Nonlinear stability for the
Ginzburg–Landau equation

Julien Guillod

Sorbonne Université

October 30, 2018

Joint work with:
G. Schneider
P. Wittwer
D. Zimmermann

1. Introduction

Ginzburg–Landau equation

Real Ginzburg–Landau equation
$$\begin{gather*} \partial_{t}A=\partial_{x}^{2}A+A\bigl(1-|A|^{2}\bigr)\\ x\in\mathbb{R}, \qquad t>0, \qquad A(x,t)\in\mathbb{C} \end{gather*}$$
Family of periodic equilibria
$$A_q(x) = \sqrt{1-q^2} \e^{i q x} \quad \text{for} \quad q \in (-1,1)$$
Main question
Asymptotic stability of $A_q$ at fixed $q$
under small localized perturbations.

Perturbation

  • Perturbation in a multiplicative way: $$A(x,t) = A_q(x) \bigl(1 + V(x,t) \bigr)$$
  • Equation for the perturbation: $$\partial_{t}V=\partial_{x}^{2}V+2iq\partial_{x}V-(1-q^{2})\bigl(V+\overline{V}+2|V|^{2}+V^{2}(1+\overline{V})\bigr)$$
  • Linear equation solved by: $$V(x,t) = \hat{V} \e^{ikx} \e^{\lambda t} \quad \text{for} \quad \hat{V}\in\mathbb{C}$$
  • Eigenvalues: $$\lambda_{1/2}(k)=-(1-q^{2}-k^{2})\pm\sqrt{(1-q^{2})^{2}+4q^{2}k^{2}}$$

Spectral stability

Spectral stability (Eckhaus, 1965)
The Ginzburg–Landau equation is spectrally stable for $q^2\leq\frac{1}{3}$ and unstable for $q^2>\frac{1}{3}$. The value $q^2=\frac{1}{3}$ is the Eckhaus boundary.
Eckhaus stability

Linear diffusion

  • Eigenvalues at $k=0$: $$\begin{align*} \lambda_{1}(k) & =-\frac{1-3q^{2}}{1-q^{2}}k^{2}-\frac{2q^{4}}{(1-q^{2})^{3}}k^{4}+O(k^{6})\\ \lambda_{2}(k) & =-2(1-q^{2})+O(k^{2}) \end{align*}$$
  • Linear diffusion when $q^2<\frac{1}{3}$: $$\partial_{t}\hat{\varphi}=-k^{2}\hat{\varphi}\implies\partial_{t}\varphi=\partial_{x}^{2}\varphi\implies\varphi(x,t)=\frac{\varphi^{*}}{t^{1/2}}\e^{-\frac{x^{2}}{4t}}+O(t^{-1})$$
  • Linear diffusion when $q^2=\frac{1}{3}$: $$\partial_{t}\hat{\varphi}=-k^{4}\hat{\varphi}\implies\partial_{t}\varphi=-\partial_{x}^{4}\varphi\implies\varphi(x,t)=\frac{\varphi^{*}}{t^{1/4}}f\bigl(\tfrac{x^{4}}{t}\bigr)+O(t^{-1/2})$$

Nonlinear stability

Nonlinear stability below the Eckhaus boundary
(Collet-Eckmann-Hepstein, 1992)
(Bricmont-Kupiainen, 1992)
For $q^2<\frac{1}{3}$, the equilibrium $A_q$ is asymptotically stable under small perturbations of the initial data $\hat{V}_0 \in L^1 \cap L^\infty$. Moreover $$\Vert V(t) \Vert_{L^\infty} \leq C (1+t)^{-1/2}$$ and the leading order is determined by the linear diffusion.
Main question
What is happening at the Eckhaus boundary $q^2=\frac{1}{3}$?

2. Stability at the
Eckhaus boundary

Main result

Nonlinear stability at the Eckhaus boundary
(Guillod-Schneider-Wittwer-Zimmermann, 2018)
At $q^2=\frac{1}{3}$, the equilibrium $A_q$ is asymptotically stable under small perturbations of the initial data $\hat{V}_0 \in L^1 \cap L^\infty$. Moreover $$\Vert V(t) \Vert_{L^\infty} \leq C (1+t)^{-1/4}$$ and the leading order is determined by the self-similar solution of $$\partial_t \varphi = -\partial_x^4 \varphi - \partial_x \bigl( (\partial_x \varphi)^2 \bigr)$$ up to a rescaling in space and time.

Steps of the proof

  1. Separate real and imaginary parts and everything in Fourier: $$V = V_r + i V_i \qquad \text{and} \qquad \hat{\mathbf{v}} = (\hat{V}_r,\hat{V}_i)$$
  2. Diagonalization of the linear operator $\hat{L}$: $$\hat{L} \hat{\mathbf{v}}_j = \lambda_j \hat{\mathbf{v}}_j$$
  3. Projection on the critical mode with truncation: $$\hat{P}_1 \hat{\mathbf{v}} = \chi \bigl\langle \hat{\mathbf{v}}_1^*,\hat{\mathbf{v}} \bigr\rangle \hat{\mathbf{v}}_1 \qquad \hat{P}_2 = 1 - \hat{P}_1$$ where $\chi(k) = 1$ for $|k|\leq1$ and $\chi(k) = 0$ for $|k|>1$
  4. Linear estimates of $\hat{L}_j = \hat{P}_j \hat{L}$ in $L^1$ and $L^\infty$
  5. Implicit function theorem to deal with the marginal terms
  6. Nonlinear estimates in weighted $L^1$ and $L^\infty$
  7. Fixed-point argument

Power counting at linear level

  • Decays of the eigenvectors: $$\begin{align*} \hat{V}_{1}\sim\e^{-k^{4}t} & \implies V_{1}\sim\frac{1}{t^{1/4}}f\left(\frac{x}{t^{1/4}}\right) \sim t^{-1/4}\\ \hat{V}_{2}\sim\e^{-t\phantom{k^{4}}} & \implies V_{2}\sim\e^{-t} \end{align*}$$
  • Power counting: $$V_1 \sim t^{-1/4} \quad V_2 \sim \e^{-t} \quad \partial_x \sim t^{-1/4} \quad \partial_t \sim t^{-1}$$
  • Eigenvectors near $k=0$: $$\hat{V}_r \approx \hat{V}_2- \tfrac{\sqrt{3}}{2} ik \hat{V}_1 \quad \hat{V}_i \approx \hat{V}_1 - \tfrac{\sqrt{3}}{2} ik \hat{V}_2$$
  • $V_r$ is slaved to $V_i$: $$V_r \sim t^{-1/2} \qquad V_i \sim t^{-1/4}$$

Power counting at nonlinear level

  • Equations for $V_r$ and $V_i$: $$\begin{align*} \underbrace{\partial_{t}V_{r}}_{\sim t^{-3/2}} & =\underbrace{\partial_{x}^{2}V_{r}}_{\sim t^{-1}}-\underbrace{\tfrac{4}{3}V_{r}}_{\sim t^{-1/2}}-\sqrt{\tfrac{4}{3}}\underbrace{\partial_{x}V_{i}}_{\sim t^{-1/2}}-\tfrac{2}{3}\Bigl(\underbrace{3V_{r}^{2}}_{\sim t^{-1}}+\underbrace{V_{i}^{2}}_{\sim t^{-1/2}}+\underbrace{V_{r}^{3}}_{\sim t^{-3/2}}+\underbrace{V_{r}V_{i}^{2}}_{\sim t^{-1}}\Bigr)\\ \underbrace{\partial_{t}V_{i}}_{\sim t^{-5/4}} & =\underbrace{\partial_{x}^{2}V_{i}}_{\sim t^{-3/4}}+\sqrt{\tfrac{2}{3}}\underbrace{\partial_{x}V_{r}}_{\sim t^{-3/4}}-\tfrac{2}{3}\Bigl(\underbrace{2V_{r}V_{i}}_{\sim t^{-3/4}}+\underbrace{V_{r}^{2}V_{i}}_{\sim t^{-5/4}}+\underbrace{V_{i}^{3}}_{\sim t^{-3/4}}\Bigr) \end{align*}$$
  • Approximate resolution for $V_r$: \begin{align*} \underbrace{V_{r}}_{\sim t^{-1/2}} & =\underbrace{-\tfrac{\sqrt{3}}{2}\partial_{x}V_{i}-\tfrac{1}{2}V_{i}^{2}}_{\sim t^{-1/2}}\\ & \quad \underbrace{-\tfrac{1}{8}V_{i}^{4}-\tfrac{15}{8}(\partial_{x}V_{i})^{2}-\tfrac{\sqrt{3}}{2}V_{i}^{2}\partial_{x}V_{i}-\tfrac{3}{4}V_{i}\partial_{x}^{2}V_{i}-\tfrac{3\sqrt{3}}{8}\partial_{x}^{3}V_{i}}_{\sim t^{-1}}+O(t^{-3/2}) \end{align*}
  • Plugging into the equation for $V_i$: $$\underbrace{\partial_{t}V_{i}}_{\sim t^{-5/4}}=\underbrace{-\tfrac{3}{4}\partial_{x}^{4}V_{i}-\tfrac{3\sqrt{3}}{2}\partial_{x}\left((\partial_{x}V_{i})^{2}\right)}_{\sim t^{-5/4}}+O(t^{-5/4})$$