We consider unidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral with a suitable kernel function. We first give a brief review of asymptotic wave models describing the unidirectional propagation of small-but-finite amplitude long waves. When the kernel function is the well-known exponential kernel, the asymptotic description is provided by the Korteweg–de Vries (KdV) equation, the Benjamin–Bona–Mahony (BBM) equation, or the Camassa–Holm (CH) equation. When the Fourier transform of the kernel function has fractional powers, it turns out that fractional forms of these equations describe unidirectional propagation of the waves. We then compare the exact solutions of the KdV equation and the BBM equation with the numerical solutions of the nonlocal model. We observe that the solution of the nonlocal model is well approximated by associated solutions of the KdV equation and the BBM equation over the time interval considered.
1. Eringen, A. Nonlocal Continuum Field Theories. Springer, New York, 2002.
2. Engelbrecht, J. and Braun, M. Nonlinear waves in nonlocal media. Appl. Mech. Rev., 1998, 51, 475–488.
http://dx.doi.org/10.1115/1.3099016
3. Lazar, M., Maugin, G. A., and Aifantis, E. C. On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct., 2006, 43, 1404–1421.
http://dx.doi.org/10.1016/j.ijsolstr.2005.04.027
4. Gopalakrishnan, S. and Narendar, S. Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations. Springer, Switzerland, 2013.
http://dx.doi.org/10.1007/978-3-319-01032-8
5. Duruk, N., Erkip, A., and Erbay, H. A. A higher-order Boussinesq equation in locally nonlinear theory of one-dimensional nonlocal elasticity. IMA J. Appl. Math., 2009, 74, 97–106.
http://dx.doi.org/10.1093/imamat/hxn020
6. Duruk, N., Erbay, H. A., and Erkip, A. Global existence and blow-up for a class of nonlocal nonlinear Cauchy problems arising in elasticity. Nonlinearity, 2010, 23, 107–118.
http://dx.doi.org/10.1088/0951-7715/23/1/006
7. Duruk, N., Erbay, H. A., and Erkip, A. Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations. J. Differ. Equations, 2011, 250, 1448–1459.
http://dx.doi.org/10.1016/j.jde.2010.09.002
8. Erbay, H. A., Erbay, S., and Erkip, A. The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials. Nonlinearity, 2011, 24, 1347–1359.
http://dx.doi.org/10.1088/0951-7715/24/4/017
9. Erbay, H. A., Erbay, S., and Erkip, A. Derivation of the Camassa–Holm equations for elastic waves. Phys. Lett. A, 2015, 379, 956–961.
http://dx.doi.org/10.1016/j.physleta.2015.01.031
10. Korteweg, D. J. and de Vries, G. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag., 1895, 39, 422–443.
http://dx.doi.org/10.1080/14786449508620739
11. Benjamin, T. B., Bona, J. L., and Mahony, J. J. Model equations for long waves in nonlinear dispersive systems. Philos. T. Roy. Soc. A, 1972, 272, 47–78.
http://dx.doi.org/10.1098/rsta.1972.0032
12. Camassa, R. and Holm, D. D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 1993, 71, 1661–1664.
http://dx.doi.org/10.1103/PhysRevLett.71.1661
13. Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys., 1983, 54, 4703–4710.
http://dx.doi.org/10.1063/1.332803
14. Dai, H.-H. Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod. Acta Mech., 1998, 127, 193–207.
http://dx.doi.org/10.1007/BF01170373
15. Chen, R. M. Some nonlinear dispersive waves arising in compressible hyperelastic plates. Int. J. Eng. Sci., 2006, 44, 1188–1204.
http://dx.doi.org/10.1016/j.ijengsci.2006.08.003
16. Johnson, R. S. Camassa–Holm, Korteweg–de Vries and related models for water waves. J. Fluid Mech., 2002, 455, 63–82.
http://dx.doi.org/10.1017/S0022112001007224
17. Constantin, A. and Lannes, D. The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. An., 2009, 192, 165–186.
http://dx.doi.org/10.1007/s00205-008-0128-2
18. Johnson, R. S. A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques. Commun. Pure Appl. Anal., 2012, 11, 1497–1522.
http://dx.doi.org/10.3934/cpaa.2012.11.1497
19. Bona, J. L., Souganidis, P. E., and Straus, W. A. Stability and instability of solitary waves of Korteweg–de Vries type. P. Roy. Soc. Lond. A Mat., 1987, 411, 395–412.
20. Kapitula, T. and Stefanov, A. A Hamiltonian–Krein (instability) index theory for solitary waves to KdV-like eigenvalue problems. Stud. Appl. Math., 2014, 132, 183–211.
http://dx.doi.org/10.1111/sapm.12031
http://dx.doi.org/10.1007/BF01238818
