2000 character limit reached
Implications of tristability on localization phenomena: a necking bifurcation's tale (2403.07814v1)
Published 12 Mar 2024 in nlin.PS
Abstract: We analyze the implication of tristability on localization phenomena in one-dimensional extended dissipative systems. In this context, localized states appear due to the interaction and locking of front waves connecting different extended states. In the tristable regime investigated here two extended uniform states coexist with one periodic Turing pattern. This scenario leads to the transition from the standard-homoclinic-snaking-related localized states associated with uniform-pattern bistability to the collapsed-homoclinic-snaking-related states which arise in a uniform-bistable configuration. We find that this transition is mediated by the emergence of hybrid states through codimension-two necking bifurcations. To perform this study we use bifurcation analysis on a non-variational mean-field model describing the spatiotemporal dynamics of light pulses in passive Kerr cavities.
- Y. Pomeau, “Front motion, metastability and subcritical bifurcations in hydrodynamics,” Physica D: Nonlinear Phenomena, vol. 23, pp. 3–11, Dec. 1986.
- P. Coullet, C. Elphick, and D. Repaux, “Nature of spatial chaos,” Physical Review Letters, vol. 58, pp. 431–434, Feb. 1987.
- O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” Journal de Physique, vol. 49, no. 11, pp. 1829–1833, 1988.
- P. Coullet, “Localized patterns and fronts in nonequilibrium systems,” International Journal of Bifurcation and Chaos, vol. 12, pp. 2445–2457, Nov. 2002.
- P. D. Woods and A. R. Champneys, “Heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible Hamiltonian–Hopf bifurcation,” Physica D: Nonlinear Phenomena, vol. 129, pp. 147–170, May 1999.
- J. Knobloch and T. Wagenknecht, “Homoclinic snaking near a heteroclinic cycle in reversible systems,” Physica D: Nonlinear Phenomena, vol. 206, pp. 82–93, June 2005.
- P. Parra-Rivas, E. Knobloch, D. Gomila, and L. Gelens, “Dark solitons in the Lugiato-Lefever equation with normal dispersion,” Physical Review A, vol. 93, p. 063839, June 2016.
- D. Gomila, A. J. Scroggie, and W. J. Firth, “Bifurcation structure of dissipative solitons,” Physica D: Nonlinear Phenomena, vol. 227, pp. 70–77, Mar. 2007.
- P. Parra-Rivas, D. Gomila, L. Gelens, and E. Knobloch, “Bifurcation structure of localized states in the Lugiato-Lefever equation with anomalous dispersion,” Physical Review E, vol. 97, p. 042204, Apr. 2018.
- P. Parra-Rivas, L. Gelens, and F. Leo, “Localized structures in dispersive and doubly resonant optical parametric oscillators,” Physical Review E, vol. 100, p. 032219, Sept. 2019.
- P. Parra-Rivas, C. Mas-Arabí, and F. Leo, “Parametric localized patterns and breathers in dispersive quadratic cavities,” Physical Review A, vol. 101, p. 063817, June 2020.
- P. Parra-Rivas, E. Knobloch, L. Gelens, and D. Gomila, “Origin, bifurcation structure and stability of localized states in Kerr dispersive optical cavities,” IMA Journal of Applied Mathematics, vol. 86, pp. 856–895, Oct. 2021.
- F. Al Saadi and A. Champneys, “Unified framework for localized patterns in reaction–diffusion systems; the Gray–Scott and Gierer–Meinhardt cases,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 379, p. 20200277, Dec. 2021. Publisher: Royal Society.
- Y. R. Zelnik, P. Gandhi, E. Knobloch, and E. Meron, “Implications of tristability in pattern-forming ecosystems,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 28, p. 033609, Mar. 2018.
- F. Al Saadi and P. Parra-Rivas, “Transitions between dissipative localized structures in the simplified Gilad–Meron model for dryland plant ecology,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 33, p. 033129, Mar. 2023.
- H. Schmidt and D. Avitabile, “Bumps and oscillons in networks of spiking neurons,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, p. 033133, Mar. 2020.
- P. Parra-Rivas, A. R. Champneys, F. A. Saadi, D. Gomila, and E. Knobloch, “Organization of Spatially Localized Structures near a Codimension-Three Cusp-Turing Bifurcation,” SIAM Journal on Applied Dynamical Systems, pp. 2693–2731, Dec. 2023. Publisher: Society for Industrial and Applied Mathematics.
- M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Reviews of Modern Physics, vol. 65, pp. 851–1112, July 1993.
- L. A. Lugiato and R. Lefever, “Spatial Dissipative Structures in Passive Optical Systems,” Physical Review Letters, vol. 58, pp. 2209–2211, May 1987.
- M. Haelterman, S. Trillo, and S. Wabnitz, “Dissipative modulation instability in a nonlinear dispersive ring cavity,” Optics Communications, vol. 91, pp. 401–407, Aug. 1992.
- M. Rowley, P.-H. Hanzard, A. Cutrona, H. Bao, S. T. Chu, B. E. Little, R. Morandotti, D. J. Moss, G.-L. Oppo, J. S. Totero Gongora, M. Peccianti, and A. Pasquazi, “Self-emergence of robust solitons in a microcavity,” Nature, vol. 608, pp. 303–309, Aug. 2022. Number: 7922 Publisher: Nature Publishing Group.
- J. Lottes, G. Biondini, and S. Trillo, “Excitation of switching waves in normally dispersive Kerr cavities,” Optics Letters, vol. 46, pp. 2481–2484, May 2021. Publisher: Optica Publishing Group.
- Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering-gallery-mode resonators,” Physical Review A, vol. 87, p. 053852, May 2013.
- M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Optics Letters, vol. 35, pp. 306–308, Feb. 2010.
- E. K. Akakpo, M. Haelterman, F. Leo, and P. Parra-Rivas, “Emergence of collapsed snaking related dark and bright Kerr dissipative solitons with quartic-quadratic dispersion,” Physical Review E, vol. 108, p. 014203, July 2023. Publisher: American Physical Society.
- P. Parra-Rivas, S. Hetzel, Y. V. Kartashov, P. F. d. Córdoba, J. A. Conejero, A. Aceves, and C. Milián, “Quartic Kerr cavity combs: bright and dark solitons,” Optics Letters, vol. 47, pp. 2438–2441, May 2022. Publisher: Optica Publishing Group.
- E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (ii): bifurcation in infinite dimensions,” International Journal of Bifurcation and Chaos, vol. 01, pp. 745–772, Dec. 1991.
- E. Doedel, H. B. Keller, and J. P. Kernevez, “Numerical analysis and control of bifurcation problems (i): bifurcation in finite dimensions,” International Journal of Bifurcation and Chaos, vol. 01, pp. 493–520, Sept. 1991.
- E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, “AUTO-07p: Software for continuation and bifurcation problems in ordinary differential equations,” Department of Computer Science, Concordia University, Montreal, 2007.
- S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, New York: Springer-Verlag, 2 ed., 2003.
- A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos, Solitons & Fractals, vol. 4, pp. 1323–1354, Aug. 1994.
- E. Makrides and B. Sandstede, “Existence and stability of spatially localized patterns,” Journal of Differential Equations, vol. 266, pp. 1073–1120, Jan. 2019.
- J. Burke and E. Knobloch, “Snakes and ladders: Localized states in the Swift–Hohenberg equation,” Physics Letters A, vol. 360, pp. 681–688, Jan. 2007.
- P. Parra-Rivas, D. Gomila, E. Knobloch, S. Coen, and L. Gelens, “Origin and stability of dark pulse Kerr combs in normal dispersion resonators,” Optics Letters, vol. 41, pp. 2402–2405, June 2016.
- J. Prat, I. Mercader, and E. Knobloch, “The 1:2 mode interaction in rayleigh–bénard convection with and without boussinesq symmetry,” International Journal of Bifurcation and Chaos, vol. 12, pp. 281–308, Feb. 2002.
- P. Parra-Rivas and C. Fernandez-Oto, “Formation of localized states in dryland vegetation: Bifurcation structure and stability,” Physical Review E, vol. 101, p. 052214, May 2020.
- J. M. T. Thompson, “Advances in Shell Buckling: Theory and Experiments,” International Journal of Bifurcation and Chaos, Feb. 2015. Publisher: World Scientific Publishing Company.
- G. Kozyreff and S. J. Chapman, “Asymptotics of Large Bound States of Localized Structures,” Physical Review Letters, vol. 97, p. 044502, July 2006.
- A. D. Dean, P. C. Matthews, S. M. Cox, and J. R. King, “Exponential asymptotics of homoclinic snaking,” Nonlinearity, vol. 24, p. 3323, Oct. 2011.
- A. F. J. Runge, D. D. Hudson, K. K. K. Tam, C. M. de Sterke, and A. Blanco-Redondo, “The pure-quartic soliton laser,” Nature Photonics, vol. 14, pp. 492–497, Aug. 2020. Number: 8 Publisher: Nature Publishing Group.
- S. McCalla and B. Sandstede, “Snaking of radial solutions of the multi-dimensional Swift–Hohenberg equation: A numerical study,” Physica D: Nonlinear Phenomena, vol. 239, pp. 1581–1592, Aug. 2010.
- Y. Sun, P. Parra-Rivas, M. Zitelli, F. Mangini, M. Ferraro, and S. Wabnitz, “2 - An introduction to guided-wave nonlinear ultrafast photonics,” in Advances in Nonlinear Photonics (G. C. Righini and L. Sirleto, eds.), Woodhead Publishing Series in Electronic and Optical Materials, pp. 27–55, Woodhead Publishing, Jan. 2023.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
