Nonlinear stability in seismic waves

Abstract

We analyse a passive system featuring a neutrally stable short-wavelength mode. The system is modelled by the Nikolaevskiy equation relevant to elastic waves, reaction-diffusion systems and convection. After quickly falling onto a centre manifold, the system then exhibits slow decay. Using the centre manifold technique, we deduce that the decay law is the inverse square root of time. The result is confirmed by direct computations of the system. References

• V. N. Nikolaevskii, Dynamics of viscoelastic media with internal oscillations. S. L. Koh et al. (eds.), Recent Advances in Engineering Science, Springer-Verlag, Berlin, pp. 210–221, 1989. doi:10.1007/978-3-642-83695-4_21
• A. J. Bernoff, Finite amplitude convection between stress-free boundaries; Ginzburg–Landau equations and modulation theory. Eur. J. Appl. Math., 5:267–282, 1994. doi:10.1017/S0956792500001467
• D. Tanaka, Chemical turbulence equivalent to Nikolaevskii turbulence. Phys. Rev. E, 70:015202(R), 2004. doi:10.1103/PhysRevE.70.015202
• D. V. Strunin, Phase equation with nonlinear excitation for nonlocally coupled oscillators. Physica D, 238:1909–1916, 2009. doi:10.1016/j.physd.2009.06.022
• D. V. Strunin, M. G. Mohammed, Range of validity and intermittent dynamics of the phase of oscillators with nonlinear self-excitation. Commun. Nonlinear Sci., 29:128–147, 2015. doi:10.1016/j.cnsns.2015.04.024
• I. A. Beresnev, V. N. Nikolaevskiy, A model for nonlinear seismic waves in a medium with instability. Physica D, 66:1–6, 1993. doi:10.1016/0167-2789(93)90217-O
• M. I. Tribelsky, M. G. Velarde, Short-wavelength instability in systems with slow long-wavelength dynamics. Phys. Rev. E, 54:4973–4981, 1996. doi:10.1103/PhysRevE.54.4973
• P. C. Matthews, S. M. Cox, One-dimensional pattern formation with Galilean invariance near a stationary bifurcation. Phys. Rev. E, 62:1473–1476(R), 2000. doi:10.1103/PhysRevE.62.R1473
• S. M. Cox, P. C. Matthews, Pattern formation in the damped Nikolaevskiy equation. Phys. Rev. E, 76:056202, 2007. doi:10.1103/PhysRevE.76.056202
• M. I. Tribelsky, K. Tsuboi, New scenario for transition to turbulence? Phys. Rev. Lett., 76:1631–1634, 1996. doi:10.1103/PhysRevLett.76.1631
• D. V. Strunin, On dissipative nature of elastic waves. J. Coup. Sys. Multiscale Dyn., 2:70–73, 2014. doi:10.1166/jcsmd.2014.1045
• J. Carr, Applications of centre manifold theory. Springer-Verlag, New York, pp. 142, 1982. http://www.springer.com/gp/book/9780387905778
• A. J. Roberts. Differential Algebraic Equation Solvers. Mathworks File Exchange. http://www.mathworks.com/matlabcentral/fileexchange/28-differential-algebraic-equation-solvers