The effect of dissipation on solutions of the generalized Korteweg-de Vries equation

Journal of Computational and Applied Mathematics

Volumn 74, Issue 1-2, 1996, Pages 127-154

  Bona, J L ^a,b   Dougalis, V A ^c,d   Karakashian, O A ^e   McKinney, W R ^f  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c UNIVERSITY OF ATHENS   (Greece)

^d INSTITUTE OF COMPUTER SCIENCE   (Greece)

^e UNIVERSITY OF TENNESSEE   (United States)

^f NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Blow up; Generalized Korteweg de Vries equation; Nonlinear, dispersive, dissipative wave equations; Singularity formation in nonlinear waves

Indexed keywords

COMPUTER SIMULATION; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS; STABILITY; WATER WAVES;

BLOW UP; DISPERSIVE; DISSIPATIVE WAVE EQUATIONS; GENERALIZED KORTEWEG-DE VRIES EQUATION; SINGULARITY FORMATION IN NONLINEAR WAVES;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0030571235     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/0377-0427(96)00021-0     Document Type: Article

References (39)

1. J.P. Albert and J.L. Bona, Comparisons between model equations for long waves, J. Nonlinear Sci. 1 (1991) 345-374.
2. J.P. Albert, J.L. Bona amd F. Felland, A criterion for the formation of singularities for the generalized Korteweg-de Vries equation, Mat. Aplic. Comp. 7 (1988) 3-11.
3. C.J. Amick, J.L. Bona and M.E. Schonbek, Decay of solutions of some nonlinear wave equations, J. Differential Equations 81 (1989) 1-49.
4. G. Baker, V.A. Dougalis and O.A. Karakashian, Convergence of Galerkin approximations for the Korteweg-de Vries equation, Math. Comp. 40 (1983) 419-433.
5. T.B. Benjamin, Lectures on Nonlinear Wave Motion, Lectures in Appl. Math. 15 (Amer. Mathematical Soc., Providence, RI, 1974).
6. T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear, dispersive media, Philos. Trans. Roy. Soc. London Ser. A 272 (1972) 47-78.
7. P. Biler, Large-time behaviour of periodic solutions to dissipative equations of Korteweg-de Vries-Burgers type, Bull. Polish Acad. Sci. Math. 32 (1984) 401-405.
8. J.L. Bona, Convergence of periodic wave trains in the limit of large wavelength, Appl. Sci. Res. 37 (1981) 21-30.
9. J.L. Bona, F. Demengel and K. Promislow, Spectral splitting and dissipation of nonlinear dispersive waves, Preprint.
10. J.L. Bona, V.A. Dougalis and O.A. Karakashian, Fully discrete Galerkin methods for the Korteweg-de Vries equation, Comput. Math. Appl. 12A (1986) 859-884.
11. J.L. Bona, V.A. Dougalis, O.A. Karakashian and W.R. McKinney, Computations of blow-up and decay for periodic solutions of the generalized Korteweg-de Vries equation, Appl. Numer. Math. 10 (1992) 335-355.
12. J.L. Bona, V.A. Dougalis, O.A. Karakashian and W.R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 351 (1995) 107-164.
13. J.L Bona and L. Luo, Decay of solutions to nonlinear, disperisve wave equations, Differential & Integral Equations, 6 (1993) 961-980.
14. J.L. Bona, W.G. Pritchard and L.R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A 302 (1981) 457-510.
15. J.L. Bona, K. Promislow and G. Wayne, Higher-order asymptotics of decaying solutions of some nonlinear, dispersive, dissipative equations, Nonlinearity 8 (1994) 1179-1206.
16. J.L. Bona, K. Promislow and G. Wayne, On the asymptotic behavior of solutions to nonlinear, dispersive, dissipative wave equations, J. Math. Comp. 37 (1994) 265-277.
17. J.L. Bona, S. Rajopadhye and M.E. Schonbek, Models for propagation of bores. i. Two-dimensional theory, Differential & Integral Equations 7 (1994) 699-734.
18. J.L. Bona and M.E. Schonbek, Travelling-wave solutions to the Kortweg-de Vries-Burgers equation, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985) 207-226.
19. J.L. Bona and M. Sciálom, The effect of change in the nonlinearity and the dispersion relation on model equations for long waves, Canad. J. Appl. Math. 3 (1993) 1-41.
20. J.L. Bona and R. Smith, The initial-value problem for the Kortweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975) 555-604.
21. J.L. Bona, P.E. Souganidis and W.A. Strauss, Stability and instability of solitary waves of KdV type, Proc. Roy. Soc. London Ser. A 411 (1987) 395-412.
22. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. ii. The KdV equations, Geom. Func. Anal. 3 (1993) 209-262.
23. D.B. Dix, The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992) 1665-1693.
24. C. Foias and J.-C. Saut, Asymptotic behavior, as t → +∞ of solutions of Navier-Stokes equations and nonlinear spectral manifolds, Indiana Univ. Math. J. 33 (1984) 459-477.
25. H. Grad and P.N. Hu, Unified shock profile in a plasma, Phys. Fluids 10 (1967) 2596-2602.
26. A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: a discussion centered around the Korteweg-de Vries equation, SIAM Rev. 14 (1972) 582-643.
27. R.S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech. 42 (1970) 42-60.
28. O.A. Karakashian and W.R. McKinney, On optimal high order in time approximations for the Korteweg-de Vries equation, Math. Comp. 55 (1990) 473-496.
29. O.A. Karakashian and W.R. McKinney, On the approximation of solutions of the generalized Korteweg-de Vries-Burgers equation, Math. Comput. Simulation 37 (1994) 405-416.
30. T. Kato, Quasilinear Equations of Evolution with Applications to Partial Differential Equations, Lecture Notes in Math. 448 (Springer, Berlin 1975).
31. T. Kato, On the Korteweg-de Vries equation, Manuscripta Math. 28 (1979) 89-99.
32. T. Kato, On the Cauchy Problem for the (Generalized) Korteweg-de Vries equation, Studies in Appl. Math., Advances in Mathematics Supplementary Studies, 8 (Academic Press, New York, 1983).
33. R.L. Pego, Remarks on the stability of shock profiles for conservation laws with dissipation, Trans. Amer. Math. Soc. 291 (1985) 353-361.
34. R.L. Pego and M.I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. Roy. Soc. London Ser. A 340 (1992) 47-94.
35. E. Schechter, Well-behaved evolutions and the Trotter product formulas, Ph.D. Thesis, University of Chicago (1978).
36. A.C. Scott, F.Y.F. Chu and D.W. McLaughlin, The soliton: a new concept in applied science, Proc. IEEE 61 (1973) 1443-1483.
37. W.A. Strauss, Dispersion of low-energy waves for two conservative equations, Arch. Rational Mech. Anal. 55 (1974) 86-92.
38. B.-Y. Zhang, Remarks on the Cauchy problem of the Korteweg-de Vries equation on a periodic domain, preprint, 1993.
39. L. Zhang, Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equation in n space dimensions, Nonlinear Anal TMA 25 (1995) 1343-1369.