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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\R^3$ be an open set and let $\\kappa \u003e 0$ be a constant, the Ginzburg–Landau parameter.\nFor a function $\\psi \\colon \\Omega \\to \\C$ and a [[vector field|0299VCW1TH]] $A \\colon \\Omega \\to \\R^3$, write $\\nabla_A \\psi = \\nabla \\psi - i A \\psi$, where $\\nabla$ is the [[gradient|0ABK9V9Y64]].\nThe *Ginzburg–Landau equations* are the system\n$$- \\nabla_A \\cdot \\nabla_A \\psi = \\kappa^2 \\big( 1 - \\abs{\\psi}^2 \\big) \\psi, \\qquad \\nabla \\times \\nabla \\times A = \\mathrm{Im}\\big( \\overline{\\psi} \\, \\nabla_A \\psi \\big),$$\nfor the unknowns $\\psi$, the order parameter, and $A$, the magnetic potential.\nHere $\\nabla_A \\cdot V = \\nabla \\cdot V - i A \\cdot V$ for a vector field $V$ with values in $\\C^3$, and $\\nabla \\times$ is the [[curl|0NF6GQNXMY]].\n\n## Remarks\n\nThe equations are the Euler–Lagrange equations of the Ginzburg–Landau energy\n$$\\mathcal{E}(\\psi, A) = \\int_\\Omega \\Big( \\tfrac{1}{2} \\abs{\\nabla_A \\psi}^2 + \\tfrac{\\kappa^2}{4} \\big( 1 - \\abs{\\psi}^2 \\big)^2 + \\tfrac{1}{2} \\abs{\\nabla \\times A}^2 \\Big) dx,$$\nwritten in dimensionless units.\nOn a bounded domain the natural boundary condition is $\\nabla_A \\psi \\cdot n = 0$.\nThe energy and the equations are invariant under the gauge change $\\psi \\mapsto e^{i\\chi} \\psi$, $A \\mapsto A + \\nabla \\chi$.\n\nExample: $\\psi = 1$ and $A = 0$ is a solution: both sides of each equation are $0$.\n\nNon-example: $\\psi = 1/2$ and $A = 0$ is not a solution, because the left side of the first equation is $0$ while the right side is $\\tfrac{3}{8} \\kappa^2 \\ne 0$.\n\nWithout magnetic field, setting $A = 0$ and dropping the second equation gives the simplified equation $-\\Delta \\psi = \\kappa^2 (1 - \\abs{\\psi}^2) \\psi$.\nThe name Ginzburg–Landau equation is also used for the time-dependent complex equation $\\partial_t \\psi = (1 + i\\alpha) \\Delta \\psi + \\psi - (1 + i\\beta) \\abs{\\psi}^2 \\psi$ of pattern formation.\n",
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