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From a PDE model to an ODE model of dynamics of synaptic depression
Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability
1. | School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ |
2. | Ship Science, University of Southampton, Southampton SO17 1BJ, United Kingdom |
References:
[1] |
N. F. Bondarenko, M. Z. Gak and F. V. Dolzhansky, Laboratory and theoretical models of a plane periodic flow, Izv. Atmos. Oceanic Phys., 15 (1979), 711-716. |
[2] |
N. E. Canney and J. D. Carter, Stability of plane waves on deep water with dissipation, Math. Comp. Simul., 74 (2007), 159-167.
doi: 10.1016/j.matcom.2006.10.010. |
[3] |
J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216.
doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2. |
[4] |
B. Chen and P. G. Saffman, Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 62 (1980), 1-21. |
[5] |
Z.-M. Chen, A vortex based panel method for potential flow simulation around a hydrofoil, J. Fluids Stuct., 28 (2012), 378-391.
doi: 10.1016/j.jfluidstructs.2011.10.003. |
[6] |
T. B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. R. Soc. A, 299 (1967), 59-75.
doi: 10.1098/rspa.1967.0123. |
[7] |
T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech., 27 (1967), 417-430.
doi: 10.1017/S002211206700045X. |
[8] |
T. J. Bridges and A. Mielke, A proof of the Benjamin-Feir instability, Arch. Ratianal Mech. Anal., 133 (1995), 145-198.
doi: 10.1007/BF00376815. |
[9] |
H. Lamb, "Hydrodynamics,'' Cambridge University Press, Cambridge, 1932. |
[10] |
M. S. Longuet-Higgins, On the stability of steep gravity waves, Proc. R. Soc. Lond. Ser. A, 396 (1984), 269-280.
doi: 10.1098/rspa.1984.0122. |
[11] |
J. C. Luke, A variational principle for a fluid with a free surface, J. Fluid Mech., 27 (1967), 395-397.
doi: 10.1017/S0022112067000412. |
[12] |
J. Pedlosky, "Geophysical Fluid Dynamics,'' Springer-Verlag, New York, 1979. |
[13] |
H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271. |
[14] |
G. G. Stokes, On the theory of osillatory waves, Trans. Cambridge Philos. Soc., 8 (1847), 441-473. |
[15] |
M. Tanaka, The stability of steep gravity waves, J. Phys. Soc. Japan, 52 (1983), 3047-3055.
doi: 10.1143/JPSJ.52.3047. |
[16] |
M. Tanaka, The stability of steep gravity waves. II, J. Fluid Mech., 156 (1985), 281-289.
doi: 10.1017/S0022112085002099. |
[17] |
G. B. Whitham, "Linear and Nonlinear Waves,'' Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
[18] |
G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. Math., 41 (1988), 19-46.
doi: 10.1002/cpa.3160410104. |
[19] |
G. Wu, Y. Liu, and D. K. P. Yue, A note on stabilizing the Benjamin-Feir instability, J. Fluid Mech., 556 (2006), 45-54.
doi: 10.1017/S0022112005008293. |
[20] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of adeep fluid, J. Appl. Mech. Tech. Phys., 2 (1968), 190-194. |
show all references
References:
[1] |
N. F. Bondarenko, M. Z. Gak and F. V. Dolzhansky, Laboratory and theoretical models of a plane periodic flow, Izv. Atmos. Oceanic Phys., 15 (1979), 711-716. |
[2] |
N. E. Canney and J. D. Carter, Stability of plane waves on deep water with dissipation, Math. Comp. Simul., 74 (2007), 159-167.
doi: 10.1016/j.matcom.2006.10.010. |
[3] |
J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216.
doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2. |
[4] |
B. Chen and P. G. Saffman, Numerical evidence for the existence of new types of gravity waves of permanent form on deep water, Stud. Appl. Math., 62 (1980), 1-21. |
[5] |
Z.-M. Chen, A vortex based panel method for potential flow simulation around a hydrofoil, J. Fluids Stuct., 28 (2012), 378-391.
doi: 10.1016/j.jfluidstructs.2011.10.003. |
[6] |
T. B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. R. Soc. A, 299 (1967), 59-75.
doi: 10.1098/rspa.1967.0123. |
[7] |
T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water, J. Fluid Mech., 27 (1967), 417-430.
doi: 10.1017/S002211206700045X. |
[8] |
T. J. Bridges and A. Mielke, A proof of the Benjamin-Feir instability, Arch. Ratianal Mech. Anal., 133 (1995), 145-198.
doi: 10.1007/BF00376815. |
[9] |
H. Lamb, "Hydrodynamics,'' Cambridge University Press, Cambridge, 1932. |
[10] |
M. S. Longuet-Higgins, On the stability of steep gravity waves, Proc. R. Soc. Lond. Ser. A, 396 (1984), 269-280.
doi: 10.1098/rspa.1984.0122. |
[11] |
J. C. Luke, A variational principle for a fluid with a free surface, J. Fluid Mech., 27 (1967), 395-397.
doi: 10.1017/S0022112067000412. |
[12] |
J. Pedlosky, "Geophysical Fluid Dynamics,'' Springer-Verlag, New York, 1979. |
[13] |
H. Segur, D. Henderson, J. Carter, J. Hammack, C.-M. Li, D. Pheiff and K. Socha, Stabilizing the Benjamin-Feir instability, J. Fluid Mech., 539 (2005), 229-271. |
[14] |
G. G. Stokes, On the theory of osillatory waves, Trans. Cambridge Philos. Soc., 8 (1847), 441-473. |
[15] |
M. Tanaka, The stability of steep gravity waves, J. Phys. Soc. Japan, 52 (1983), 3047-3055.
doi: 10.1143/JPSJ.52.3047. |
[16] |
M. Tanaka, The stability of steep gravity waves. II, J. Fluid Mech., 156 (1985), 281-289.
doi: 10.1017/S0022112085002099. |
[17] |
G. B. Whitham, "Linear and Nonlinear Waves,'' Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
[18] |
G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Comm. Pure Appl. Math., 41 (1988), 19-46.
doi: 10.1002/cpa.3160410104. |
[19] |
G. Wu, Y. Liu, and D. K. P. Yue, A note on stabilizing the Benjamin-Feir instability, J. Fluid Mech., 556 (2006), 45-54.
doi: 10.1017/S0022112005008293. |
[20] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of adeep fluid, J. Appl. Mech. Tech. Phys., 2 (1968), 190-194. |
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