Relevant bibliographies by topics / Complex Ginzburg-Landau equation

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Academic literature on the topic 'Complex Ginzburg-Landau equation'

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Published: 4 June 2021
Last updated: 16 February 2022

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Journal articles on the topic "Complex Ginzburg-Landau equation"

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Gao, Hongjun, and Keng-Huat Kwek. "Global existence for the generalised 2D Ginzburg-Landau equation." ANZIAM Journal 44, no. 3 (January 2003): 381–92. http://dx.doi.org/10.1017/s1446181100008099.

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AbstractGinzburg-Landau type complex partial differential equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. Most work so far has concentrated on Ginzburg-Landau type equations with one spatial variable (1D). In this paper, the authors study a complex generalised Ginzburg-Landau equation with two spatial variables (2D) and fifth-order and cubic terms containing derivatives. Based on detail analysis, sufficient conditions for the existence and uniqueness of global solutions are obtained.
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Huang, Chunyan. "On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/607028.

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We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation(1/2⩽α⩽1)in any spatial dimensionn⩾1with rough initial data. For1/2<α⩽1, we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spacesMp,11-2α(1⩽p⩽∞). Forα=1/2, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data inB˙∞,10(ℝn)∩M∞,10(ℝn). The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroupe-a+it-Δαto overcome the derivative in the nonlinear term.
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Ipsen, M., F. Hynne, and P. G. Sørensen. "Amplitude Equations and Chemical Reaction–Diffusion Systems." International Journal of Bifurcation and Chaos 07, no. 07 (July 1997): 1539–54. http://dx.doi.org/10.1142/s0218127497001217.

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The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.
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García-Morales, Vladimir, and Katharina Krischer. "The complex Ginzburg–Landau equation: an introduction." Contemporary Physics 53, no. 2 (March 2012): 79–95. http://dx.doi.org/10.1080/00107514.2011.642554.

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Graham, R., and T. Tél. "Potential for the Complex Ginzburg-Landau Equation." Europhysics Letters (EPL) 13, no. 8 (December 15, 1990): 715–20. http://dx.doi.org/10.1209/0295-5075/13/8/008.

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Zubrzycki, Andrzej. "A complex Ginzburg-Landau equation in magnetism." Journal of Magnetism and Magnetic Materials 150, no. 2 (October 1995): L143—L145. http://dx.doi.org/10.1016/0304-8853(95)00391-6.

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Mirzazadeh, Mohammad, Mehmet Ekici, Abdullah Sonmezoglu, Mostafa Eslami, Qin Zhou, Abdul H. Kara, Daniela Milovic, Fayequa B. Majid, Anjan Biswas, and Milivoj Belić. "Optical solitons with complex Ginzburg–Landau equation." Nonlinear Dynamics 85, no. 3 (May 9, 2016): 1979–2016. http://dx.doi.org/10.1007/s11071-016-2810-5.

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Salete, Eduardo, Antonio M. Vargas, Ángel García, Mihaela Negreanu, Juan J. Benito, and Francisco Ureña. "Complex Ginzburg–Landau Equation with Generalized Finite Differences." Mathematics 8, no. 12 (December 20, 2020): 2248. http://dx.doi.org/10.3390/math8122248.

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In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods.
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Rosier, Lionel, and Bing-Yu Zhang. "Null controllability of the complex Ginzburg–Landau equation." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 26, no. 2 (February 2009): 649–73. http://dx.doi.org/10.1016/j.anihpc.2008.03.003.

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Cazenave, Thierry, Flávio Dickstein, and Fred B. Weissler. "Standing waves of the complex Ginzburg–Landau equation." Nonlinear Analysis: Theory, Methods & Applications 103 (July 2014): 26–32. http://dx.doi.org/10.1016/j.na.2014.03.001.

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Dissertations / Theses on the topic "Complex Ginzburg-Landau equation"

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Liu, Weigang. "A General Study of the Complex Ginzburg-Landau Equation." Diss., Virginia Tech, 2019. http://hdl.handle.net/10919/90886.

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In this dissertation, I study a nonlinear partial differential equation, the complex Ginzburg-Landau (CGL) equation. I first employed the perturbative field-theoretic renormalization group method to investigate the critical dynamics near the continuous non-equilibrium transition limit in this equation with additive noise. Due to the fact that time translation invariance is broken following a critical quench from a random initial configuration, an independent ``initial-slip'' exponent emerges to describe the crossover temporal window between microscopic time scales and the asymptotic long-time regime. My analytic work shows that to first order in a dimensional expansion with respect to the upper critical dimension, the extracted initial-slip exponent in the complex Ginzburg-Landau equation is identical to that of the equilibrium model A. Subsequently, I studied transient behavior in the CGL through numerical calculations. I developed my own code to numerically solve this partial differential equation on a two-dimensional square lattice with periodic boundary conditions, subject to random initial configurations. Aging phenomena are demonstrated in systems with either focusing and defocusing spiral waves, and the related aging exponents, as well as the auto-correlation exponents, are numerically determined. I also investigated nucleation processes when the system is transiting from a turbulent state to the ``frozen'' state. An extracted finite dimensionless barrier in the deep-quenched case and the exponentially decaying distribution of the nucleation times in the near-transition limit are both suggestive that the dynamical transition observed here is discontinuous. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308
Doctor of Philosophy
The complex Ginzburg-Landau equation is one of the most studied nonlinear partial differential equation in the physics community. I study this equation using both analytical and numerical methods. First, I employed the field theory approach to extract the critical initial-slip exponent, which emerges due to the breaking of time translation symmetry and describes the intermediate temporal window between microscopic time scales and the asymptotic long-time regime. I also numerically solved this equation on a two-dimensional square lattice. I studied the scaling behavior in non-equilibrium relaxation processes in situations where defects are interactive but not subject to strong fluctuations. I observed nucleation processes when the system under goes a transition from a strongly fluctuating disordered state to the relatively stable “frozen” state where its dynamics cease. I extracted a finite dimensionless barrier for systems that are quenched deep into the frozen state regime. An exponentially decaying long tail in the nucleation time distribution is found, which suggests a discontinuous transition. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-SC0002308.
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Braun, Robert, and Fred Feudel. "Supertransient chaos in the two-dimensional complex Ginzburg-Landau equation." Universität Potsdam, 1996. http://opus.kobv.de/ubp/volltexte/2007/1409/.

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We have shown that the two-dimensional complex Ginzburg-Landau equation exhibits supertransient chaos in a certain parameter range. Using numerical methods this behavior is found near the transition line separating frozen spiral solutions from turbulence. Supertransient chaos seems to be a common phenomenon in extended spatiotemporal systems. These supertransients are characterized by an average transient lifetime which depends exponentially on the size of the system and are due to an underlying nonattracting chaotic set.
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Cruz-Pacheco, Gustavo. "The nonlinear Schroedinger limit of the complex Ginzburg-Landau equation." Diss., The University of Arizona, 1995. http://hdl.handle.net/10150/187238.

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This work consists of a study of the complex Ginzburg-Landau equation (CGL) as a perturbation of the nonlinear Schrodinger equation (NLS) in one dimension under periodic boundary conditions. Using an averaging technique which is similar to a Melnikov method for pde's, necessary conditions are derived for the persistence of NLS solutions under the CGL perturbation. For the traveling wave solutions, these conditions are derived for a general nonlinearity and written explicitly as two equations for the two continuous parameters which determine the NLS traveling wave. It is shown using a Melnikov argument that in this case these two conditions are sufficient provided they satisfy a transversality condition. As a concrete example, the equations for the parameters are solved numerically in the important case of the CGL equation with a cubic nonlinearity. For the case of the CGL equation with a general power nonlinearity, it is proved that the NLS homoclinic orbits to rotating waves are destroyed by the CGL perturbation. Special attention is dedicated to the cubic case. For this nonlinearity, the NLS equation is a completely integrable Hamiltonian system and a much larger family of its solutions can be written explicitly. The necessary conditions for the persistence of the NLS isospectral manifold are written explicitly as a system of equations for the simple periodic eigenvalues. As an example, the conditions for an even genus two solution are written down as a system of three equations with three unknowns.
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Horsch, Karla 1968. "Attractors for Lyapunov cases of the complex Ginzburg-Landau equation." Diss., The University of Arizona, 1997. http://hdl.handle.net/10150/282419.

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A special case of the complex Ginzburg-Landau (CGL) equation possessing a Lyapunov functional is identified. The global attractor of this Lyapunov CGL (LCGL) is studied in one spatial dimension with periodic boundary conditions. The LCGL may be viewed as a dissipative perturbation of the nonlinear Schrodinger equation (NLS), a completely integrable Hamiltonian system. The o-limit sets of the LCGL are identified as compact, connected unions of subsets of the stationary points of the flow. The stationary points do not depend on the strength of the perturbation, and so neither do the o-limit sets. However, the basins of attraction do depend sensitively on the perturbation strength. To determine the stability of the o-limit sets, the global Lyapunov functional is studied. Using the integrable NLS machinery, the second variation of the Lyapunov functional is diagonalized. An analysis of the diagonal elements yields that certain LCGL stationary points are stable. We are able to analyze the basins of attraction for a planar toy problem, which like the LCGL, is a dissipative perturbation of a Hamiltonian system. For this problem, almost every phase point is in a basin of attraction of an asymptotically stable stationary point. As the perturbation tends to zero, these basins become intermingled and the event of a fixed phase point being captured into a particular basin becomes probabilistic. Formulas for computing the probabilities of capture are given. These formulas are substantiated through a formal asymptotic analysis and numerical experiments. Such a probabilistic description of the basins of attraction is not completed for the infinite dimensional LCGL.
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Aguareles, Carrero Maria. "Interaction of spiral waves in the general complex Ginzburg-Landau equation." Doctoral thesis, Universitat Politècnica de Catalunya, 2007. http://hdl.handle.net/10803/5854.

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Molts sistemes físics tenen la propietat que la seva dinàmica ve definida per algun tipus de difussió espaial en competició amb un fenòmen de reacció, com per exemple en el cas de dos components químics que reaccionen al mateix temps que es difon l'un en el si de l'altre. La presència d'aquests dos fenòmens, la difusió i la reacció, sovint dóna lloc a patrons no homogenis de gran riquesa. Els models matemàtics que descriuen aquest tipus de comportament són normalment equacions en derivades parcials les solucions de les quals representen aquests patrons.

En aquesta tesi s'analitza l'equació de Ginzburg-Landau complexa general, que és una equació en derivades parcials de reacció-difusió que s'utilitza sovint com a model matemàtic per a descriure sistemes oscil·latoris en dominis extensos. En particular estudiem els patrons que sorgeixen en el pla quan s'imposa que el grau de Brouwer de la solució no sigui nul. Aquests patrons estan formats per ones de rotació en forma d'espirals, és a dir, les corbes de nivell de la solució formen espirals que emanen dels punts on la funció s'anul·la. Quan la solució s'anul·la només en un punt i per tant només hi ha una espiral, tota la dependència temporal apareix en el terme de freqüència. Així doncs, la funció solució es pot expressar com a funció del radi polar i en termes del seu grau topològic i la freqüència de l'ona. Per tant, aquestes solucions es poden expressar en termes d'un sistema d'equacions diferencials ordinàries. Aquestes solucions només existeixen per una certa freqüència que depèn unívocament dels paràmetres de l'equació i, com a conseqüència i degut a la relació de dispersió entre el nombre d'ones i la freqüència, el nombre d'ones a l'infinit, l'anomenat nombre d'ones asimptòtic, ve també determinat unívocament pels paràmetres. Quan les solucions tenen més d'un zero aïllat la condició sobre el grau de la funció fa que de cada zero sorgeixi una espiral diferent i aquestes es mouen en el pla mantenint la seva estructura local. En aquest treball s'usen tècniques d'anàlisi asimptòtica per trobar equacions del moviment per als centres de les espirals i es troba que aquesta evolució temporal és lenta. En concret, per la distàncies relatives grans entre els centres de les espirals, l'escala de temps per a la seva dinàmica ve donada pel logaritme de l'invers d'aquesta distància. Es demostra que aquestes equacions del moviment són diferents en funció de la relació entre els paràmetres de l'equació de Ginzburg-Landau complexa i la separació entre els centres de les espirals, i que la forma com es passa d'unes equacions a les altres és molt singular. També es demostra que el nombre d'ones asimptòtic per al cas de sistemes amb diverses espirals també està unívocament determinat pels paràmetres però no obstant, el cas de sistemes amb diverses espirals es diferencia del cas d'una única ona en què deixa de ser constant i evoluciona al mateix ritme que la velocitat dels centres de les espirals.
Many physical systems have the property that its dynamics is driven by some kind of spatical diffusion that is in competition with a reaction, like for instance two chemical species that react at the same time that there is a diffusion of each of them into the other. This interplay between reaction and diffusion produce non-homogeneous patterns that can sometimes be very rich. The mathematical models that describe this kind of behaviours are usually nonlinear partial differential equations whose solutions represent these patterns.

In this thesis we focus on an especific reaction-diffusion equation that is the so-called general complex Ginzburg-Landau equation that is used as a model for oscillatory systems in extended domains. In particular we are interested in the type of patterns in the plane that arise when the solutions have a non-vanishing Brouwer degree. These patterns have the property that they exhibit rotating waves in the shape of spirals, which means that the contour lines arrange in the shape of spirals that emerge from the points where the solution vanishes. When the solution vanishes only at one point all the time dependence appears as a frequency term so the solutions can be expressed as a function of the polar radius and in terms of the topological degree of the solution and the frequency of the wave. Therefore, these solutions can be expressed in terms of a system of ordinary differential equations. These solutions do only exist with a given frequency, and as a consequence and due to the existence of a dispresion relation, the wavenumber far from the origin, the so-called asymptotic wavenumber, is also unique. When the solutions have more than one isolated zero, the condition on the degree of the function has the effect of producing several spirals that emerge from the different zeros of the solution. These spirals evolve in time keeping their structure but moving around on the plane. In this work we use asymptotic analysis techniques to derive laws of motion for the centres of the spirals and we show that the time evolution of these patterns is slow and, for large relative separations of the centres of the spirals, the time scale for the their dynamics is logarithmic in the inverse of this distance. These laws of motion are different depending on the relation between the parameters of the complex Ginzburg-Landau equation and the relative separation of the spirals. We show that the way these laws change as the spirals separate or approach is highly singular. We also show that the asymptotic wavenumber in the case of multiple spirals is as well unique and that it evolves in time at the same rate as the velocity of the centres.
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Mancas, Ciprian. "DISSIPATIVE SOLITONS IN THE CUBIC–QUINTIC COMPLEX GINZBURG–LANDAU EQUATION:BIFURCATIONS AND SPATIOTEMPORAL STRUCTURE." Doctoral diss., University of Central Florida, 2007. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2912.

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Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubic--quintic Ginzburg--Landau equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non--integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse--type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this dissertation, we develop a theoretical framework for these novel classes of solutions. In the first part, we use a traveling wave reduction or a so--called spatial approximation to comprehensively investigate the bifurcations of plane wave and periodic solutions of the CGLE. The primary tools used here are Singularity Theory and Hopf bifurcation theory respectively. Generalized and degenerate Hopf bifurcations have also been considered to track the emergence of global structure such as homoclinic orbits. However, these results appear difficult to correlate to the numerical bifurcation sequences of the dissipative solitons. In the second part of this dissertation, we shift gears to focus on the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. For this part, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the two alternative starting formulations for the Lagrangian are recent and not well explored. Also, after extensive discussions with David Kaup, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler--Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well--known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors constructed via the method of multiple scales yield solitons whose amplitudes are non--stationary or time dependent. In particular, pulsating, snake (and, less easily, creeping) dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves --- their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Finally, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Results will be presented for the pulsating and snake soliton cases. Chaotic evolution of the trial function parameters in chaotic regimes identified using dynamical systems analysis would yield chaotic solitary waves. The method also holds promise for detailed modeling of chaotic solitons as well. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.
Ph.D.
Department of Mathematics
Sciences
Mathematics PhD
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Stark, Donald Richard. "Structure and turbulence in the complex Ginzburg-Landau equation with a nonlinearity of arbitrary order." Diss., The University of Arizona, 1995. http://hdl.handle.net/10150/187363.

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Numerical and analytical studies are undertaken for the "inviscid" limit of the complex Ginzburg-Landau (CGL) equation with the objective of studying the applicability of paradigms from finite dimensional dynamical systems and statistical mechanics to the case of an infinite dimensional dynamical system. The analytical results rely on exploiting the structure of this limit, which becomes the nonlinear Schrodinger (NLS) equation. In the NLS limit the CGL equation can exhibit strong spatio-temporal chaos. The initial growth of the bursts closely mimics the blowup solutions of the NLS equation. The study of this turbulent behavior focuses on the inertial range of the time-averaged wavenumber spectrum. Analytical estimates of the decay rate are constructed assuming both structure driven and homogeneous turbulence, and are compared with numerical simulations. The quintic case is observed to have a stronger decay rate than what is predicted by either theory. This reflects the dominance of dissipation in the dynamics. In the septic case, two distinct inertial ranges are observed. This combination suggests that the evolution of a single burst, on average, is predominantly due to the self-focusing mechanism of blowup NLS in the initial stage, and regularization effects of dissipation in the final stage. Because the initial stage is primarily influenced by the NLS structure, the rate of decay for this range is close to the decay predicted for the structure driven turbulence. In a numerical experiment it is observed that some NLS solutions survive the deformation due to a CGL perturbation. In some cases the question of persistence can be addressed analytically using an averaging technique similar to a Melnikov method for pde's.
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Brusch, Lutz. "Complex Patterns in Extended Oscillatory Systems." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2001. http://nbn-resolving.de/urn:nbn:de:swb:14-1006416783250-74051.

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Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert.
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MacKenzie, Tony. "Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation." University of Southern Queensland, Faculty of Sciences, 2005. http://eprints.usq.edu.au/archive/00001466/.

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This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.
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Blockley, Edward William. "Nonlinear solutions of the amplitude equations governing fluid flow in rotating spherical geometries." Thesis, University of Exeter, 2008. http://hdl.handle.net/10036/41950.

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We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.
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Book chapters on the topic "Complex Ginzburg-Landau equation"

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Akhmediev, Nail, and Adrian Ankiewicz. "Solitons of the Complex Ginzburg—Landau Equation." In Springer Series in Optical Sciences, 311–41. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-540-44582-1_12.

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Elphick, Christian, and Ehud Meron. "Vortex Interactions in the Complex Ginzburg-Landau Equation." In Growth and Form, 263–70. Boston, MA: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4684-1357-1_25.

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Dattagupta, Sushanta, and Sanjay Puri. "Phase Ordering Dynamics in the Complex Ginzburg-Landau Equation." In Dissipative Phenomena in Condensed Matter, 147–72. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-06758-1_6.

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Okazawa, Noboru. "Semilinear Elliptic Problems Associated with the Complex Ginzburg-Landau Equation." In Partial Differential Equations and Functional Analysis, 169–87. Basel: Birkhäuser Basel, 2006. http://dx.doi.org/10.1007/3-7643-7601-5_11.

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Gibbon, J. D. "Weak and Strong Turbulence in the Complex Ginzburg Landau Equation." In Turbulence in Fluid Flows, 33–48. New York, NY: Springer New York, 1993. http://dx.doi.org/10.1007/978-1-4612-4346-5_2.

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Pumir, A., B. I. Shraiman, W. van Saarloos, P. C. Hohenberg, H. Chaté, and M. Holen. "Phase VS. Defect Turbulence in the 1D Complex Ginzburg-Landau Equation." In NATO ASI Series, 173–77. Boston, MA: Springer US, 1992. http://dx.doi.org/10.1007/978-1-4615-3438-9_18.

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Conte, R., and M. Musette. "Exact Solutions to the Complex Ginzburg-Landau Equation from a Linear System." In Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, 281–86. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2082-1_27.

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Zhang, Lu, Lei Chen, and Xiao Song. "Preconditioned Iteration Method for the Nonlinear Space Fractional Complex Ginzburg-Landau Equation." In Simulation Tools and Techniques, 351–62. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72795-6_29.

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Zhang, Lu, Lei Chen, and Wenyu Zhou. "Fast Preconditioned Iterative Method for the Space Fractional Complex Ginzburg-Landau Equation." In Simulation Tools and Techniques, 78–89. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72792-5_8.

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Grossauer, Harald, and Otmar Scherzer. "Using the Complex Ginzburg-Landau Equation for Digital Inpainting in 2D and 3D." In Scale Space Methods in Computer Vision, 225–36. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/3-540-44935-3_16.

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Conference papers on the topic "Complex Ginzburg-Landau equation"

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Brand, Helmut R., Orazio Descalzi, and Jaime Cisternas. "Hole Solutions in the Cubic Complex Ginzburg-Landau Equation versus Holes in the Cubic-Quintic Complex Ginzburg-Landau Equation." In NONEQUILIBRIUM STATISTICAL MECHANICS AND NONLINEAR PHYSICS: XV Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics. AIP, 2007. http://dx.doi.org/10.1063/1.2746737.

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Li, Jing, and Zhixiong Zhang. "Complex Ginzburg-Landau equation with boundary control and observation." In 2016 2nd International Conference on Control Science and Systems Engineering (ICCSSE). IEEE, 2016. http://dx.doi.org/10.1109/ccsse.2016.7784365.

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Xiang, Chunhuan, and Honglei Wang. "An approximate method for solving complex Ginzburg-Landau equation." In 2018 8th International Conference on Manufacturing Science and Engineering (ICMSE 2018). Paris, France: Atlantis Press, 2018. http://dx.doi.org/10.2991/icmse-18.2018.111.

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Bazhenov, M., M. Rabinovich, and L. Rubchinsky. "Time periodic spatial disorder in a complex Ginzburg–Landau equation." In Chaotic, fractal, and nonlinear signal processing. AIP, 1996. http://dx.doi.org/10.1063/1.51051.

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BARASHENKOV, I. V., and S. D. CROSS. "LOCALISED SOLUTIONS OF THE PARAMETRICALLY DRIVEN COMPLEX GINZBURG-LANDAU EQUATION." In Proceedings of the International Conference. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812810175_0043.

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Gutiérrez, Pablo, and Orazio Descalzi. "Existence Range of Pulses in the Quintic Complex Ginzburg-Landau Equation." In NONEQUILIBRIUM STATISTICAL MECHANICS AND NONLINEAR PHYSICS: XV Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics. AIP, 2007. http://dx.doi.org/10.1063/1.2746736.

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Latas, Sofia C. V., Mário F. S. Ferreira, and Margarida V. Facão. "Dynamics of ultrashort pulse solutions of the complex Ginzburg-Landau equation." In SPIE LASE, edited by Peter E. Powers. SPIE, 2010. http://dx.doi.org/10.1117/12.841634.

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Turaev, D., S. Zelik, and A. G. Vladimirov. "Chaotic bound state of localized structures in the complex Ginzburg-Landau equation." In Nonlinear Guided Waves and Their Applications. Washington, D.C.: OSA, 2005. http://dx.doi.org/10.1364/nlgw.2005.wd9.

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Hawick, Kenneth A., and Daniel P. Playne. "Numerical Simulation of the Complex Ginzburg-Landau Equation on GPUs with CUDA." In Parallel and Distributed Computing and Networks / Software Engineering. Calgary,AB,Canada: ACTAPRESS, 2011. http://dx.doi.org/10.2316/p.2011.719-036.

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Gitterman, M. "Wave Propagation in Inhomogeneous Media: From the Helmholtz to the Ginzburg-Landau Equation." In Wave Propagation: Scattering and Emission in Complex Media - International Workshop. CO-PUBLISHED WITH WORLD SCIENTIFIC PUBLISHING CO AND SCIENCE PRESS, CHINA, 2005. http://dx.doi.org/10.1142/9789812702869_0018.

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Reports on the topic "Complex Ginzburg-Landau equation"

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Takac, P. Dynamics on the attractor for the complex Ginzburg-Landau equation. Office of Scientific and Technical Information (OSTI), August 1994. http://dx.doi.org/10.2172/10174640.

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Vernov, Sergey Yu. Construction of Elliptic Solutions to the Quintic Complex One-dimensional Ginzburg–Landau Equation. GIQ, 2012. http://dx.doi.org/10.7546/giq-8-2007-322-333.

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