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 Asymptotic analysis of a new type of multi-bump, self-similar, blowup solutions of the Ginzburg–Landau equation

2012 ◽  
Vol 24 (1) ◽  
pp. 103-129 ◽  
Author(s):  
V. ROTTSCHÄFER
Keyword(s):  
Asymptotic Analysis ◽  
Landau Equation ◽  
Asymptotic Methods ◽  
Geometric Construction ◽  
The Other ◽  
Ginzburg Landau ◽  
Geometric Methods ◽  
Ginzburg Landau Equation ◽  
New Type ◽  
Self Similar

We study of a new type of multi-bump blowup solutions of the Ginzburg–Landau equation. Multi-bump blowup solutions have previously been found in numeric simulations, asymptotic analysis and were proved to exist via geometric construction. In the geometric construction of the solutions, the existence of two types of multi-bump solutions was shown. One type is exponentially small at ξ=0, and the other type of solutions is algebraically small at ξ=0. So far, the first type of solutions were studied asymptotically. Here, we analyse the solutions which are algebraically small at ξ=0 by using asymptotic methods. This construction is essentially different from the existing one, and ideas are obtained from the geometric construction. Hence, this is a good example of where both asymptotic analysis and geometric methods are needed for the overall picture.

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.

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