# American Institute of Mathematical Sciences

2016, 36(11): 6101-6116. doi: 10.3934/dcds.2016066

## The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations

 1 Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794, Istanbul, Turkey, Turkey 2 Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

Received  November 2015 Revised  December 2015 Published  August 2016

In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters $\epsilon$ and $\delta$ measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation.
Citation: 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
##### References:
 [1] A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system,, Advances in Differential Equations, 11 (2006), 121. [2] T. B. Benjamin, J. 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. doi: 10.1098/rsta.1972.0032. [3] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves,, Arch. Rational Mech. Anal., 178 (2005), 373. 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. doi: 10.1103/PhysRevLett.71.1661. [5] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229. doi: 10.1007/BF02392586. [6] A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. Lond. A, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. [7] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rational Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. [8] A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation,, Differential and Integral Equations, 15 (2002), 1061. [9] N. Duruk, A. 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. doi: 10.1093/imamat/hxn020. [10] N. Duruk, H.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. doi: 10.1088/0951-7715/23/1/006. [11] H. A. Erbay, S. Erbay and A. Erkip, Derivation of the Camassa-Holm equations for elastic waves,, Physics Letters A, 379 (2015), 956. doi: 10.1016/j.physleta.2015.01.031. [12] H. A. Erbay, S. Erbay and A. Erkip, Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: The KdV, BBM and CH equations,, Proceedings of the Estonian Academy of Sciences, 64 (2015), 256. doi: 10.3176/proc.2015.3.08. [13] T. Gallay and G. Schneider, KP description of unidirectional long waves. The model case,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 885. doi: 10.1017/S0308210500001165. [14] D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation,, J. Non-linear Math. Phys., 14 (2007), 303. doi: 10.2991/jnmp.2007.14.3.1. [15] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. [16] 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. [17] D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics,, AMS Mathematical Surveys and Monographs, (2013). doi: 10.1090/surv/188. [18] G. Schneider, The long wave limit for a Boussinesq equation,, SIAM J. Appl. Math., 58 (1998), 1237. doi: 10.1137/S0036139995287946.

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##### References:
 [1] A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a Boussinesq system,, Advances in Differential Equations, 11 (2006), 121. [2] T. B. Benjamin, J. 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. doi: 10.1098/rsta.1972.0032. [3] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves,, Arch. Rational Mech. Anal., 178 (2005), 373. 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. doi: 10.1103/PhysRevLett.71.1661. [5] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Mathematica, 181 (1998), 229. doi: 10.1007/BF02392586. [6] A. Constantin, On the scattering problem for the Camassa-Holm equation,, Proc. R. Soc. Lond. A, 457 (2001), 953. doi: 10.1098/rspa.2000.0701. [7] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Rational Mech. Anal., 192 (2009), 165. doi: 10.1007/s00205-008-0128-2. [8] A. Constantin and L. Molinet, The initial value problem for a generalized Boussinesq equation,, Differential and Integral Equations, 15 (2002), 1061. [9] N. Duruk, A. 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. doi: 10.1093/imamat/hxn020. [10] N. Duruk, H.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. doi: 10.1088/0951-7715/23/1/006. [11] H. A. Erbay, S. Erbay and A. Erkip, Derivation of the Camassa-Holm equations for elastic waves,, Physics Letters A, 379 (2015), 956. doi: 10.1016/j.physleta.2015.01.031. [12] H. A. Erbay, S. Erbay and A. Erkip, Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: The KdV, BBM and CH equations,, Proceedings of the Estonian Academy of Sciences, 64 (2015), 256. doi: 10.3176/proc.2015.3.08. [13] T. Gallay and G. Schneider, KP description of unidirectional long waves. The model case,, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 885. doi: 10.1017/S0308210500001165. [14] D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation,, J. Non-linear Math. Phys., 14 (2007), 303. doi: 10.2991/jnmp.2007.14.3.1. [15] R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. doi: 10.1017/S0022112001007224. [16] 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. [17] D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics,, AMS Mathematical Surveys and Monographs, (2013). doi: 10.1090/surv/188. [18] G. Schneider, The long wave limit for a Boussinesq equation,, SIAM J. Appl. Math., 58 (1998), 1237. doi: 10.1137/S0036139995287946.
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