Oscillons in the planar Ginzburg–Landau equation with 2 : 1 forcing

Oscillons are spatially localized, time-periodic structures that have been observed in many natural processes, often under temporally periodic forcing. Near Hopf bifurcations, such systems can be formally reduced to forced complex Ginzburg–Landau equations, with oscillons then corresponding to stationary localized patterns. In this manuscript, stationary localized structures of the planar 2 : 1 forced Ginzburg–Landau equation are investigated analytically and numerically. The existence of these patterns is proved in regions where two spatial eigenvalues collide at zero. A numerical study complements these analytical results away from onset.

[1]  Nader Masmoudi,et al.  Global solutions for 2D quadratic Schrödinger equations , 2010 .

[2]  Jay Fineberg,et al.  OSCILLONS AND PROPAGATING SOLITARY WAVES IN A VERTICALLY VIBRATED COLLOIDAL SUSPENSION , 1999 .

[3]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[4]  Shu-Ming Chang,et al.  Spectra of Linearized Operators for NLS Solitary Waves , 2006, SIAM J. Math. Anal..

[5]  James Serrin,et al.  Uniqueness of positive radial solutions of Δu+f(u)=0 in ℝn , 1987 .

[6]  Irving R Epstein,et al.  Stationary and oscillatory localized patterns, and subcritical bifurcations. , 2004, Physical review letters.

[7]  P. Germain.  Global existence for coupled Klein-Gordon equations with different speeds , 2010, 1005.5238.

[8]  Nader Masmoudi,et al.  Global Solutions for 3D Quadratic Schrödinger Equations , 2008, 1001.5158.

[9]  Irving R Epstein,et al.  Resonance-induced oscillons in a reaction-diffusion system. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Lennart Stenflo,et al.  Plasma oscillons in spherically bounded magnetized plasmas , 1998 .

[11]  Nader Masmoudi,et al.  Global solutions for the gravity water waves equation in dimension 3 , 2009 .

[12]  G. Schneider,et al.  Hopf Bifurcation in Spatially Extended Reaction—Diffusion Systems , 1998 .

[13]  P. Zamankhan,et al.  Localized Structures in Vertically Vibrated Granular Materials , 2007 .

[14]  P. Umbanhowar,et al.  Localized excitations in a vertically vibrated granular layer , 1996, Nature.

[15]  M. Khonsari,et al.  Analytical Formulation for the Temperature Profile by Duhamel’s Theorem in Bodies Subjected to an Oscillatory Heat Source , 2007 .

[16]  Hermann Riecke,et al.  Oscillon-type structures and their interaction in a Swift-Hohenberg model , 1998 .

[17]  A. Zhabotinsky,et al.  Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction. , 2001, Physical review letters.

[18]  C. Elphick,et al.  Normal form reduction for time-periodically driven differential equations , 1987 .

[19]  Milos Dolnik,et al.  Oscillatory cluster patterns in a homogeneous chemical system with global feedback , 2000, Nature.

[20]  Valery Petrov,et al.  Resonant pattern formation in achemical system , 1997, Nature.

[21]  Björn Sandstede,et al.  Localized radial solutions of the Swift–Hohenberg equation , 2009 .

[22]  Arnd Scheel,et al.  Radially Symmetric Patterns of Reaction-Diffusion Systems , 2003 .

[23]  Hezi Yizhaq,et al.  Why do plants in resource-deprived environments form rings? , 2007 .

[24]  André Vanderbauwhede,et al.  Centre Manifolds, Normal Forms and Elementary Bifurcations , 1989 .

[25]  Fineberg,et al.  Temporally harmonic oscillons in newtonian fluids , 2000, Physical review letters.

[26]  Reciprocal oscillons and nonmonotonic fronts in forced nonequilibrium systems. , 2006, Physical review letters.

[27]  Jonathan H. P. Dawes,et al.  Localized States in a Model of Pattern Formation in a Vertically Vibrated Layer , 2010, SIAM J. Appl. Dyn. Syst..

[28]  I. Rudnick,et al.  Observation of a nonpropagating hydrodynamic soliton , 1984 .

[29]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[30]  Scott G. McCalla,et al.  Spots in the Swift-Hohenberg Equation , 2013, SIAM J. Appl. Dyn. Syst..

[31]  Peter Szmolyan,et al.  Extending Geometric Singular Perturbation Theory to Nonhyperbolic Points - Fold and Canard Points in Two Dimensions , 2001, SIAM J. Math. Anal..

[32]  Tsimring,et al.  Patterns in thin vibrated granular layers: interfaces, hexagons, and superoscillons , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[33]  B. Matkowsky,et al.  A complex Swift–Hohenberg equation coupled to the Goldstone mode in the nonlinear dynamics of flames , 2003 .

[34]  Freddy Dumortier,et al.  Canard Cycles and Center Manifolds , 1996 .

[35]  Mark D. Shattuck,et al.  Patterns in 3D Vertically Oscillated Granular Layers: Simulation and Experiment , 1998 .

[36]  Edgar Knobloch,et al.  Classification of Spatially Localized Oscillations in Periodically Forced Dissipative Systems , 2008, SIAM J. Appl. Dyn. Syst..

[37]  Edward Ott,et al.  Spatiotemporal bifurcation phenomena with temporal period doubling: Patterns in vibrated sand , 1998 .

[38]  M. Bordbar,et al.  Dynamical states of bubbling in vertical vibrated granular materials. Part II: Theoretical analysis and simulations , 2007 .

[39]  Daniel H. Rothman,et al.  Oscillons, spiral waves, and stripes in a model of vibrated sand , 1998 .

[40]  Pere Colet,et al.  Excitability mediated by localized structures , 2005 .

[41]  Lev Tsimring,et al.  Localized and Cellular Patterns in a Vibrated Granular Layer , 1997 .

[42]  A. Nepomnyashchy,et al.  Feedback control of subcritical oscillatory instabilities. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  E. Cerda,et al.  MODEL FOR SUBHARMONIC WAVES IN GRANULAR MATERIALS , 1997 .

[44]  Shui-Nee Chow,et al.  Smooth Invariant Foliations in Infinite Dimensional Spaces , 1991 .

[45]  Kjartan Pierre Emilsson,et al.  Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects , 1992 .

[46]  I. V. Barashenkov,et al.  Two-dimensional solitons on the surface of magnetic fluids. , 2005, Physical review letters.

[47]  John W. Miles,et al.  Parametrically excited solitary waves , 1984, Journal of Fluid Mechanics.

[48]  T. Maimbourg,et al.  Oscillon dynamics and rogue wave generation in Faraday surface ripples. , 2012, Physical review letters.

[49]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[50]  Hermann Riecke,et al.  Continuum description of vibrated sand , 1998, patt-sol/9801004.