fluids," Proc. Royal Soc. London, A 453 (1997), 1233-1260.
27. J. P. Albert, "Concentration compactness and the stability of solitary-wave solutions to nonlocal equations," in Contemporary Mathematics #221, J. R. Dorroh, G. R. Goldstein, J. A. Goldstein & M. M. Tom, ed., American Math. Soc., Providence, 1999, 1-29.
28. J. Angulo, "Existence and stability of solitary wave solutions for the Benjamin equation," J. Differential Eq. 152 (1999), 136-159.
29. H. Chen & J. L. Bona, "Existence and asymptotic properties of solitary-wave solutions of Benjamin-type equations," Adv. Differential Eq. 3 (1998), 51-84.
30. P. D. Lax, "Integrals of nonlinear equations of evolution and solitary waves," Comm. Pure Appl. Math. 21 (1968), 467-490.
31. T. B. Benjamin, "The stability of solitary waves," Proc. Royal Soc. London, A 328 (1972), 153-183.
32. J. L. Bona, "On the stability theory of solitary waves," Proc. Royal Soc. London, A 344 (1975), 363-374.
33. J. L. Bona, P. E. Souganidis, & W. A. Strauss, "Stability and instability of solitary waves of Korteweg-de Vries type," Proc. Royal Soc. London, A 411 (1987), 395-412.
34. J. L. Bona, V. A. Dougalis, O. A. Karakashian, & W. R. McKinney, "Numerical simulation of singular solutions of the generalized Korteweg-de Vries equation," in Contemporary Mathematics #200, F. Dias, J.-M. Ghidaglia, J.-C. Saut, ed., American Math. Soc., Providence, 1996, 17-29.
35. B. Fornberg & G. B. Whitham, "A numerical and theoretical study of certain nonlinear wave phenomena," Phil. Trans. Royal Soc. London, A 289 (1978), 373-404.
36. R. L. Pego & M. I. Weinstein, "Asymptotic stability of solitary waves," Comm. Math. Phys. 164 (1994), 305-349.
37. J. L. Bona, V. A. Dougalis, O. A. Karakashian, & W. R. McKinney, "Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation," Phil. Trans. Royal Soc. London, A 351 (1995), 107-164.
38. Y. Liu, "Instability and blow-up of solutions to generalized Boussinesq equations," SIAM J. Math. Anal. 26 (1995), 1527-1546.
39. Y. Liu, "Strong instability of solitary-wave solutions of a generalized Boussinesq equation," J. Differential Eq. 164 (2000), 223-239.
40. J. P. Albert, J. L. Bona, & D. Henry, "Sufficient conditions for stability of solitary-wave solutions of model equations for long waves," Physica D 24 (1987), 343-366.
41. J. R. Miller & M. I. Weinstein, "Asymptotic stability of solitary waves for the regularized long-wave equation," Comm. Pure Appl. Math. 49 (1996), 299-441.
42. P. E. Souganidis & W. A. Strauss, "Instability of a class of dispersive solitary waves," Proc. Royal Soc. Edinburgh 114A (1990), 195-212.
43. D. M. Sloan, "Fourier pseudospectral solution of the regularised long wave equation," J. Computational and Applied Math. 36 (1991), 159-179.
44. J. Pasciak, "Spectral methods for a nonlinear initial value problems involving pseudo-differential operators," SIAM J. Numer. Anal. 19 (1982), 142-154.
45. J. L. Bona, "Convergence of periodic wavetrains in the limit of large wavelength," Applied Science Research 37 (1981), 21-30.
46. C. Canuto, M. Y. Hussaini, A. Quarteroni, & T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York-Heidelberg-Berlin, 1988.
47. P. N. Swartztrauber, NCAR Software, available at www.netlib.org, 1989.
48. L. F. Shampine & M. K. Gordon, Computer Solution of Ordinary Differential Equations: The Initial Value Problem, W. H. Freeman, San Francisco, 1975.
49. I. S. Gradshteyn & I. M. Rysik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.
50. J. P. Albert, "Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation," J. Differential Eq. 63 (1986), 117-134.
51. J. P. Albert, "On the decay of solutions of the generalized Benjamin-Bona-Mahony equation," J. Math. Anal. Appl. 141 (1989), 527-537.