View source for Ginzburg-Landau equation

A modulation (or amplitude/envelope equation) that describes the evolution of small perturbations of a marginally unstable basic state of a system of non-linear partial differential equations on an unbounded domain (cf. also [[Perturbation theory|Perturbation theory]]; [[Perturbation of a linear system|Perturbation of a linear system]]; [[Linear partial differential equation|Linear partial differential equation]]). The stationary problem associated to the Ginzburg–Landau equation with real coefficients also has a different background as the [[Euler–Lagrange equation|Euler–Lagrange equation]] associated to the Ginzburg–Landau functional (see below). To obtain the Ginzburg–Landau equation as modulation equation, one lets

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

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describe an underlying problem (with certain boundary conditions), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200503.png" /> is an elliptic linear operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200504.png" /> a non-linear operator of order less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200506.png" /> a (bifurcation) parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200507.png" /> a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200508.png" />. The linearized stability of the basic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g1200509.png" /> of (a1) is determined by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005010.png" /> and solving an eigenvalue problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005012.png" /> for any pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005014.png" />). Under certain conditions on the eigenvalue problem, one can define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005015.png" /> as the critical eigenvalue (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005019.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005020.png" /> as the critical value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005022.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005024.png" />, i.e. the neutral manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005025.png" /> has a minimum at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005027.png" />-space: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005028.png" /> is linearly stable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005029.png" />). Introduce <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005031.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005033.png" /> as the critical eigenfunction at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005035.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005038.png" /> can be expanded both as an [[Asymptotic series|asymptotic series]] and as a [[Fourier series|Fourier series]]:

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Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005040.png" /> is an unknown amplitude and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005042.png" /> are rescaled variables:

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The Ginzburg–Landau equation describes the evolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005045.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

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(at leading order). The equation is obtained (formally) by inserting the above expansion into (a1) and applying an orthogonality condition. The second-order differential operator is elliptic when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005048.png" /> is an isolated non-degenerate minimum of the neutral manifold. The Landau constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005049.png" /> can be expressed in terms of information obtained from the linear eigenvalue problem and its adjoint. In most cases studied in the literature, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005051.png" />; (a2) can then be rescaled into

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005053.png" />. Historically, the Ginzburg–Landau equation was first derived as a modulation equation for two classical hydrodynamic stability problems: Rayleigh–Bénard convection [[#References|[a4]]] and [[Poiseuille flow|Poiseuille flow]] [[#References|[a5]]]. Several aspects of the mathematical validity of this formal approximation scheme have been studied in [[#References|[a2]]], [[#References|[a7]]], [[#References|[a6]]].

By its nature, the Ginzburg–Landau equation appears as leading-order approximation in many systems. Therefore, there is much literature on the behaviour of its solutions. Its most simple solutions are periodic: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005054.png" />. Under certain conditions on the coefficients there is a subfamily of stable periodic solutions; it is called the Eckhaus band in (a3). The existence and stability of more complicated  "localized"  (homoclinic, heteroclinic) solutions to (a3) (also with more general non-linear terms) is considered in [[#References|[a8]]] (with mostly formal results). Up to now (1998), there is no mathematical text (book or survey paper) that gives an overview of what is known about the behaviour of solutions of the Ginzburg–Landau equation as evolution equation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005055.png" /> or even on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005056.png" />.

However, much is known about the solutions of the stationary (elliptic) Ginzburg–Landau equation with real coefficients on bounded domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g120/g120050/g12005057.png" /> [[#References|[a1]]]. The Ginzburg–Landau equation with real coefficients has a variational structure: the Ginzburg–Landau functional

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The Ginzburg–Landau functional appears in various parts of science; in general, it is not related to the above sketched modulation equation interpretation of the Ginzburg–Landau equation [[#References|[a1]]]. The name  "Ginzburg–Landau" , both of the equation and of the functional, comes from a paper on superconductivity [[#References|[a3]]]. However, in this context the (real, stationary) equation and/or the functional is part of a larger system of equations/functionals.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bethuel,   H. Brezis, F. Hélein,   "Ginzburg–Landau vortices" , Birkhäuser  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Collet,   J.P. Eckmann,   "The time-dependent amplitude equation for the Swift–Hohenberg problem"  ''Comm. Math. Phys.'' , '''132'''  (1990)  pp. 139–153</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.L. Ginzburg,   L.D. Landau,   "On the theory of superconductivity"  ''Zh. Eksper. Teor. Fiz.'' , '''20'''  (1950)  pp. 1064–1082  (In Russian)  (English transl.: Men of Physics: L.D. Landau (D. ter Haar, ed.), Pergamon, 1965,138-167)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.C. Newell,   J.A. Whitehead,   "Finite bandwidth, finite amplitude convection"  ''J. Fluid Mech.'' , '''38'''  (1969)  pp. 279–303</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Stewartson,   J.T. Stuart,   "A non-linear instability theory for a wave system in plane Poiseuille flow"  ''J. Fluid Mech.'' , '''48'''  (1971)  pp. 529–545</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Schneider,   "Global existence via Ginzburg–Landau formalism and pseudo-orbits of the Ginzburg–Landau approximations"  ''Comm. Math. Phys.'' , '''164'''  (1994)  pp. 159–179</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. van Harten,   "On the validity of the Ginzburg–Landau's equation"  ''J. Nonlinear Sci.'' , '''1'''  (1991)  pp. 397–422</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W. van Saarloos,   P.C. Hohenberg,   "Fronts, pulses, sources and sinks in generalised complex Ginzburg–Landau equations"  ''Physica D'' , '''56'''  (1992)  pp. 303–367</TD></TR></table>