---

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 195--205 | DOI:10.5890/DNC.2021.06.003

Adham A. Ali , Fatima Z. Ahmed

Department of Mathematics, Kirkuk University, Kirkuk, Iraq

Download Full Text PDF

 

Abstract

A characteristic feature of seismic waves is the presence of dominant frequency/wave number in the spectrum. A well-known model for such waves is the Nikolaevskiy equation, which is also applicable to some reaction-diffusion systems and Rayleigh-Benard convection. For the critical case when there is one neutral mode, we describe the dynamics of the Fourier modes (elastic waves) under the Nikolaevskiy equation using the centre manifold technique. After quickly attracted to the surface (manifold), the modes then evolve slow algebraic decay. An inverse square-root law for the decaying regime is obtained. The result is confirmed by direct computations of the dynamical system for the modes.

References

1. [1]	Nikolaevskii, V.N. (1989), Dynamics of viscoelastic media with internal oscillators. In Recent Advances in Engineering Science (pp. 210-221). Springer, Berlin, Heidelberg.

2. [2]	Bernoff, A.J. (1994), Finite amplitude convection between stress-free boundaries; Ginzburg-Landau equations and modulation theory, European Journal of Applied Mathematics, 5(3), 267-282.

3. [3]	Fujisaka, H. and Yamada, T. (2001), Possibility of soft-mode turbulence in reaction-diffusion systems, Progress of Theoretical Physics, 106(2), 315-322.

4. [4]	Tanaka, D. (2004), Chemical turbulence equivalent to Nikolavskii turbulence, Physical Review E, 70(1), 015202.

5. [5]	Hidaka, Y., Huh, J.H., Hayashi, K.I., Kai, S., and Tribelsky, M.I. (1997), Soft-mode turbulence in electrohydrodynamic convection of a homeotropically aligned nematic layer, Physical Review E, 56(6), R6256.

6. [6]	Cox, S.M. and Matthews, P.C. (2007), Pattern formation in the damped Nikolaevskiy equation, Physical Review E, 76(5), 056202.

7. [7]	Simbawa, E., Matthews, P.C., and Cox, S.M. (2010), Nikolaevskiy equation with dispersion, Physical Review E, 81(3), 036220.

8. [8]	Wittenberg, R.W. (2014), Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations, Discrete and Continuous Dynamical Systems-A, 34(12), 5325-5357.

9. [9]	Misbah, C. and Valance, A. (1994), Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation, Physical Review E, 49(1), 166.

10. [10]	Tribelsky, M.I. and Velarde, M.G. (1996), Short-wavelength instability in systems with slow long-wavelength dynamics, Physical Review E, 54(5), 4973.

11. [11]	Tribelsky, M.I. (1997), Short-wavelength instability and transition to chaos in distributed systems with additional symmetry, Physics-Uspekhi, 40(2), 159.

12. [12]	Tribelsky, M.I. (2000), New type of turbulence, or how symmetry results in chaos, In Macromolecular Symposia, 160(1), pp. 225-232, Weinheim: WILEY-VCH Verlag.

13. [13]	Matthews, P.C. and Cox, S.M. (2000), One-dimensional pattern formation with Galilean invariance near a stationary bifurcation, { \textit Physical Review E}, 62, R1473-R1476.

14. [14]	Landa, P.S. (2013), Nonlinear oscillations and waves in dynamical systems, Springer Science and Business Media.

15. [15]	Strunin, D.V. (2014), On dissipative nature of elastic waves, Journal of Coupled Systems and Multiscale Dynamics, 2(2), 70-73.

16. [16]	Strunin, D.V. and Ali, A.A. (2016), On nonlinear dynamics of neutral modes in elastic waves in granular media, Journal of Coupled Systems and Multiscale Dynamics, 4(3), 163-169.

17. [17]	Ali, A.A. and Strunin, D.V. (2019), The role of rheology in modelling elastic waves with gas bubbles in granular fluid-saturated media, Journal of Mechanics of Materials and Structures, 14(1), pp.1-24.

18. [18]	Strunin, D.V. (2009), Phase equation with nonlinear excitation for nonlocally coupled oscillators, Physica D: Nonlinear Phenomena, 238(18), 1909-1916.

19. [19]	Haskovec, J., Hittmeir, S., Markowich, P., and Mielke, A. (2018), Decay to equilibrium for energy-reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 50(1), 1037-1075.

20. [20]	Carr, J. (2012), Applications of centre manifold theory, Vol. 35, Springer Science and Business Media.

21. [21]	http://www.mathworks.com/matlabcentral/fileexchange/28-differential-algebraic-equation-solvers.