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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[21]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[21]

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

[1]

H. A. Erbay, S. Erbay, A. Erkip. The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6101-6116. doi: 10.3934/dcds.2016066

[2]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065

[3]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047

[4]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883

[5]

Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159

[6]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

[7]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067

[8]

Stephen C. Anco, Elena Recio, María L. Gandarias, María S. Bruzón. A nonlinear generalization of the Camassa-Holm equation with peakon solutions. Conference Publications, 2015, 2015 (special) : 29-37. doi: 10.3934/proc.2015.0029

[9]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230

[10]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305

[11]

Shouming Zhou, Chunlai Mu. Global conservative and dissipative solutions of the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1713-1739. doi: 10.3934/dcds.2013.33.1713

[12]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026

[13]

Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181

[14]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483

[15]

Priscila Leal da Silva, Igor Leite Freire. An equation unifying both Camassa-Holm and Novikov equations. Conference Publications, 2015, 2015 (special) : 304-311. doi: 10.3934/proc.2015.0304

[16]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194

[17]

David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629

[18]

Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45

[19]

Alberto Bressan, Geng Chen, Qingtian Zhang. Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 25-42. doi: 10.3934/dcds.2015.35.25

[20]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (11)
  • HTML views (1)
  • Cited by (0)

Other articles
by authors

[Back to Top]