💼 Case: consider the following ODE with real solutions

$$ \frac{\partial u}{\partial t}=D\frac{\partial ^2 u}{\partial x^2}+f(u) $$

where $f(u)$ is a nonlinear function (💃 example: $f(u)=ru-u^3$), $u=u(x,t)$ and $D>0$

with periodic boundary conditions

$$ \frac{\partial u}{\partial x} (0,t)=\frac{\partial u}{\partial x} (L,t)=0 $$

where $L$ is the length of the periodic domain

📖 Definitions:

• $D\frac{\partial^2 u}{\partial x^2}$ diffusive term with $x$ spatial-coor

• PDE’s require boundary conditions

🗒️ Note: this setup could easily be expanded to 3D

⚠️ Warning: when it is general I will put 🗺️ and when we are doing our example 💃

First solution

• 🗺️ Lets try to find a homogeneous solution in space and time meaning $u(x,y)=u_h$ with

  $$ \frac{\partial u_h}{\partial t} =\frac{\partial^2 u_h}{\partial x^2}=0 $$

  so $u_h$ must satisfy $f(u_h)=0$

• 💃 which allows us to write

  $$ ru_h-u_h^3 =0 \qquad \Rightarrow \qquad u_h=0 \;~ \text{or}\;~ u_h=\pm \sqrt{r} \;~ \text{for}\;~ r>0 $$

👩‍👩‍👦‍👦 Stability analysis

• 🗺️ Consider a perturbation $v(x,t)$ so that $u(x,t)=u_h+v(x,t)$

  $$ \begin{aligned} \frac{\partial v}{\partial t} & = D \frac{\partial^2 v}{\partial x^2} + f(u_h+v(x,t)) \\ &\simeq D\frac{\partial ^2 v}{\partial x^2} + f'(u_h)v(x,t)

  \end{aligned} $$

  • where we used the first order Taylor expansion

  This means our PDE becomes

  $$ \frac{\partial v}{\partial t} = D\frac{\partial ^2 v}{\partial x^2} +f'(u_h)v(x,t) $$

  where $f'(u_h)$ is a number since $f(u_h)=0$

• Substituting our ansatz $v(x,t)=e^{\lambda t} [e^{ikx}\pm e^{-ikx}]$ we get

  $$ \begin{aligned} \lambda v(x,t)&=-k^2 Dv(x,t)+f'(u_h)v(x,t) \\ \lambda &=-k^2 D+f'(u_h)

  \end{aligned} $$

  🗒️ Note: here $e^{\lambda t} \cos(kx)$, $e^{\lambda t}\sin(kx)$, $e^{\lambda t}e^{kx}$ and $e^{\lambda t}e^{-kx}$ are solutions

• The solution must satisfy the boundary conditions:

  $$ \frac{\partial v}{\partial x} (0,t)=\frac{\partial v}{\partial x} (L,t)=0 $$

  which can only be satisfied by $e^{\lambda t} \cos(kx)$ where $k=\frac{n\pi}{L}$ where $n\in \N$

  🗒️ Note: this puts a restriction on the eigenvalues

🦙 Case 1:

if we have $f'(u_h)>0$

• The modes near $k=0$ are unstable because their growth rate is

  $$ \lambda (k)=f'(u_h)-k^2 D>0 $$

• $k=0$ is the most unstable mode as it has the largest growth rate (wavelength → $\infin$)