Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: Primary: 35Q53, 35Q74; Secondary: 74J30, 35C20.

 Citation:

•  [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), 121-166. [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), 47-78.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), 373-410.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 J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Mathematica, 181 (1998), 229-243.doi: 10.1007/BF02392586. [6] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. Lond. A, 457 (2001), 953-970.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), 165-186.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), 1061-1072. [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), 97-106.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), 107-118.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), 956-961.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), 256-262.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), 885-898.doi: 10.1017/S0308210500001165. [14] D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Non-linear Math. Phys., 14 (2007), 303-312.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), 63-82.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), 422-443. [17] 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. [18] G. Schneider, The long wave limit for a Boussinesq equation, SIAM J. Appl. Math., 58 (1998), 1237-1245.doi: 10.1137/S0036139995287946.