On three-dimensional generalizations of the Boussinesq and Korteweg-de Vries equations
Author:
E. Infeld
Journal:
Quart. Appl. Math. 38 (1980), 277-287
MSC:
Primary 35Q20; Secondary 76B25
DOI:
https://doi.org/10.1090/qam/592196
MathSciNet review:
592196
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Three-dimensional generalizations of two different forms of the Boussinesq equation are derived. They are investigated for stability of slowly varying nonlinear wavetrains. The results obtained are then compared with the stability properties following from the full water wave equations. Agreement is found to be good for ${h_0}{k_0}$ (depth times wavenumber) of order one. This is very satisfactory, as the Boussinesq equations are only supposed to be valid for small ${h_0}{k_0}$. In particular, one version of the Boussinesq equation is found to yield instability with respect to one-dimensional perturbations for ${h_0}{k_0} > 1.5$ (as against 1.36 for the full equations). Finally, a similar comparison is performed for the three-dimensional Korteweg—de Vries equation.
J. Boussinesq, Comptes Rendus 12, 755 (1871)
D. J. Korteweg and G. de Vries, Phil. Mag. (5) 39, 422 (1895)
- G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954
- C. H. Su and C. S. Gardner, Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation, J. Mathematical Phys. 10 (1969), 536–539. MR 271526, DOI https://doi.org/10.1063/1.1664873
B. B. Kadomtsev and V. I. Pitvyashvili, Dokl. Akad. Nauk SSSR 192, 757 (1970)
M. Kako and G. Rowlands, Plasma Phys. 18, 165 (1976)
- I. A Kunin, Teoriya uprugikh sred s mikrostrukturoĭ : nelokal′naya teoriya uprugosti, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0426567
- W. D. Hayes, Group velocity and nonlinear dispersive wave propagation, Proc. Roy. Soc. London Ser. A 332 (1973), 199–221. MR 337134, DOI https://doi.org/10.1098/rspa.1973.0021
N. N. Bogolyubov and Y. A Mitropolski, Asymptotic methods in the theory of nonlinear oscillations, Hindustani, Delhi, 1961
T. B. Benjamin, Proc. R. Soc. Lond. A299, 59 (1967)
- S. Kogelman and R. C. Di Prima, Stability of spatially periodic supercritical flows in hydrodynamics, Phys. Fluids 13 (1970), 1–11. MR 266494, DOI https://doi.org/10.1063/1.1692775
G. B. Whitham, Proc. R. Soc. Lond. A283, 238 (1965)
- E. Infeld and G. Rowlands, Stability of nonlinear ion sound waves and solitons in plasmas, Proc. Roy. Soc. London Ser. A 366 (1979), no. 1727, 537–554. MR 547762, DOI https://doi.org/10.1098/rspa.1979.0068
- G. B. Whitham, Non-linear dispersion of water waves, J. Fluid Mech. 27 (1967), 399–412. MR 208903, DOI https://doi.org/10.1017/S0022112067000424
- E. Infeld, G. Rowlands, and M. Hen, Three-dimensional stability of Korteweg-de Vries waves and solitons, Acta Phys. Polon. A 54 (1978), no. 2, 131–139. MR 510223
J. Boussinesq, Comptes Rendus 12, 755 (1871)
D. J. Korteweg and G. de Vries, Phil. Mag. (5) 39, 422 (1895)
G. B. Whitham, Linear and nonlinear waves, John Wiley, 1974
C. H. Su and C. S. Gardner, J. Math. Phys. 10, 536 (1969)
B. B. Kadomtsev and V. I. Pitvyashvili, Dokl. Akad. Nauk SSSR 192, 757 (1970)
M. Kako and G. Rowlands, Plasma Phys. 18, 165 (1976)
I. A. Kunin, Teoria uprugyh sryed s mikrostrukturoy, Izdat. Nauka, Moskva, 1975
W. D. Hayes, Proc. R. Soc. Lond. A332, 199 (1973)
N. N. Bogolyubov and Y. A Mitropolski, Asymptotic methods in the theory of nonlinear oscillations, Hindustani, Delhi, 1961
T. B. Benjamin, Proc. R. Soc. Lond. A299, 59 (1967)
C. H. Su, Phys. Fluids 13, 1275 (1970)
G. B. Whitham, Proc. R. Soc. Lond. A283, 238 (1965)
E. Infeld and G. Rowlands, Proc. R. Soc. Lond. A366, 537 (1979)
G. B. Whitham, J. Fluid Mech. 27, 399 (1967)
E. Infeld, G. Rowlands and M. Hen, Acta Phys. Polon. A54, 131 (1978)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
35Q20,
76B25
Retrieve articles in all journals
with MSC:
35Q20,
76B25
Additional Information
Article copyright:
© Copyright 1980
American Mathematical Society