Various continuum limits of the original discrete hexagonal lattice model are used to obtain transverse weakly nonlinear equations for longitudinal waves. It is shown, that the long wavelength continuum limit gives rise to the Kadomtsev–Petviashvili equation, while another continuum limit results in obtaining two-dimensional generalization of the nonlinear Schrödinger equation.
1. Askar, A. Lattice Dynamical Foundations of Continuum Theories. World Scientific, Singapore, 1985.
2. Maugin, G. A. Nonlinear Waves in Elastic Crystals. Oxford University Press, UK, 1999.
3. Zabusky, N. J. and Deem, G. S. Dynamics of nonlinear lattices. I. Localized optical excitations, acoustic radiation, and strong nonlinear behavior. J. Comput. Phys., 1967, 2, 126–153.
http://dx.doi.org/10.1016/0021-9991(67)90031-9
4. Porubov, A. V. and Berinskii, I. E. Non-linear plane waves in materials having hexagonal internal structure. Int. J. Non-Linear Mech., 2014, 67, 27–33.
http://dx.doi.org/10.1016/j.ijnonlinmec.2014.07.003
5. Ablowitz, M. J. and Segur, H. Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981.
http://dx.doi.org/10.1137/1.9781611970883
6. Peterson, P., Soomere, T., Engelbrecht, J., and van Groesen, E. Soliton interaction as a possible model for extreme waves in shallow water. Nonlinear Proc. Geoph., 2003, 10, 503–510.
http://dx.doi.org/10.5194/npg-10-503-2003
7. Belashov, V. Yu. and Vladimirov, S. V. Solitary Waves in Dispersive Complex Media: Theory, Simulation, Applications. Springer, Berlin, 2005.
http://dx.doi.org/10.1007/b138237
8. Porubov, A. V., Tsuji, H., Lavrenov, I. V., and Oikawa, M. Formation of the rogue wave due to nonlinear two-dimensional waves interaction. Wave Motion, 2005, 42, 202–210.
http://dx.doi.org/10.1016/j.wavemoti.2005.02.001
9. Yuen, H. C. and Lake, B. M. Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech., 1982, 22, 67–229.
http://dx.doi.org/10.1016/S0065-2156(08)70066-8
10. Kivshar, Yu. S. and Pelinovsky, D. E. Self-focusing and transverse instabilities of solitary waves. Physics Reports, 2000, 331, 117–195.
http://dx.doi.org/10.1016/S0370-1573(99)00106-4
11. Aero, E. L. and Bulygin, A. N. Strongly nonlinear theory of nanostructure formation owing to elastic and nonelastic strains in crystalline solids. Mech. Solids, 2007, 42, 807–822.
http://dx.doi.org/10.3103/S0025654407050147
12. Porubov, A. V., Aero, E. L., and Maugin, G. A. Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials. Phys. Rev. E, 2009, 79, 046608.
http://dx.doi.org/10.1103/PhysRevE.79.046608
