Stuart–Landau equation

${\displaystyle \frac{dA}{dt}=\sigma A-\frac{l}{2}A|A{|}^2.}$

where

• ${\displaystyle A=|A|e^{i\varphi }}$ is a complex quantity describing the disturbance,
• ${\displaystyle \sigma ={\sigma }_r+i{\sigma }_i}$ is the complex growth rate,
• ${\displaystyle l=l_r+i{l}_i}$ is a complex number and ${\displaystyle l_r}$ is the Landau constant.

The evolution of the actual disturbance is given by the real part of ${\displaystyle A(t)}$ i.e., by ${\displaystyle |A|\cos \varphi }$. Here the real part of the growth rate is taken to be positive, i.e., ${\displaystyle {\sigma }_r>0}$ because otherwise the system is stable in the linear sense, that is to say, for infinitesimal disturbances (${\displaystyle |A|}$ is a small number), the nonlinear term in the above equation is negligible in comparison to the other two terms in which case the amplitude grows in time only if ${\displaystyle {\sigma }_r>0}$. The Landau constant is also taken to be positive, ${\displaystyle l_r>0}$ because otherwise the amplitude will grow indefinitely (see below equations and the general solution in the next section). The Landau equation is the equation for the magnitude of the disturbance,

${\displaystyle \frac{d|A{|}^2}{dt}=2{\sigma }_r|A{|}^2-l_r|A{|}^4,}$

which can also be re-written as

${\displaystyle \frac{d|A|}{dt}={\sigma }_r|A|-\frac{l_r}{2}|A{|}^3.}$

Similarly, the equation for the phase is given by

${\displaystyle \frac{d\varphi }{dt}={\sigma }_i-\frac{l_i}{2}|A{|}^2.}$

For non-homogeneous systems, i.e., when ${\displaystyle A}$ depends on spatial coordinates, see Ginzburg–Landau equation. Due to the universality of the equation, the equation finds its application in many fields such as hydrodynamic stability, Belousov–Zhabotinsky reaction, etc.

General solution

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 ${\displaystyle {\sigma }_r\ne 0}$ 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 ${\displaystyle t\to \mathrm{\infty }}$, the magnitude of the disturbance ${\displaystyle |A|}$ approaches a constant value that is independent of its initial value, i.e., ${\displaystyle |A{|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\to (2{\sigma }_r/l_r{)}^{1/2}}$ 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 ${\displaystyle \varphi (t)}$ is given by

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

As ${\displaystyle t\gg 1/{\sigma }_r}$, the phase varies linearly with time, ${\displaystyle \varphi \sim ({\sigma }_i/{\sigma }_r-l_i/l_r){\sigma }_{r}t.}$

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 ${\displaystyle Re\le R{e}_{\mathrm{c}\mathrm{r}}}$ and unstable otherwise, where ${\displaystyle Re}$ is the Reynolds number and the ${\displaystyle R{e}_{\mathrm{c}\mathrm{r}}}$ is the critical Reynolds number; a familiar example that is applicable here is the critical Reynolds number, ${\displaystyle R{e}_{\mathrm{c}\mathrm{r}}\approx 50}$, corresponding to the transition to Kármán vortex street in the problem of flow past a cylinder. The growth rate ${\displaystyle {\sigma }_r}$ is negative when ${\displaystyle Re<R{e}_{\mathrm{c}\mathrm{r}}}$ and is positive when ${\displaystyle Re>R{e}_{\mathrm{c}\mathrm{r}}}$ and therefore in the neighbourhood ${\displaystyle Re\to R{e}_{\mathrm{c}\mathrm{r}}}$, it may written as ${\displaystyle {\sigma }_r=\text{const}.\times (Re-R{e}_{\mathrm{c}\mathrm{r}})}$ wherein the constant is positive. Thus, the limiting amplitude is given by

${\displaystyle |A{|}_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto \sqrt{Re-R{e}_{\mathrm{c}\mathrm{r}}}.}$