# American Institute of Mathematical Sciences

February  2013, 33(2): 663-700. doi: 10.3934/dcds.2013.33.663

## Solitary waves of the rotation-generalized Benjamin-Ono equation

 1 School of Mathematics and Computer Science, Damghan University, Damghan 36715-364, Iran 2 Mathematics and Computer Science Department, College of the Holy Cross, Worcester, MA, 01610

Received  May 2011 Revised  August 2012 Published  September 2012

This work studies the rotation-generalized Benjamin-Ono equation which is derived from the theory of weakly nonlinear long surface and internal waves in deep water under the presence of rotation. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.
Citation: Amin Esfahani, Steve Levandosky. Solitary waves of the rotation-generalized Benjamin-Ono equation. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 663-700. doi: 10.3934/dcds.2013.33.663
##### References:
 [1] J. P. Albert, Concentration compactness and the stability of solitary wave solutions to nonlocal equations, Contemp. Math, 221 (1999), 1-29. doi: 10.1090/conm/221/03116. [2] J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Comm. Partial Differential Equations, 17 (1992), 1-22. doi: 10.1080/03605309208820831. [3] J. P. Albert, Positivity properties and uniqueness of solitary wave solutions of the intermediate long-wave equation. Evolution equations, Lecture Notes in Pure and Appl. Math, Dekker, New York, 168 (1995), 11-20. [4] J. P. Albert and J. L. Bona, Total positivity and the stability of internal waves in stratified fluids of finite depth, IMA J. Appl. Math, 46 (1991), 1-19. doi: 10.1093/imamat/46.1-2.1. [5] J. P. Albert, J. L. Bona and J. M. Restrepo, Solitary-wave solutions of the Benjamin equation, SIAM J. Appl. Math, 59 (1999), 2139-2161. doi: 10.1137/S0036139997321682. [6] J. P. Albert, J. L. Bona and J. C. Saut, Model equations for waves in startified fluids, Proc. Royal Soc. Edinburgh Sec. A, 453 (1997), 1233-1260. [7] C. J. Amick and J. F. Toland, Uniqueness of Benjamin's solitary wave solution of the Benjamin-Ono, IMA J. Appl. Math, 46 (1991), 21-28. doi: 10.1093/imamat/46.1-2.21. [8] J. Angulo, On the instability of solitary waves solutions of the generalized Benjamin equation, Adv. Differential Equations, 8 (2003), 55-82. [9] J. Angulo, On the instability of solitary wave solutions for fifith-order water wave models, Elec. J. Diff. Equations, 2003 (2003), 1-18. [10] E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math., 87 (1992), 1-4. [11] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X. [12] J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. [13] R. M. Chen, V. M. Hur and Y. Liu, Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation, Nonlinearity, 21 (2008), 2949-979. doi: 10.1088/0951-7715/21/12/012. [14] M. Chen, Y. Liu and P. Zhang, Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc., 364 (2012), 3395-3425. doi: 10.1090/S0002-9947-2012-05383-9. [15] A. Esfahani, Decay properties of the traveling waves of the rotation-generalized Kadomtsev-Petviashvili equation, J. Phys. A: Math. Theor., 43 (2010), 395201. doi: 10.1088/1751-8113/43/39/395201. [16] V. N. Galkin and Y. A. Stepanyants, On the existence of stationary solitary waves in a Rotating fluid, J. Appl. Maths. Mechs., 55 (1991), 1051-1055. doi: 10.1016/0021-8928(91)90148-N. [17] O. A. Gilman, R. Grimshaw and Y. A. Stepanyants, Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math, 95 (1995), 115-126. [18] J. Gonçcalves Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. H. Poincaré, Phys. Théor., 54 (1991), 403-433. [19] R. Grimshaw, Evolution equations for weakly nonlinear long internal waves in a rotating fluid, Stud. Appl. Math, 73 (1985), 1-33. [20] S. Levandosky, A stability analysis of fifth-order water wave models, Phys. D, 125 (1999), 222-240. doi: 10.1016/S0167-2789(98)00245-0. [21] S. Levandosky, Stability and instability of fourth order solitary waves, J. Dynam. Differential Equations, 10 (1998), 151-188. doi: 10.1023/A:1022644629950. [22] S. Levandosky and Y. Liu, Stability of solitary waves of a generalized Ostrovsky equation, SIAM J. Math. Anal., 38 (2006), 985-1011. doi: 10.1137/050638722. [23] S. P. Levandosky and Y. Liu, Stability and weak rotation limit of solitary waves of the Ostrovsky equation, Discrete Contin. Dynam. Systems-B, 7 (2007), 793-806. [24] F. Linares and A. Milanes, A note on solutions to a model for long internal waves in a rotating fluid, Mat. Contemp., 27 (2004), 101-115. [25] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré, Anal. Non linéare， 1 (1984), 109-145. [26] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré, Anal. Non linéare， 4 (1984), 223-283. [27] Y. Liu, On the stability of solitary waves for the Ostrovsky equation, Quart. Appl. Math, 65 (2007), 571-589. [28] Y. Liu and M. Ohta, Stability of solitary waves for the Ostrovsky equation, Proc. Amer. Math. Soc., 136 (2008), 511-517. doi: 10.1090/S0002-9939-07-09191-5. [29] Y. Liu and V. Varlamov, Stability of solitary waves and weak rotation limit for the Ostrovsky equation, J. Differential Equations, 203 (2004), 159-183. doi: 10.1016/j.jde.2004.03.026. [30] Y. Liu and M. M. Tom, Blow-up and instability of a regularized long-wave-KP equation, Differential Integral Equations, 19 (2003), 1131-1152. [31] L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologiya, 18 (1978), 181-191. [32] L. A. Ostrovsky and Y. A. Stepanyants, "Nonlinear Surface and Internal Waves in Rotating Fluids," Research Reports in Physics, Nonlinear wave 3, Springer, Berlin, Heidelberg, 1990. [33] D. E. Pelinovsky and Y. A. Stepanyants, Convergence of petviashvili's iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM J. Numer. Anal., 42 (2004), 1110-1127. doi: 10.1137/S0036142902414232. [34] L. G. Redekopp, Nonlinear waves in geophysics: Long internal waves, Lectures in Appl. Math, 20 (1983), 59-78.

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##### References:
 [1] J. P. Albert, Concentration compactness and the stability of solitary wave solutions to nonlocal equations, Contemp. Math, 221 (1999), 1-29. doi: 10.1090/conm/221/03116. [2] J. P. Albert, Positivity properties and stability of solitary-wave solutions of model equations for long waves, Comm. Partial Differential Equations, 17 (1992), 1-22. doi: 10.1080/03605309208820831. [3] J. P. Albert, Positivity properties and uniqueness of solitary wave solutions of the intermediate long-wave equation. Evolution equations, Lecture Notes in Pure and Appl. Math, Dekker, New York, 168 (1995), 11-20. [4] J. P. Albert and J. L. Bona, Total positivity and the stability of internal waves in stratified fluids of finite depth, IMA J. Appl. Math, 46 (1991), 1-19. doi: 10.1093/imamat/46.1-2.1. [5] J. P. Albert, J. L. Bona and J. M. Restrepo, Solitary-wave solutions of the Benjamin equation, SIAM J. Appl. Math, 59 (1999), 2139-2161. doi: 10.1137/S0036139997321682. [6] J. P. Albert, J. L. Bona and J. C. Saut, Model equations for waves in startified fluids, Proc. Royal Soc. Edinburgh Sec. A, 453 (1997), 1233-1260. [7] C. J. Amick and J. F. Toland, Uniqueness of Benjamin's solitary wave solution of the Benjamin-Ono, IMA J. Appl. Math, 46 (1991), 21-28. doi: 10.1093/imamat/46.1-2.21. [8] J. Angulo, On the instability of solitary waves solutions of the generalized Benjamin equation, Adv. Differential Equations, 8 (2003), 55-82. [9] J. Angulo, On the instability of solitary wave solutions for fifith-order water wave models, Elec. J. Diff. Equations, 2003 (2003), 1-18. [10] E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom, Stud. Appl. Math., 87 (1992), 1-4. [11] T. B. Benjamin, Internal waves of permanent form in fluids of great depth, J. Fluid Mech., 29 (1967), 559-592. doi: 10.1017/S002211206700103X. [12] J. L. Bona and Y. A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl., 76 (1997), 377-430. [13] R. M. Chen, V. M. Hur and Y. Liu, Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation, Nonlinearity, 21 (2008), 2949-979. doi: 10.1088/0951-7715/21/12/012. [14] M. Chen, Y. Liu and P. Zhang, Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation, Trans. Amer. Math. Soc., 364 (2012), 3395-3425. doi: 10.1090/S0002-9947-2012-05383-9. [15] A. Esfahani, Decay properties of the traveling waves of the rotation-generalized Kadomtsev-Petviashvili equation, J. Phys. A: Math. Theor., 43 (2010), 395201. doi: 10.1088/1751-8113/43/39/395201. [16] V. N. Galkin and Y. A. Stepanyants, On the existence of stationary solitary waves in a Rotating fluid, J. Appl. Maths. Mechs., 55 (1991), 1051-1055. doi: 10.1016/0021-8928(91)90148-N. [17] O. A. Gilman, R. Grimshaw and Y. A. Stepanyants, Approximate and numerical solutions of the stationary Ostrovsky equation, Stud. Appl. Math, 95 (1995), 115-126. [18] J. Gonçcalves Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. H. Poincaré, Phys. Théor., 54 (1991), 403-433. [19] R. Grimshaw, Evolution equations for weakly nonlinear long internal waves in a rotating fluid, Stud. Appl. Math, 73 (1985), 1-33. [20] S. Levandosky, A stability analysis of fifth-order water wave models, Phys. D, 125 (1999), 222-240. doi: 10.1016/S0167-2789(98)00245-0. [21] S. Levandosky, Stability and instability of fourth order solitary waves, J. Dynam. Differential Equations, 10 (1998), 151-188. doi: 10.1023/A:1022644629950. [22] S. Levandosky and Y. Liu, Stability of solitary waves of a generalized Ostrovsky equation, SIAM J. Math. Anal., 38 (2006), 985-1011. doi: 10.1137/050638722. [23] S. P. Levandosky and Y. Liu, Stability and weak rotation limit of solitary waves of the Ostrovsky equation, Discrete Contin. Dynam. Systems-B, 7 (2007), 793-806. [24] F. Linares and A. Milanes, A note on solutions to a model for long internal waves in a rotating fluid, Mat. Contemp., 27 (2004), 101-115. [25] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I., Ann. Inst. H. Poincaré, Anal. Non linéare， 1 (1984), 109-145. [26] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II., Ann. Inst. H. Poincaré, Anal. Non linéare， 4 (1984), 223-283. [27] Y. Liu, On the stability of solitary waves for the Ostrovsky equation, Quart. Appl. Math, 65 (2007), 571-589. [28] Y. Liu and M. Ohta, Stability of solitary waves for the Ostrovsky equation, Proc. Amer. Math. Soc., 136 (2008), 511-517. doi: 10.1090/S0002-9939-07-09191-5. [29] Y. Liu and V. Varlamov, Stability of solitary waves and weak rotation limit for the Ostrovsky equation, J. Differential Equations, 203 (2004), 159-183. doi: 10.1016/j.jde.2004.03.026. [30] Y. Liu and M. M. Tom, Blow-up and instability of a regularized long-wave-KP equation, Differential Integral Equations, 19 (2003), 1131-1152. [31] L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Okeanologiya, 18 (1978), 181-191. [32] L. A. Ostrovsky and Y. A. Stepanyants, "Nonlinear Surface and Internal Waves in Rotating Fluids," Research Reports in Physics, Nonlinear wave 3, Springer, Berlin, Heidelberg, 1990. [33] D. E. Pelinovsky and Y. A. Stepanyants, Convergence of petviashvili's iteration method for numerical approximation of stationary solutions of nonlinear wave equations, SIAM J. Numer. Anal., 42 (2004), 1110-1127. doi: 10.1137/S0036142902414232. [34] L. G. Redekopp, Nonlinear waves in geophysics: Long internal waves, Lectures in Appl. Math, 20 (1983), 59-78.
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