scholarly journals Global existence for the generalised 2D Ginzburg-Landau equation

2003 ◽  
Vol 44 (3) ◽  
pp. 381-392 ◽  
Author(s):  
Hongjun Gao ◽  
Keng-Huat Kwek
Keyword(s):  
Existence And Uniqueness ◽  
Landau Equation ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Ginzburg Landau ◽  
Spatial Variables ◽  
Type Complex ◽  
Ginzburg Landau Equation ◽  
Fifth Order ◽  
Formation Systems

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.

scholarly journals Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation

1995 ◽  
Vol 38 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Jinqiao Duan ◽  
Philip Holmes
Keyword(s):  
Domain Walls ◽  
Landau Equation ◽  
Order Term ◽  
Traveling Wave Solutions ◽  
Ginzburg Landau ◽  
Existence Proofs ◽  
Ginzburg Landau Equation ◽  
Pulse Type ◽  
Spatial Derivatives ◽  
Fifth Order

We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.

scholarly journals ASYMPTOTIC COMPACTNESS OF STOCHASTIC COMPLEX GINZBURG–LANDAU EQUATION ON AN UNBOUNDED DOMAIN

2010 ◽  
Vol 10 (04) ◽  
pp. 613-636 ◽  
Author(s):  
DIRK BLÖMKER ◽  
YONGQIAN HAN
Keyword(s):  
Dynamical System ◽  
Pattern Formation ◽  
White Noise ◽  
Unbounded Domain ◽  
Landau Equation ◽  
Random Dynamical System ◽  
Ginzburg Landau ◽  
Type Complex ◽  
Regularity Properties ◽  
Formation Systems

The Ginzburg–Landau-type complex equations are simplified mathematical models for various pattern formation systems in mechanics, physics and chemistry. In this paper, we consider the complex Ginzburg–Landau (CGL) equations on the whole real line perturbed by an additive spacetime white noise. Our main result shows that it generates an asymptotically compact stochastic or random dynamical system. This is a crucial property for the existence of a stochastic attractor for such CGL equations. We rely on suitable spaces with weights, due to the regularity properties of spacetime white noise, which gives rise to solutions that are unbounded in space.

Cauchy problem for the Ginzburg-Landau equation for the superconductivity model

1997 ◽  
Vol 127 (6) ◽  
pp. 1181-1192 ◽  
Author(s):  
Boling Guo ◽  
Guangwei Yuan
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Smooth Solution ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Landau Model ◽  
Lorentz Gauge ◽  
Global Smooth Solution ◽  
Ginzburg Landau Equation

In this paper, the existence and uniqueness of the global smooth solution are proved for an evolutionary Ginzburg–Landau model for superconductivity under the Coulomb and Lorentz gauge.

scholarly journals Optical solitons of fractional complex Ginzburg–Landau equation with conformable, beta, and M-truncated derivatives: a comparative study

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Amjad Hussain ◽  
Adil Jhangeer ◽  
Naseem Abbas ◽  
Ilyas Khan ◽  
El-Syed M. Sherif
Keyword(s):  
Landau Equation ◽  
Sufficient Conditions ◽  
Detailed Comparison ◽  
Optical Solitons ◽  
Comparative Approach ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Kerr Law Nonlinearity ◽  
Algebraic Scheme

Abstract In this paper, we investigate the optical solitons of the fractional complex Ginzburg–Landau equation (CGLE) with Kerr law nonlinearity which shows various phenomena in physics like nonlinear waves, second-order phase transition, superconductivity, superfluidity, liquid crystals, and strings in field theory. A comparative approach is practised between the three suggested definitions of derivative viz. conformable, beta, and M-truncated. We have constructed the optical solitons of the considered model with a new extended direct algebraic scheme. By utilization of this technique, obtained solutions carry a variety of new families including dark-bright, dark, dark-singular, and singular solutions of Type 1 and 2, and sufficient conditions for the existence of these structures are given. Further, graphical representations of the obtained solutions are depicted. A detailed comparison of solutions to the considered problem, obtained by using different definitions of derivatives, is reported as well.

scholarly journals On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Chunyan Huang
Keyword(s):  
Initial Data ◽  
Landau Equation ◽  
Global Solutions ◽  
Spatial Dimension ◽  
Modulation Spaces ◽  
Ginzburg Landau ◽  
Small Initial Data ◽  
Ginzburg Landau Equation ◽  
Rough Initial Data ◽  
Analytic Regularity

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.

scholarly journals Transforming Analytically Intractable Dynamical Systems with a Control Parameter into a Tractable Ginzburg-Landau Equation: Few Illustrations

2018 ◽  
Vol 35 (1-2) ◽  
pp. 35-44 ◽  
Author(s):  
C. Kanchana ◽  
P.G. Siddheshwar
Keyword(s):  
Dynamical Systems ◽  
Control Parameter ◽  
Landau Equation ◽  
Chen System ◽  
Third Order ◽  
Ginzburg Landau ◽  
First Order ◽  
Multi Scale ◽  
Ginzburg Landau Equation ◽  
Fifth Order

In the paper a means of making a simplified study of dynamical systems with a control parameter is presented. The intractable, third-order classical Lorenz system, the Lorenz-like Chen system and two topologically dissimilar fifth-order Lorenz systems are considered for illustration. Using the multi-scale method, these systems are reduced to an analytically tractable first-order Ginzburg-Landau equation (GLE) in one of the amplitudes. The analytical solution of the GLE is used to find the remaining amplitudes.

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