# American Institute of Mathematical Sciences

October  2012, 17(7): 2329-2341. doi: 10.3934/dcdsb.2012.17.2329

## 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

Received  November 2010 Revised  March 2012 Published  July 2012

The Benjamin-Feir instability describes the instability of a uniform oscillatory wave train in an irrotational flow subject to small perturbation of wave number, amplitude and frequency. Their instability analysis is based on the perturbation around the second order Stokes wave which satisfies the dynamic and kinematic free-surface boundary conditions up to the second order. In the same irrotational flow and perturbation framework of the Benjamin-Feir analysis, the perturbation in the present paper is around a nonlinear oscillatory wave train which solves exactly the dynamic free-surface boundary condition and satisfies the kinematic free-surface boundary condition up to the third order. It is shown that the nonlinear oscillatory wave train is stable with respect to the perturbation when the irrotational flow involves small Rayleigh energy dissipation.
Citation: Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
##### 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.
 [1] Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 [2] Yang Liu, Wenke Li. A family of potential wells for a wave equation. Electronic Research Archive, 2020, 28 (2) : 807-820. doi: 10.3934/era.2020041 [3] Alan Compelli, Rossen Ivanov. Benjamin-Ono model of an internal wave under a flat surface. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4519-4532. doi: 10.3934/dcds.2019185 [4] Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525 [5] Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, 2021, 3 (2) : 99-131. doi: 10.3934/fods.2021009 [6] Shu Wang, Yixuan Zhao. Asymptotic stability of planar rarefaction wave to a multi-dimensional two-phase flow. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022091 [7] Paolo Baiti, Alberto Bressan, Helge Kristian Jenssen. Instability of travelling wave profiles for the Lax-Friedrichs scheme. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 877-899. doi: 10.3934/dcds.2005.13.877 [8] Guido Schneider, Matthias Winter. The amplitude system for a Simultaneous short-wave Turing and long-wave Hopf instability. Discrete and Continuous Dynamical Systems - S, 2022, 15 (9) : 2657-2672. doi: 10.3934/dcdss.2021119 [9] Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215 [10] José R. Quintero, Alex M. Montes. Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type. Evolution Equations and Control Theory, 2022, 11 (3) : 681-709. doi: 10.3934/eect.2021021 [11] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427 [12] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 [13] Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615 [14] P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029 [15] Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075 [16] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [17] Yavar Kian. Stability of the determination of a coefficient for wave equations in an infinite waveguide. Inverse Problems and Imaging, 2014, 8 (3) : 713-732. doi: 10.3934/ipi.2014.8.713 [18] Lili Fan, Hongxia Liu, Huijiang Zhao, Qingyang Zou. Global stability of stationary waves for damped wave equations. Kinetic and Related Models, 2013, 6 (4) : 729-760. doi: 10.3934/krm.2013.6.729 [19] Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control and Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010 [20] Qingqing Liu, Xiaoli Wu. Stability of rarefaction wave for viscous vasculogenesis model. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022034

2021 Impact Factor: 1.497