The limit behavior of solutions for the Cauchy problem of the complex Ginzburg-Landau equation

2002 ◽  
Vol 55 (4) ◽  
pp. 481-508 ◽  
Author(s):  
Baoxiang Wang
Keyword(s):  
Cauchy Problem ◽  
Landau Equation ◽  
Limit Behavior ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
The Cauchy Problem

scholarly journals On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation

1995 ◽  
Vol 36 (3) ◽  
pp. 313-324 ◽  
Author(s):  
Charles Bu
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Analytical Methods ◽  
Critical Case ◽  
Landau Equation ◽  
Local Existence ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Spatial Dimensions ◽  
The Cauchy Problem

AbstractWe present analytical methods to investigate the Cauchy problem for the complex Ginzburg-Landau equation u1 = (v + iα)Δu − (κ + iβ) |u|2qu + γu in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, κ ≥ 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when .

scholarly journals On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law

2019 ◽  
Vol 22 (07) ◽  
pp. 1950054
Author(s):  
João-Paulo Dias ◽  
Filipe Oliveira ◽  
Hugo Tavares
Keyword(s):  
Cauchy Problem ◽  
Fluid Flow ◽  
Conservation Law ◽  
Landau Equation ◽  
Coupled System ◽  
Global Weak Solutions ◽  
Ginzburg Landau ◽  
Wave Solutions ◽  
Ginzburg Landau Equation ◽  
The World

We study a coupled system of a complex Ginzburg–Landau equation with a quasilinear conservation law [Formula: see text] which can describe the interaction between a laser beam and a fluid flow (see [I.-S. Aranson and L. Kramer, The world of the complex Ginzburg–Landau equation, Rev. Mod. Phys. 74 (2002) 99–143]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore, we prove the existence of standing wave solutions of the form [Formula: see text] in several cases.

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.

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.

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