Stuart–Landau equation

The Landau equation is linear when it is written for the dependent variable ${\displaystyle |A|^{-2}}$,

${\displaystyle {\frac {d|A|^{-2}}{dt}}+2\sigma _{r}|A|^{-2}=l_{r}.}$

The general solution for ${\displaystyle \sigma _{r}\neq 0}$ of the above equation is

${\displaystyle |A(t)|^{-2}={\frac {l_{r}}{2\sigma _{r}}}+\left(|A(0)|^{-2}-{\frac {l_{r}}{2\sigma _{r}}}\right)e^{-2\sigma _{r}t}.}$

As ${\displaystyle t\rightarrow \infty }$, the magnitude of the disturbance ${\displaystyle |A|}$ approaches a constant value that is independent of its initial value, i.e., ${\displaystyle |A|_{\mathrm {max} }\rightarrow (2\sigma _{r}/l_{r})^{1/2}}$ when ${\displaystyle t\gg 1/\sigma _{r}}$. The above solution implies that ${\displaystyle |A|}$ does not have a real solution if ${\displaystyle l_{r}<0}$ and ${\displaystyle \sigma _{r}>0}$. The associated solution for the phase function ${\displaystyle \phi (t)}$ is given by

${\displaystyle \phi (t)-\phi (0)=\sigma _{i}t-{\frac {l_{i}}{2l_{r}}}\ln \left[1+{\frac {|A(0)|^{2}l_{r}}{2\sigma _{r}}}(e^{2\sigma _{r}t}-1)\right].}$

As ${\displaystyle t\gg 1/\sigma _{r}}$, the phase varies linearly with time, ${\displaystyle \phi \sim (\sigma _{i}/\sigma _{r}-l_{i}/l_{r})\sigma _{r}t.}$

It is instructive to consider a hydrodynamic stability case where it is found that, according to the linear stability analysis, the flow is stable when ${\displaystyle Re\leq Re_{\mathrm {cr} }}$ and unstable otherwise, where ${\displaystyle Re}$ is the Reynolds number and the ${\displaystyle Re_{\mathrm {cr} }}$ is the critical Reynolds number; a familiar example that is applicable here is the critical Reynolds number, ${\displaystyle Re_{\mathrm {cr} }\approx 50}$, corresponding to the transition to Kármán vortex street in the problem of flow past a cylinder.[9][10] The growth rate ${\displaystyle \sigma _{r}}$ is negative when ${\displaystyle Re<Re_{\mathrm {cr} }}$ and is positive when ${\displaystyle Re>Re_{\mathrm {cr} }}$ and therefore in the neighbourhood ${\displaystyle Re\rightarrow Re_{\mathrm {cr} }}$, it may written as ${\displaystyle \sigma _{r}={\text{const}}.\times (Re-Re_{\mathrm {cr} })}$ wherein the constant is positive. Thus, the limiting amplitude is given by

${\displaystyle |A|_{\mathrm {max} }\propto {\sqrt {Re-Re_{\mathrm {cr} }}}.}$

When the Landau constant is negative, ${\displaystyle l_{r}<0}$, we must include a negative term of higher order to arrest the unbounded increase of the perturbation. In this case, the Landau equation becomes[11]

${\displaystyle {\frac {d|A|^{2}}{dt}}=2\sigma _{r}|A|^{2}-l_{r}|A|^{4}-\beta _{r}|A|^{6},\quad \beta _{r}>0.}$

The limiting amplitude then becomes

${\displaystyle |A|_{\mathrm {max} }\rightarrow {\frac {|l_{r}|}{2\beta _{r}}}\pm {\sqrt {{\frac {l_{r}^{2}}{4\beta _{r}^{2}}}+{\frac {2|l_{r}|\sigma _{r}}{\beta _{r}}}}},\quad {\text{as}}\quad t\gg 1/\sigma _{r}}$

where the plus sign corresponds to the stable branch and the minus sign to the unstable branch. There exists a value of a critical value ${\displaystyle Re_{\mathrm {cr} }'}$ where the above two roots are equal (${\displaystyle \sigma _{r}=-|l_{r}|/8\beta _{r}}$) such that ${\displaystyle Re_{\mathrm {cr} }'<Re_{\mathrm {cr} }}$, indicating that the flow in the region ${\displaystyle Re_{\mathrm {cr} }'<Re<Re_{\mathrm {cr} }}$ is metastable, that is to say, in the metastable region, the flow is stable to infinitesimal perturbations, but not to finite amplitude perturbations.

• Landau's phase transition theory
• Ginzburg–Landau theory