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 & Continuous Dynamical Systems - A, 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. 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. 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, 168 (1995), 11.

[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. 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. 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.

[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. 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.

[9]

J. Angulo, On the instability of solitary wave solutions for fifith-order water wave models,, Elec. J. Diff. Equations, 2003 (2003), 1.

[10]

E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom,, Stud. Appl. Math., 87 (1992), 1.

[11]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. 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.

[13]

R. M. Chen, V. M. Hur and Y. Liu, Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation,, Nonlinearity, 21 (2008), 2949. 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. 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). 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. 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.

[18]

J. Gonçcalves Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field,, Ann. Inst. H. Poincaré, 54 (1991), 403.

[19]

R. Grimshaw, Evolution equations for weakly nonlinear long internal waves in a rotating fluid,, Stud. Appl. Math, 73 (1985), 1.

[20]

S. Levandosky, A stability analysis of fifth-order water wave models,, Phys. D, 125 (1999), 222. 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. 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. 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.

[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.

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincaré, 1 (1984), 109.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II.,, Ann. Inst. H. Poincaré, 4 (1984), 223.

[27]

Y. Liu, On the stability of solitary waves for the Ostrovsky equation,, Quart. Appl. Math, 65 (2007), 571.

[28]

Y. Liu and M. Ohta, Stability of solitary waves for the Ostrovsky equation,, Proc. Amer. Math. Soc., 136 (2008), 511. 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. 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.

[31]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologiya, 18 (1978), 181.

[32]

L. A. Ostrovsky and Y. A. Stepanyants, "Nonlinear Surface and Internal Waves in Rotating Fluids,", Research Reports in Physics, (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. doi: 10.1137/S0036142902414232.

[34]

L. G. Redekopp, Nonlinear waves in geophysics: Long internal waves,, Lectures in Appl. Math, 20 (1983), 59.

show all references

References:
[1]

J. P. Albert, Concentration compactness and the stability of solitary wave solutions to nonlocal equations,, Contemp. Math, 221 (1999), 1. 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. 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, 168 (1995), 11.

[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. 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. 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.

[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. 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.

[9]

J. Angulo, On the instability of solitary wave solutions for fifith-order water wave models,, Elec. J. Diff. Equations, 2003 (2003), 1.

[10]

E. S. Benilov, On the surface waves in a shallow channel with an uneven bottom,, Stud. Appl. Math., 87 (1992), 1.

[11]

T. B. Benjamin, Internal waves of permanent form in fluids of great depth,, J. Fluid Mech., 29 (1967), 559. 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.

[13]

R. M. Chen, V. M. Hur and Y. Liu, Solitary waves of the rotation-modified Kadomtsev-Petviashvili equation,, Nonlinearity, 21 (2008), 2949. 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. 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). 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. 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.

[18]

J. Gonçcalves Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field,, Ann. Inst. H. Poincaré, 54 (1991), 403.

[19]

R. Grimshaw, Evolution equations for weakly nonlinear long internal waves in a rotating fluid,, Stud. Appl. Math, 73 (1985), 1.

[20]

S. Levandosky, A stability analysis of fifth-order water wave models,, Phys. D, 125 (1999), 222. 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. 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. 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.

[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.

[25]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincaré, 1 (1984), 109.

[26]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II.,, Ann. Inst. H. Poincaré, 4 (1984), 223.

[27]

Y. Liu, On the stability of solitary waves for the Ostrovsky equation,, Quart. Appl. Math, 65 (2007), 571.

[28]

Y. Liu and M. Ohta, Stability of solitary waves for the Ostrovsky equation,, Proc. Amer. Math. Soc., 136 (2008), 511. 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. 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.

[31]

L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean,, Okeanologiya, 18 (1978), 181.

[32]

L. A. Ostrovsky and Y. A. Stepanyants, "Nonlinear Surface and Internal Waves in Rotating Fluids,", Research Reports in Physics, (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. doi: 10.1137/S0036142902414232.

[34]

L. G. Redekopp, Nonlinear waves in geophysics: Long internal waves,, Lectures in Appl. Math, 20 (1983), 59.

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