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.

NEW PROPERTIES OF QUASIPERIODIC SOLUTIONS OF THE COMPLEX GINZBURG-LANDAU EQUATION

1992 ◽  
Vol 02 (04) ◽  
pp. 955-972 ◽  
Author(s):  
TATIANA S. AKHROMEYEVA ◽  
GEORGE G. MALINETSKII ◽  
ALEXEY B. POTAPOV ◽  
GEORGE Z. TSERTSVADZE
Keyword(s):  
Numerical Methods ◽  
Initial Data ◽  
Landau Equation ◽  
Mathematical Physics ◽  
Ginzburg Landau ◽  
New Class ◽  
A Value ◽  
Ginzburg Landau Equation ◽  
New Type ◽  
No Fluxes

By using analytical and numerical methods the authors study one of the basic models of mathematical physics—the so-called complex Ginzburg-Landau equation [Formula: see text] with the provision that no fluxes exist at the segment boundaries. A new class of solutions is found for this equation. It is shown that among its solutions there are analogs of limiting cycles of the second kind. A value describing these analogs is introduced, and a scenario of its variation depending on the parameters of the problem is given. A new type of spontaneous appearance of symmetry is shown when we go from initial data in the general form to spatially symmetrical solutions describing quasiperiodic regimes.

The Inviscid Limit of the Fractional Complex Ginzburg–Landau Equation

2016 ◽  
Vol 17 (6) ◽  
pp. 333-341
Author(s):  
Lijun Wang ◽  
Jingna Li ◽  
Li Xia
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Convergence Rate ◽  
Schrodinger Equation ◽  
Landau Equation ◽  
Limit Behavior ◽  
Inviscid Limit ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

AbstractIn this paper, the inviscid limit behavior of solution of the fractional complex Ginzburg–Landau (FCGL) equation$${\partial _t}u + (a + i\nu){\Lambda ^{2\alpha}}u + (b + i\mu){\left| u \right|^{2\sigma}}u = 0, \quad (x, t) \in {{\Cal T}^n} \times (0, \infty)$$is considered. It is shown that the solution of the FCGL equation converges to the solution of nonlinear fractional complex Schrödinger equation, while the initial data${u_0}$is taken in${L^2}, $${H^\alpha}$, and${L^{2\sigma + 2}}$as$a,\, b$tends to zero, and the convergence rate is also obtained.

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.

Resolving weak internal layer interactions for the Ginzburg–Landau equation

1994 ◽  
Vol 5 (4) ◽  
pp. 495-523 ◽  
Author(s):  
Luis G. Reyna ◽  
Michael J. Ward
Keyword(s):  
Steady State ◽  
Landau Equation ◽  
Spatial Dimension ◽  
Internal Layer ◽  
Equilibrium Solutions ◽  
Singular Perturbation Method ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Mass Constraint ◽  
Unconstrained Problem

The internal layer behaviour, in one spatial dimension, associated with two classes of Ginzbug–Landau equation with double-well nonlinearities and small diffusivities is investigated. The problems that are examined are the Ginzburg–Landau equation with and without a constant mass constraint. For the constrained problem, steady-state internal layer solutions are constructed using a formal projection method. This method is also used to derive a differential-algebraic system describing the slow dynamics of the constrained internal layer motion. The dynamics of a two-layer evolution is studied in detail. For the unconstrained problem, a nonlinear WKB-type transformation is introduced that magnifies exponentially weak layer interactions and leads to well-conditioned steady problems. A conventional singular perturbation method, without the need for exponential asymptotics, is used on the resulting transformed problem as an alternative method to construct equilibrium solutions and metastable patterns. Exponentially sensitive steady-state internal layer solutions as well as a one-layer evolution are computed accurately using the transformed problem.

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