On the validity of the Ginzburg-Landau equation |
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Authors: | A van Harten |
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Institution: | (1) Mathematical Institute, University of Utrecht, Budapestlaan 6, 3508 TA Utrecht, The Netherlands |
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Abstract: | Summary The famous Ginzburg-Landau equation describes nonlinear amplitude modulations of a wave perturbation of a basic pattern when
a control parameterR lies in the unstable regionO(ε
2) away from the critical valueR
c for which the system loses stability. Hereε>0 is a small parameter. G-L's equation is found for a general class of nonlinear evolution problems including several classical
problems from hydrodynamics and other fields of physics and chemistry. Up to now, the rigorous derivation of G-L's equation
for general situations is not yet completed. This was only demonstrated for special types of solutions (steady, time periodic)
or for special problems (the Swift-Hohenberg equation). Here a mathematically rigorous proof of the validity of G-L's equation
is given for a general situation of one space variable and a quadratic nonlinearity. Validity is meant in the following sense.
For each given initial condition in a suitable Banach space there exists a unique bounded solution of the initial value problem
for G-L's equation on a finite interval of theO(1/ε2)-long time scale intrinsic to the modulation. For such a finite time interval of the intrinsic modulation time scale on which
the initial value problem for G-L's equation has a bounded solution, the initial value problem for the original evolution
equation with corresponding initial conditions, has a unique solutionO(ε2) — close to the approximation induced by the solution of G-L's equation. This property guarantees that, for rather general
initial conditions on the intrinsic modulation time scale, the behavior of solutions of G-L's equation is really inherited
from solutions of the original problem, and the other way around: to a solution of G-L's equation corresponds a nearby exact
solution with a relatively small error. |
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Keywords: | nonlinear stability theory modulation equations approximation on a long time scale error estimates |
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