Non-Magnetic Complex Ginzburg-Landau Systems

• The non-magnetic complex Ginzburg–Landau system is a family of PDE and variational models for complex order parameters without magnetic vector coupling.
• Key analyses include elliptic vortex profiles, phase diagram transitions in dissipative regimes, and noise-induced aging with measurable scaling exponents.
• Rigorous micro-to-macro derivations, feedback control techniques, and stability criteria integrate to model spatiotemporal patterns in nonequilibrium systems.

Searching arXiv for recent and foundational work on non-magnetic complex Ginzburg-Landau systems, including vortex structure, stability, stochastic dynamics, phase diagrams, control, and derivations. The non-magnetic complex Ginzburg–Landau system is a class of PDEs and variational models for complex-valued scalar or vector order parameters in which no magnetic vector potential appears. In the supplied literature, the term covers static elliptic Ginzburg–Landau equations, dissipative and dispersive complex Ginzburg–Landau evolution equations, stochastic variants with additive white noise, and multicomponent vortex models. Representative forms include the scalar equation

$-\Delta u=u(1-|u|^2)\qquad \text{in }\mathbb R^3,$

the two-component elliptic system for $\Psi=(\psi_+,\psi_-)$,

$$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$

$$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$

and the dissipative two-dimensional CGL

$$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$

In the non-magnetic setting, the order parameter is an ordinary complex scalar or vector field, rather than a field coupled to gauge-covariant derivatives or an electromagnetic potential (Alama et al., 2012).

1. Model classes and non-magnetic structure

A central two-component non-magnetic model is the Ginzburg–Landau energy

$$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$

with parameters

$$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$

The condition $B^2<A_+A_-$ makes the quadratic potential strictly positive definite and is used throughout the elliptic theory (Alama et al., 2012).

A second two-component formulation uses

$$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$

with vacuum manifold

$|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$

so that the vacuum states are

$\Psi=(\psi_+,\psi_-)$0

and carry degree pairs $\Psi=(\psi_+,\psi_-)$1 (Alama et al., 2012).

On the dynamical side, the standard non-magnetic dissipative CGL in two dimensions is written as

$\Psi=(\psi_+,\psi_-)$2

while a noisy reduced form is

$\Psi=(\psi_+,\psi_-)$3

augmented by weak additive noise (Chaté et al., 2016). A variable-coefficient version is

$\Psi=(\psi_+,\psi_-)$4

which the cited work treats as a model of spatially extended nonequilibrium systems (Uchiyama, 2019).

The non-magnetic qualifier has a precise technical role in the microscopic derivation from Bogoliubov–de Gennes theory: there is no vector potential, no gauge-covariant derivatives, the BdG operator remains translation invariant, and the macroscopic GL equation has constant coefficients $\Psi=(\psi_+,\psi_-)$5 (Frank et al., 25 May 2026). This distinguishes the scalar non-magnetic setting from the magnetic time-dependent Ginzburg–Landau system with vortex filaments and a London-type outer field equation discussed only as a separate extension (Jin et al., 15 Apr 2026).

2. Equivariant vortices and static elliptic theory

For the planar two-component system, symmetric vortex solutions are sought in the equivariant form

$\Psi=(\psi_+,\psi_-)$6

with $\Psi=(\psi_+,\psi_-)$7. The radial profiles satisfy

$\Psi=(\psi_+,\psi_-)$8

$\Psi=(\psi_+,\psi_-)$9

with $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$0 as $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$1 and Frobenius behavior $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$2 near the origin (Alama et al., 2012).

Under $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$3, for every degree pair $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$4 there exists a unique equivariant solution. The profiles have the asymptotic expansion

$$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$5

where

$$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$6

The sign of $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$7 determines whether $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$8 approaches $$-\Delta \psi_+ + \Big(A_+\big(|\psi_+|^2-t_+^2\big)+B\big(|\psi_-|^2-t_-^2\big)\Big)\psi_+ = 0,$$9 from below or from above; this is the mechanism behind monotonic and non-monotonic vortex tails (Alama et al., 2012).

The monotonicity theory differs sharply from the scalar GL case. If $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$0, then

$$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$1

If $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$2, $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$3, and $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$4, then

$$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$5

For any nontrivial degree pair with $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$6, there exists $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$7 such that for $$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$8 both profiles are monotone, while in the regime

$$-\Delta \psi_- + \Big(A_-\big(|\psi_-|^2-t_-^2\big)+B\big(|\psi_+|^2-t_+^2\big)\Big)\psi_- = 0,$$9

for example in the $$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$0 case, one component may approach its limit from above and therefore cannot be monotone (Alama et al., 2012).

A related degree-$$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$1 problem on the disk uses symmetric boundary data

$$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$2

For the symmetric branch

$$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$3

the coupling term vanishes and $$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$4 solves the scalar degree-one Ginzburg–Landau profile equation. The resulting scalar-based compound vortex

$$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$5

is the global minimizer for every $$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$6 when $$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$7, but for $$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$8 and $$\partial_t A = A + (1+i b_1)\nabla^2 A - (b_3-i)|A|^2A.$$9 sufficiently small it is not the minimizer (Alama et al., 2012).

The weak-coupling regime $$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$0 produces the split-core phenomenon: the two components do not vanish at the same point; instead,

$$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$1

and $$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$2 stays bounded away from zero in the vortex core region. By contrast, when $$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$3, the locally minimizing degree-$$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$4 entire solutions are exactly

$$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$5

with $$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$6 the unique scalar degree-one entire vortex (Alama et al., 2012).

The spectral stability theory for symmetric degree-one vortices on the unit disk yields a parallel dichotomy. For the energy

$$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$7

the symmetric equivariant degree-one vortex is stable for every $$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$8 when $$E(\Psi)=\int_{\mathbb{R}^2}\left( \frac12|\nabla \psi_+|^2+\frac12|\nabla \psi_-|^2 +\frac{A_+}{4}\big(|\psi_+|^2-t_+^2\big)^2 +\frac{A_-}{4}\big(|\psi_-|^2-t_-^2\big)^2 +\frac{B}{2}\big(|\psi_+|^2-t_+^2\big)\big(|\psi_-|^2-t_-^2\big) \right)\,dx,$$9. If $$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$0, there exists a unique threshold $$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$1 such that the symmetric vortex is unstable for sufficiently large $$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$2. The unstable direction is the splitting mode

$$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$3

which corresponds to moving the two component vortices in opposite directions (Alama et al., 2013).

The static non-magnetic theory also admits genuinely three-dimensional singular sets. A 2025 construction gives a smooth entire solution of

$$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$4

in $$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$5 such that $$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$6, the zero set is exactly

$$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$7

and

$$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$8

Its blow-down measures satisfy

$$A_+>0,\qquad A_->0,\qquad B^2<A_+A_-,\qquad t_+>0,\qquad t_->0.$$9

and loops linking either axis have degree $B^2<A_+A_-$0 in absolute value, showing that each line carries unit multiplicity (Caselli et al., 17 Sep 2025).

3. Spatiotemporal regimes, defects, and synchronization

The two-dimensional deterministic CGL phase diagram is organized in the $B^2<A_+A_-$1 plane. Plane-wave solutions

$B^2<A_+A_-$2

change stability at the Benjamin–Feir line

$B^2<A_+A_-$3

For $B^2<A_+A_-$4, all such plane waves are unstable; for $B^2<A_+A_-$5, a band of wavenumbers remains stable (Chaté et al., 2016).

The numerical phase diagram contains three main disordered regimes. Phase turbulence occupies the region between the BF line and line $B^2<A_+A_-$6, with no defects and a field that never reaches zero. Defect turbulence occurs to the left of line $B^2<A_+A_-$7, with zeros of $B^2<A_+A_-$8 created and destroyed continuously in spacetime. Frozen states occupy the region to the right of line $B^2<A_+A_-$9, where spiral defects and shock-line boundaries form a quasi-stationary cellular structure (Chaté et al., 2016).

The transitions are described as nucleation-like rather than as ordinary equilibrium phase transitions. Line $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$0 marks a breakdown of sustained phase turbulence by creation of a defect pair and growth of a defect-turbulence bubble. Line $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$1 marks the loss of sustained defect turbulence and the nucleation of a frozen spiral structure. Both transitions are reported as hysteretic, and the paper explicitly notes that it is not clear whether phase turbulence survives in the infinite-size/infinite-time limit, whether line $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$2 merges with the BF line, or whether line $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$3 differs asymptotically from the local spiral-growth threshold $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$4 (Chaté et al., 2016).

A distinct dynamical phenomenon is anticipated synchronization in a unidirectionally coupled master–slave pair,

$$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$5

$$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$6

with delayed slave variable $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$7. The anticipated synchronization manifold

$$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$8

is an exact solution because $$E_\varepsilon(\Psi;\Omega) = \int_\Omega \left\{ \frac12 |\nabla \Psi|^2 + \frac1{4\varepsilon^2}\Big(|\Psi|^2-1\Big)^2 + \frac{\beta}{4\varepsilon^2}\Big(|\psi_+|^2-|\psi_-|^2\Big)^2 \right\}\,dx,$$9 on that manifold. Stability is not automatic and depends on the regime, the delay, and the complex coupling phase (Ciszak et al., 2014).

The reported maximum anticipation times scale with the linear autocorrelation time of the uncoupled master. For the parameter sets studied numerically, the paper gives

• defect turbulence: $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$0, $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$1,
• bichaos: $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$2, $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$3,
• phase turbulence: $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$4, $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$5.

The same study reports that the largest anticipation times are obtained for complex-valued coupling constants, and that nonzero positive $|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$6 enlarges the stable anticipated-synchronization region. In two dimensions, anticipated synchronization persists, but the maximum anticipation time is smaller than in one dimension (Ciszak et al., 2014).

These results make the non-magnetic CGL system a standard setting for the coexistence of phase instability, defect creation and annihilation, coherent spiral emission, delayed feedback phenomena, and hysteretic transitions between attractors. A plausible implication is that “non-magnetic” does not denote a narrow equilibrium limit; in the cited literature it includes fully nonequilibrium spatiotemporal chaos.

4. Noise, critical relaxation, and aging

The noisy complex Ginzburg–Landau equation studied in the renormalization-group literature is written in a Gross–Pitaevskii-like form as

$|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$7

or equivalently in relaxational form as

$|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$8

The noise is additive complex Gaussian white noise with

$|\Psi|^2=1,\qquad |\psi_+|=|\psi_-|,$9

The formulation is interpreted simultaneously as a noisy dissipative Gross–Pitaevskii equation, a time-dependent complex Ginzburg–Landau equation, and a generalization of equilibrium model A for a non-conserved complex order parameter (Liu et al., 2016).

Near the continuous non-equilibrium phase transition, the short-time relaxation from a fully randomized Gaussian initial state exhibits critical aging and an independent initial-slip exponent $\Psi=(\psi_+,\psi_-)$00. In the aging regime $\Psi=(\psi_+,\psi_-)$01, the scaling forms are

$\Psi=(\psi_+,\psi_-)$02

$\Psi=(\psi_+,\psi_-)$03

To one loop in the $\Psi=(\psi_+,\psi_-)$04 expansion, the initial-slip exponent is

$\Psi=(\psi_+,\psi_-)$05

which is exactly the equilibrium model A result. The cited analysis attributes this to the infrared-stable equilibrium fixed point and argues, using the RG flow and a complex spherical model extension, that the conclusion likely remains true to all orders in the perturbation expansion (Liu et al., 2016).

A complementary numerical study of the two-dimensional noisy CGL uses

$\Psi=(\psi_+,\psi_-)$06

with weak additive noise of strength

$\Psi=(\psi_+,\psi_-)$07

The focusing spiral quadrant is defined by $\Psi=(\psi_+,\psi_-)$08, the defocusing spiral quadrant by $\Psi=(\psi_+,\psi_-)$09. Coarsening is tracked through the defect-density length scale

$\Psi=(\psi_+,\psi_-)$10

and aging through

$\Psi=(\psi_+,\psi_-)$11

Only $\Psi=(\psi_+,\psi_-)$12 defects are stable, with topological charge

$\Psi=(\psi_+,\psi_-)$13

defined from the phase singularity of $\Psi=(\psi_+,\psi_-)$14 (Liu et al., 2019).

The measured exponents are non-universal across parameter sets. In the focusing quadrant, at $\Psi=(\psi_+,\psi_-)$15 the study reports

$\Psi=(\psi_+,\psi_-)$16

and at $\Psi=(\psi_+,\psi_-)$17,

$\Psi=(\psi_+,\psi_-)$18

In the defocusing quadrant, at $\Psi=(\psi_+,\psi_-)$19,

$\Psi=(\psi_+,\psi_-)$20

Near $\Psi=(\psi_+,\psi_-)$21, representative values are $\Psi=(\psi_+,\psi_-)$22 and $\Psi=(\psi_+,\psi_-)$23 to $\Psi=(\psi_+,\psi_-)$24, close to the 2D XY-model value cited for $\Psi=(\psi_+,\psi_-)$25 (Liu et al., 2019).

The authors of that study conclude that physical aging in the noisy CGL is governed by non-universal aging scaling exponents, and propose heuristic criteria for slow coarsening: in the focusing quadrant, proximity to the real Ginzburg–Landau limit $\Psi=(\psi_+,\psi_-)$26; in the defocusing quadrant, proximity to $\Psi=(\psi_+,\psi_-)$27. The main obstacle is the formation of stable shock fronts, which screen defect interactions and lead to spatial freezing (Liu et al., 2019).

5. Exact reductions, feedback control, and vortex-regime algorithms

For the variable-coefficient equation

$\Psi=(\psi_+,\psi_-)$28

an exact transformation scheme removes gain/loss and frequency modulation by

$\Psi=(\psi_+,\psi_-)$29

With

$\Psi=(\psi_+,\psi_-)$30

the equation becomes

$\Psi=(\psi_+,\psi_-)$31

The paper then introduces the imaginary-time advection equation

$\Psi=(\psi_+,\psi_-)$32

and the reparametrization

$\Psi=(\psi_+,\psi_-)$33

to reduce the problem to the standard focusing NLSE

$\Psi=(\psi_+,\psi_-)$34

In this framework, one-soliton, Peregrine-soliton, and Akhmediev-breather solutions generate exact solutions of the original variable-coefficient CGLE (Uchiyama, 2019).

The non-magnetic dissipative CGLE also admits finite-parameter feedback stabilization. The uncontrolled model is

$\Psi=(\psi_+,\psi_-)$35

with $\Psi=(\psi_+,\psi_-)$36, $\Psi=(\psi_+,\psi_-)$37, $\Psi=(\psi_+,\psi_-)$38, and $\Psi=(\psi_+,\psi_-)$39. One feedback law uses finitely many volume averages,

$\Psi=(\psi_+,\psi_-)$40

and yields exponential stabilization under

$\Psi=(\psi_+,\psi_-)$41

In that case,

$\Psi=(\psi_+,\psi_-)$42

Related controllers based on Fourier modes and nodal observables also give exponential $\Psi=(\psi_+,\psi_-)$43-stabilization, and in the Fourier-mode case $\Psi=(\psi_+,\psi_-)$44-decay under the stated restrictions (Kalantarova et al., 2017).

The same paper formulates a tracking problem with controller

$\Psi=(\psi_+,\psi_-)$45

where $\Psi=(\psi_+,\psi_-)$46 solves the uncontrolled equation. Under

$\Psi=(\psi_+,\psi_-)$47

the tracking estimate is

$\Psi=(\psi_+,\psi_-)$48

This places the non-magnetic CGLE within a control-theoretic framework based on finitely many observables rather than full-state data (Kalantarova et al., 2017).

A more recent computational development treats the strongly nonlinear vortex regime asymptotically. For the non-magnetic scalar Ginzburg–Landau-type potential

$\Psi=(\psi_+,\psi_-)$49

the paper assumes

$\Psi=(\psi_+,\psi_-)$50

and decomposes the phase as

$\Psi=(\psi_+,\psi_-)$51

For fixed vortex locations $\Psi=(\psi_+,\psi_-)$52, the harmonic correction satisfies

$\Psi=(\psi_+,\psi_-)$53

while the dissipative Ginzburg–Landau motion law is

$\Psi=(\psi_+,\psi_-)$54

The paper proposes hybrid quantum-classical algorithms that advance the vortex ODE classically and solve the outer linear elliptic problem with quantum algorithms, and states that this yields “an exponential improvement in the dependence on the spatial problem size, while the dependence on the target accuracy remains essentially linear up to polylogarithmic factors” (Jin et al., 15 Apr 2026).

6. Microscopic derivation and conceptual scope

A rigorous derivation near the critical temperature connects the non-magnetic Ginzburg–Landau equation to the Bogoliubov–de Gennes equation for a BCS model without external fields. The microscopic Hamiltonian is

$\Psi=(\psi_+,\psi_-)$55

and critical points of the BCS free energy satisfy the BdG equation

$\Psi=(\psi_+,\psi_-)$56

The near-critical scaling is

$\Psi=(\psi_+,\psi_-)$57

with small solutions obeying

$\Psi=(\psi_+,\psi_-)$58

The main asymptotic factorization is

$\Psi=(\psi_+,\psi_-)$59

where $\Psi=(\psi_+,\psi_-)$60 is the microscopic Cooper-pair profile and $\Psi=(\psi_+,\psi_-)$61 is a macroscopic order parameter (Frank et al., 25 May 2026).

The macroscopic field satisfies the GL equation up to an error that vanishes in the scaling limit: $\Psi=(\psi_+,\psi_-)$62 with

$\Psi=(\psi_+,\psi_-)$63

The associated GL energy is

$\Psi=(\psi_+,\psi_-)$64

with $\Psi=(\psi_+,\psi_-)$65, $\Psi=(\psi_+,\psi_-)$66, and $\Psi=(\psi_+,\psi_-)$67 positive definite (Frank et al., 25 May 2026).

This derivation gives a precise micro-to-macro meaning to the non-magnetic GL equation: near $\Psi=(\psi_+,\psi_-)$68, every sufficiently small BdG critical point in the relevant energy regime admits an asymptotic decomposition into a microscopic pair profile and a macroscopic complex order parameter. It also clarifies the scope of the non-magnetic assumption: there is no vector potential, no magnetic field term, and the limiting GL equation is an ordinary complex scalar equation with constant coefficients (Frank et al., 25 May 2026).

Taken together, the cited works show that the non-magnetic complex Ginzburg–Landau system is not a single equation but a coherent family of scalar and vector models. Across that family, several structural themes recur: vortex degrees and phase singularities, variational coercivity from positive-definite quartic potentials, reduction to low-dimensional defect dynamics in singular regimes, and delicate dependence of stability or coarsening on the sign and size of coupling terms. The literature also shows that the absence of magnetic coupling does not eliminate complexity; it relocates it into vortex geometry, defect kinetics, delayed synchronization, stochastic aging, and multiscale reduction.