June  2017, 37(6): 3111-3122. doi: 10.3934/dcds.2017133

On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations

1. 

Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794, Istanbul, Turkey

2. 

Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

* Corresponding author: H. A. Erbay

Received  December 2016 Revised  January 2017 Published  February 2017

We rigorously establish that, in the long-wave regime characterized by the assumptions of long wavelength and small amplitude, bidirectional solutions of the improved Boussinesq equation tend to associated solutions of two uncoupled Camassa-Holm equations. We give a precise estimate for approximation errors in terms of two small positive parameters measuring the effects of nonlinearity and dispersion. Our results demonstrate that, in the present regime, any solution of the improved Boussinesq equation is split into two waves propagating in opposite directions independently, each of which is governed by the Camassa-Holm equation. We observe that the approximation error for the decoupled problem considered in the present study is greater than the approximation error for the unidirectional problem characterized by a single Camassa-Holm equation. We also consider lower order approximations and we state similar error estimates for both the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.

Citation: H. A. Erbay, S. Erbay, A. Erkip. On the decoupling of the improved Boussinesq equation into two uncoupled Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3111-3122. doi: 10.3934/dcds.2017133
References:
[1]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Advances in Differential Equations, 11 (2006), 121-166.   Google Scholar

[2]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

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J. L. BonaT. Colin and D. Lannes, Long wave approximations for water waves, Arch. Rational Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1.  Google Scholar

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[6]

A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential and Integral Equations, 15 (2002), 1061-1072.   Google Scholar

[7]

W. Craig, An existence theory for water waves and the Boussinesq and Kortewegde Vries scaling limits, Commun. Part. Diff. Eqns., 10 (1985), 787-1003.  doi: 10.1080/03605308508820396.  Google Scholar

[8]

V. Duchene, Decoupled and unidirectional asymptotic models for the propagation of internal waves, M3AS: Math. Models Methods Appl. Sci., 24 (2014), 1-65.  doi: 10.1142/S0218202513500462.  Google Scholar

[9]

N. DurukA. Erkip and H. A. Erbay, A higher-order Boussinesq equation in locally nonlinear theory of one-dimensional nonlocal elasticity, IMA J. Appl. Math., 74 (2009), 97-106.  doi: 10.1093/imamat/hxn020.  Google Scholar

[10]

N. DurukH. A. Erbay and A. Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.  doi: 10.1088/0951-7715/23/1/006.  Google Scholar

[11]

H. A. ErbayS. Erbay and A. Erkip, Derivation of the Camassa-Holm equations for elastic waves, Phys. Lett. A, 379 (2015), 956-961.  doi: 10.1016/j.physleta.2015.01.031.  Google Scholar

[12]

H. A. ErbayS. Erbay and A. Erkip, The Camassa-Holm equation as the long-wave limit of the improved Bousssinesq equation and of a class of nonlocal wave equations, Discrete Contin. Dyn. Syst., 36 (2016), 6101-6116.  doi: 10.3934/dcds.2016066.  Google Scholar

[13]

T. Gallay and G. Schneider, KP description of unidirectional long waves. The model case, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 885-898.  doi: 10.1017/S0308210500001165.  Google Scholar

[14]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[15]

D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics AMS Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.  Google Scholar

[16]

L. A. Ostrovskii and A. M. Sutin, Nonlinear elastic waves in rods, PMM J. Appl. Math. Mech., 41 (1977), 543-549.   Google Scholar

[17]

G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475-1535.  doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.  Google Scholar

[18]

G. Schneider, The long wave limit for a Boussinesq equation, SIAM J. Appl. Math., 58 (1998), 1237-1245.  doi: 10.1137/S0036139995287946.  Google Scholar

[19]

M. P. SoerensenP. L. Christiansen and P. S. Lomdahl, Solitary.waves on nonlinear elastic rods. I, J. Acoust. Soc. Am., 76 (1984), 871-879.  doi: 10.1121/1.391312.  Google Scholar

[20]

C. E. Wayne and J. D. Wright, Higher order modulation equations for a Boussinesq equation, SIAM J. Appl. Dyn. Sys., 1 (2002), 271-302.  doi: 10.1137/S1111111102411298.  Google Scholar

[21]

J. D. Wright, Corrections to the KdV Approximation for Water Waves, SIAM J. Math. Anal., 37 (2005), 1161-1206.  doi: 10.1137/S0036141004444202.  Google Scholar

show all references

References:
[1]

A. A. AlazmanJ. P. AlbertJ. L. BonaM. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system, Advances in Differential Equations, 11 (2006), 121-166.   Google Scholar

[2]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci., 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[3]

J. L. BonaT. Colin and D. Lannes, Long wave approximations for water waves, Arch. Rational Mech. Anal., 178 (2005), 373-410.  doi: 10.1007/s00205-005-0378-1.  Google Scholar

[4]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[6]

A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation, Differential and Integral Equations, 15 (2002), 1061-1072.   Google Scholar

[7]

W. Craig, An existence theory for water waves and the Boussinesq and Kortewegde Vries scaling limits, Commun. Part. Diff. Eqns., 10 (1985), 787-1003.  doi: 10.1080/03605308508820396.  Google Scholar

[8]

V. Duchene, Decoupled and unidirectional asymptotic models for the propagation of internal waves, M3AS: Math. Models Methods Appl. Sci., 24 (2014), 1-65.  doi: 10.1142/S0218202513500462.  Google Scholar

[9]

N. DurukA. Erkip and H. A. Erbay, A higher-order Boussinesq equation in locally nonlinear theory of one-dimensional nonlocal elasticity, IMA J. Appl. Math., 74 (2009), 97-106.  doi: 10.1093/imamat/hxn020.  Google Scholar

[10]

N. DurukH. A. Erbay and A. Erkip, Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity, Nonlinearity, 23 (2010), 107-118.  doi: 10.1088/0951-7715/23/1/006.  Google Scholar

[11]

H. A. ErbayS. Erbay and A. Erkip, Derivation of the Camassa-Holm equations for elastic waves, Phys. Lett. A, 379 (2015), 956-961.  doi: 10.1016/j.physleta.2015.01.031.  Google Scholar

[12]

H. A. ErbayS. Erbay and A. Erkip, The Camassa-Holm equation as the long-wave limit of the improved Bousssinesq equation and of a class of nonlocal wave equations, Discrete Contin. Dyn. Syst., 36 (2016), 6101-6116.  doi: 10.3934/dcds.2016066.  Google Scholar

[13]

T. Gallay and G. Schneider, KP description of unidirectional long waves. The model case, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 885-898.  doi: 10.1017/S0308210500001165.  Google Scholar

[14]

D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[15]

D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics AMS Mathematical Surveys and Monographs, vol. 188, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/surv/188.  Google Scholar

[16]

L. A. Ostrovskii and A. M. Sutin, Nonlinear elastic waves in rods, PMM J. Appl. Math. Mech., 41 (1977), 543-549.   Google Scholar

[17]

G. Schneider and C. E. Wayne, The long-wave limit for the water wave problem. I. The case of zero surface tension, Comm. Pure Appl. Math., 53 (2000), 1475-1535.  doi: 10.1002/1097-0312(200012)53:12<1475::AID-CPA1>3.0.CO;2-V.  Google Scholar

[18]

G. Schneider, The long wave limit for a Boussinesq equation, SIAM J. Appl. Math., 58 (1998), 1237-1245.  doi: 10.1137/S0036139995287946.  Google Scholar

[19]

M. P. SoerensenP. L. Christiansen and P. S. Lomdahl, Solitary.waves on nonlinear elastic rods. I, J. Acoust. Soc. Am., 76 (1984), 871-879.  doi: 10.1121/1.391312.  Google Scholar

[20]

C. E. Wayne and J. D. Wright, Higher order modulation equations for a Boussinesq equation, SIAM J. Appl. Dyn. Sys., 1 (2002), 271-302.  doi: 10.1137/S1111111102411298.  Google Scholar

[21]

J. D. Wright, Corrections to the KdV Approximation for Water Waves, SIAM J. Math. Anal., 37 (2005), 1161-1206.  doi: 10.1137/S0036141004444202.  Google Scholar

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