Ginzburg–Landau equation

Remarks

The equations are the Euler–Lagrange equations of the Ginzburg–Landau energy $$\mathcal{E}(\psi, A) = \int_\Omega \Big( \tfrac{1}{2} \abs{\nabla_A \psi}^2 + \tfrac{\kappa^2}{4} \big( 1 - \abs{\psi}^2 \big)^2 + \tfrac{1}{2} \abs{\nabla \times A}^2 \Big) dx,$$ written in dimensionless units. On a bounded domain the natural boundary condition is $\nabla_A \psi \cdot n = 0$. The energy and the equations are invariant under the gauge change $\psi \mapsto e^{i\chi} \psi$, $A \mapsto A + \nabla \chi$.

Example: $\psi = 1$ and $A = 0$ is a solution: both sides of each equation are $0$.

Non-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 $\tfrac{3}{8} \kappa^2 \ne 0$.

Without magnetic field, setting $A = 0$ and dropping the second equation gives the simplified equation $-\Delta \psi = \kappa^2 (1 - \abs{\psi}^2) \psi$. The 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.