Instability of solutions to the Ginzburg–Landau equation on S and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="double-struck">C</mml:mi><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">P</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>

Cheng, Da Rong

doi:10.1016/j.jfa.2020.108669

Cited by 7 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• First, the finitely many exceptions of ζ ∈ S 1 for stabilization are determined by the unstable dimension of the spiral waves; see the proof of Lemma 4.5 and Lemma 4.8. Lower bound estimates of the unstable dimension have been investigated (see [3,5] for instance), but in general the exact value of the unstable dimension remains unknown. • Second, pure temporal controls (i.e., ι = + and ζ = 0 in (3.3)) cannot achieve stabilization, as we will prove in Lemma 4.4.…”

Section: Symmetry-breaking Controls and Main Resultsmentioning

confidence: 99%

“…We aim to understand pattern formation, dynamical behavior, and feedback controls of Ginzburg-Landau spiral waves on the surface M. To this end we present a trilogy of research: existence, stability analysis, and feedback stabilization. The first two episodes regarding existence and stability analysis have been investigated extensively in [3,19,33] and also by Dai in [4,5]. This article serves as the third episode in which we stabilize certain classes of unstable spiral waves by introducing noninvasive symmetry-breaking feedback controls with spatio-temporal delays.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Schneider¹,

Wolff²,

Dai³

2022

Preprint

View full text Add to dashboard Cite

The complex Ginzburg-Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg-Landau m-armed spiral waves have been investigated extensively. However, most spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

show abstract

Section: Symmetry-breaking Controls and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Schneider¹,

Wolff²,

Dai³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…where t ≥ 0, coupling coefcients a, b, c, and m are constants, μ is the chemical potential, 2v is the threshold energy of the Feshbach resonance, and d generally is a complex number with d � d r + id i . Te Ginzburg-Landau equation has been particularly favored by scientists for its ability to efectively capture various types of features in the model and has yielded fruitful results [4][5][6][7][8][9][10]. Tese results are both interpretations of the existence results for the solutions [4,5,[7][8][9] and explorations of the attractor problem for the Ginzburg-Landau model [6,10].…”

Section: Introductionmentioning

confidence: 99%

“…Te Ginzburg-Landau equation has been particularly favored by scientists for its ability to efectively capture various types of features in the model and has yielded fruitful results [4][5][6][7][8][9][10]. Tese results are both interpretations of the existence results for the solutions [4,5,[7][8][9] and explorations of the attractor problem for the Ginzburg-Landau model [6,10]. Particularly interesting is the long-time behavior, i.e., the limiting graph of the state points moving in phase space as time changes and eventually converging, also known as attractors.…”

Section: Introductionmentioning

confidence: 99%

Attractors of Modified Coupled Ginzburg–Landau Model

Chen

Liu

2023

Mathematical Problems in Engineering

View full text Add to dashboard Cite

Bose–Einstein condensation is a gaseous, superfluid state of matter exhibited by bosons as they cool to near absolute zero, which was discovered as early as 1924 but was not experimentally realized until 1995. In 2006, Machida and Koyama developed the corresponding Ginzburg–Landau model for superfluid and Bose–Einstein condensation-spanning phenomena. We mainly consider the global attractor for the initial boundary value problem of the modified coupled Ginzburg–Landau equations, which come from the BCS-BEC crossover model. Combining Gronwall inequality, properties of the binomial function, with some suitable a priori estimates, we establish the existence of global attractors. The attractor results obtained in this paper can provide a strong theoretical basis for the experimental realization of the BCS-BEC spanning phenomenon, and the adopted research method can also serve as a reference for analysing other types of partial differential equation attractors.

show abstract

“…However the argument we use is quite different from the ones in the above mentioned articles which rests upon the use of Reshetnyak's Theorem for the Allen-Cahn part or on the constancy Theorem for varifolds for the GL part. We note that passing to the limit in the second inner variations for these problems was later shown to be possible without the assumption of convergence of energies in [16] and [9].…”

mentioning

confidence: 95%

Description of limiting vorticities for the magnetic 2D Ginzburg–Landau equations

Rodiac¹

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

Let Ω be a bounded open set in R 2 . The aim of this article is to describe the functions h in H 1 (Ω) and the Radon measures µ which satisfy −∆h + h = µ and div(T h ) = 0 in Ω, where T h is a 2 × 2 matrix given by (T h ) ij = 2∂ i h∂ j h − (|∇h| 2 + h 2 )δ ij for i, j = 1, 2. These equations arise as equilibrium conditions satisfied by limiting vorticities and limiting induced magnetic fields of solutions of the magnetic Ginzburg-Landau equations. This was shown by Sandier-Serfaty in [32,33]. Let us recall that they obtained that |∇h| is continuous in Ω. We prove that if x 0 in Ω is in the support of µ and is such that |∇h(x 0 )| = 0 then µ is absolutely continuous with respect to the 1D-Hausdorff measure restricted to a C 1 -curve near x 0 whereas µ ⌊{|∇h|=0} = h |{|∇h|=0} L 2 . We also prove that if Ω is smooth bounded and star-shaped and if h = 0 on ∂Ω then h ≡ 0 in Ω. This rules out the possibility of having critical points of the Ginzburg-Landau energy with a number of vortices much larger than the applied magnetic field h ex in that case.

show abstract

Instability of solutions to the Ginzburg–Landau equation on S and CPn

Cited by 7 publications

References 17 publications

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Attractors of Modified Coupled Ginzburg–Landau Model

Description of limiting vorticities for the magnetic 2D Ginzburg–Landau equations

Contact Info

Product

Resources

About