Bulletin of the Korean Mathematical Society (대한수학회보)

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SPECTRAL INSTABILITY OF ROLLS IN THE 2-DIMENSIONAL GENERALIZED SWIFT-HOHENBERG EQUATION

  • Myeongju Chae (School of Computer Engineering & Applied Mathematics Hankyong National University) ;
  • Soyeun Jung (Division of International Studies Kongju National University)
  • Received : 2022.10.06
  • Accepted : 2023.01.27
  • Published : 2023.09.30
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Abstract

The aim of this paper is to investigate the spectral instability of roll waves bifurcating from an equilibrium in the 2-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist, while they are Eckhaus stable in 1 dimension [13]. As compared to [18], showing that the stability of the rolls in the 2-dimensional Swift-Hohenberg equation without a quadratic nonlinearity is determined by Eckhaus and zigzag curves, our result says that the quadratic nonlinearity of the equation is the cause of such instability of the rolls.

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Acknowledgement

The first author was supported by a research grant from Hankyong National University for an academic exchange program in 2022. The second author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 2022R1F1A1074414).

References

  1. D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch, and B. Sandstede, To snake or not to snake in the planar Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 9 (2010), no. 3, 704-733. https://doi.org/10.1137/100782747 
  2. F. H. Busse, Stability regions of cellular fluid flow. In Instability of Continuous Systems, H. Leipholz(eds), Proc. of the IUTAM Symposium in Bad Herrenalb 1969, pp. 41-47, Berlin-Heidelberg-New York: Springer-Verlag, 1971. 
  3. M. Chae and S. Jung Nonlinear instability of rolls in the two dimensional gSHE, in preparation. 
  4. P. Chossat and G. Faye, Pattern formation for the Swift-Hohenberg equation on the hyperbolic plane, J. Dynam. Differential Equations 27 (2015), no. 3-4, 485-531. https://doi.org/10.1007/s10884-013-9308-3 
  5. S.-N. Chow and J. K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften, 251, Springer, New York, 1982. 
  6. D. L. Coelho, E. Vitral, J. Pontes, and N. Mangiavacchi, Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics, Phys. D 427 (2021), Paper No. 133000, 16 pp. https://doi.org/10.1016/j.physd.2021.133000 
  7. M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65 (1993). 
  8. W. Eckhaus, Studies in non-linear stability theory, Springer Tracts in Natural Philosophy, Vol. 6, Springer-Verlag New York, Inc., New York, 1965. 
  9. E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math. 53 (2000), no. 9, 1067-1091.  https://doi.org/10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q
  10. M'F. Hilali, S. M'etens, P. Borckmans, and G. Dewel, Pattern selection in the generalized Swift-hohenberg model, Phys. Rev. E 51 (1995), no 3. 
  11. J. Jin, S. Liao, and Z. Lin, Nonlinear modulational instability of dispersive PDE models, Arch. Ration. Mech. Anal. 231 (2019), no. 3, 1487-1530. https://doi.org/10.1007/s00205-018-1303-8 
  12. M. A. Johnson and K. R. Zumbrun, Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case, J. Differential Equations 249 (2010), no. 5, 1213-1240. https://doi.org/10.1016/j.jde.2010.04.015 
  13. S. Jung, Stability of bifurcating stationary periodic solutions of the generalized Swift-Hohenberg equation, Bull. Korean Math. Soc. 60 (2023), no. 1, 257-279. https://doi.org/10.4134/BKMS.b220116 
  14. T. M. Kapitula and K. S. Promislow, Spectral and dynamical stability of nonlinear waves, Applied Mathematical Sciences, 185, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-6995-7 
  15. E. Knobloch, Pattern formation and dynamics in nonequilibrium systems [book review, Cambridge Univ. Press, Cambridge, 2009], SIAM Rev. 52 (2010), no. 3, 567-572.  https://doi.org/10.1137/SIREAD000052000003000567000001
  16. D. J. B. Lloyd and A. Scheel, Continuation and bifurcation of grain boundaries in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst. 16 (2017), no. 1, 252-293. https://doi.org/10.1137/16M1073212 
  17. A. Mielke, A new approach to sideband-instabilities using the principle of reduced instability, in Nonlinear dynamics and pattern formation in the natural environment (Noordwijkerhout, 1994), 206-222, Pitman Res. Notes Math. Ser., 335, Longman, Harlow, 1995. 
  18. A. Mielke, Instability and stability of rolls in the Swift-Hohenberg equation, Comm. Math. Phys. 189 (1997), no. 3, 829-853. https://doi.org/10.1007/s002200050230 
  19. F. Rousset and N. Tzvetkov, Transverse instability of the line solitary water-waves, Invent. Math. 184 (2011), no. 2, 257-388. https://doi.org/10.1007/s00222-010-0290-7 
  20. A. Scheel and J. Weinburd, Wavenumber selection via spatial parameter jump, Philos. Trans. Roy. Soc. A 376 (2018), no. 2117, 20170191, 20 pp. https://doi.org/10.1098/rsta.2017.0191 
  21. A. Z. Shapira, H. Uecker, and A. Yochelis, Stripes on finite domains: why the zigzag instability is only a partial story, Chaos 30 (2020), no. 7, 073104, 7 pp. https://doi.org/10.1063/5.0006126 
  22. P. Subramanian, A. J. Archer, E. Knobloch, and A. M. Rucklidge, Snaking without subcriticality: grain boundaries as non-topological defects, IMA J. Appl. Math. 86 (2021), no. 5, 1164-1180. https://doi.org/10.1093/imamat/hxab032 
  23. A. Sukhtayev, K. Zumbrun, S. Jung, and R. Venkatraman, Diffusive stability of spatially periodic solutions of the Brusselator model, Comm. Math. Phys. 358 (2018), no. 1, 1-43. https://doi.org/10.1007/s00220-017-3056-x 
  24. M. M. Sushchik and L. S. Tsimring, The Eckhaus instability in hexagonal patterns, Phys. D. 74 (1994), 90-106.  https://doi.org/10.1016/0167-2789(94)90028-0
  25. L. S. Tuckerman and D. Barkley, Bifurcation analysis of the Eckhaus instability, Phys. D. 46 (1990), no. 1, 57-86. https://doi.org/10.1016/0167-2789(90)90113-4