Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation
Pages: 15 - 50, Volume 4, Issue 1, February 2011

doi:10.3934/dcdss.2011.4.15      Abstract        References        Full text (251.1K)           Related Articles

Jerry L. Bona - Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, United States (email)
Laihan Luo - Department of Mathematics, New York Institute of Technology, 1855 Broadway, New York, NY 10023, United States (email)

1 M. J. Ablowitz and A. S. Fokas, The inverse scattering transform for the Benjamin-Ono equation, A pivot to multidimensional problems, Stud. Appl. Math., 68 (1983), 1-10.
2 L. Abdelouhab, J. L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear dispersive waves, Phys. D, 40 (1989), 360-392.
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 T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592.
5 T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Royal Soc. London Ser. A, 272 (1972), 47-78.
6 P. Biler, Asymptotic behavior in time of some equations generalizing Korteweg-de Vries-Burgers, Bull. Polish Acad. Sci., 32 (1984), 275-281.
7 P. Biler, Large-time behavior of periodic solutions of two dispersive equations of Korteweg-de Vries-Burgers type, Bull. Polish Acad. Sci., 32 (1984), 401-405.
8 J. L. Bona, On solitary waves and their role in the evolution of long waves, in "Applications of Nonlinear Analysis in the Physical Sciences" (eds. H. Amann, N. Bazley & K. Kirchgässner), Pitman, London, (1981), 183-205.
9 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-Burgers equation, Applied Numerical Math., 10 (1992), 335-355.
10 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. Royal Soc. London, Series A, 351 (1995), 107-164.
11 J. L. Bona and H. Kalisch, Singularity formation in the generalized Benjamin-Ono equation, Discrete Cont. Dyn. Systems Series A, 11 (2004), 27-45.
12 J. L. Bona and L. Luo, Decay of the solutions to nonlinear, dispersive wave equations, Diff. & Int. Equations, 6 (1993), 961-980.
13 J. L. Bona and L. Luo, More results on the decay of solutions to nonlinear, dispersive wave equations, Discrete and Continuous Dynamical Systems, 1 (1995), 151-193.
14 J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Royal Soc. London Ser. A, 302 (1981), 457-510.
15 J. L. Bona, S. Rajopadhye and M. E. Schonbek, Models for propagation of bores I. Two-dimensional theory, Differential & Int. Equations, 7 (1994), 699-734.
16 J. L. Bona and M. E. Schonbek, Travelling-wave solutions of the Korteweg-de Vries-Burgers equation, Proc. Royal Soc. Edinburgh A, 101 (1985), 207-226.
17 J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Royal Soc. London Ser A, 278 (1975), 555-601.
18 H. Brezis and T. Gallouët, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.
19 H. Brezis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. P.D.E, 5 (1980), 773-789.
20 D. Derks, "Coherent Structures in the Dynamics of Perturbed Hamiltonian Systems," Ph.D. Thesis, University of Twente, 1992.
21 D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. P.D.E, 17 (1992), 1665-1693.
22 D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono -Burgers equation, J. Differential Equations, 90 (1991), 238-287.
23 P. M. Edwin and B. Roberts, The Benjamin-Ono-Burgers equation: An application in solar physics, Wave Motion, 8 (1986), 151-158.
24 J. Ginibre and G. Velo, Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation, J. Differential Equations, 93 (1991), 150-212.
25 N. Hayashi and P. I. Naumkin, Large time asymptotics of solutions to the generalized Benjamin-Ono equation, Trans. American Math. Soc., 351 (1999), 109-130.
26 R. J. Iório, On the Cauchy problem for the Benjamin-Ono equation, Comm. P.D.E., 11 (1986), 1031-1081.
27 R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J. Fluid Mech., 42 (1970), 49-60.
28 R. S. Johnson, Shallow water waves on a viscous fluid--The undular bore, Phys. of Fluids, 15 (1972), 1693-1699.
29 H. Kalisch and J. L. Bona, Models for internal waves in deep water, Discrete Cont. Dyn. Systems, 6 (2000), 1-20.
30 T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in Applied Math., Advances in Math. Suppl. Studies, 8 (1983), 93-128.
31 C. E. Kenig, G. Ponce and L. Vega, On the generalized Benjamin-Ono equation, Trans. Amer. Math. Soc., 342 (1994), 155-172.
32 D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag., 39 (1895), 422-443.
33 J. J. Mahony and W. G. Pritchard, Wave reflexion from beaches, J. Fluid Mech., 101 (1980), 809-832.
34 Y. Mammeri, On the decay in time of solutions of some generalized regularized long wave equations, Comm. Pure Appl. Anal., 7 (2008), 513-532.
35 C. C. Mei and L. F. Liu, The damping of surface gravity waves in a bounded liquid, J. Fluid Mech., 59 (1973), 239-256.
36 J. W. Miles, Surface-wave damping in a closed basin, Proc. Royal Soc. London Ser. A, 297 (1967), 459-475.
37 P. I. Naumkin and I. A. Shishmarev, "Nonlinear Nonlocal Equations in the Theory of Waves," in "Translation of Mathematical Monographs," 133, Amer. Math. Soc., Providence, 1994.
38 H. Ono, Algebraic solitary waves in stratified fluids, J. Phys. Soc. Japan, 39 (1975), 1082-1091.
39 E. Ott and R. N. Sudan, Damping of solitary waves, Phys. of Fluids, 13 (1970), 1432-1434.
40 T. Ozawa, On critical cases of Sobolev's inequality, J. Functional Anal., 127 (1995), 259-269.
41 D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321-330.
42 E. Schechter, "Well-Behaved Evolutions and the Trotter Product Formulas," Ph.D. Thesis, University of Chicago, 1978.
43 M. M. Tom, Smoothing properties of some weak solutions of the Benjamin-Ono equation, Differential & Int. Equations, 3 (1990), 683-694.
44 S. Vento, Well-posedness for the generalized Benjamin-Ono equation with arbitrary large initial data in the critical space, Inter. Math. Res. Notices, (2009), (to appear).
45 L. Zhang, Decay of solutions to generalized Benjamin-Bona-Mahony-Burgers equations in n-space dimensions, Nonlinear Analysis, 25 (1995), 1343-1369.
46 L. Zhang, Initial value problem for a nonlinear parabolic equation with singular integral-differential term, ACTA Math. Appl. Sinica, 8 (1992), 367-376.
47 Y. Zhou and B. Guo, Initial value problems for a nonlinear singular integral-differential equation of deep water, Lecture notes in Mathematics, 1306 (1986), 278-290.

Go to top