Kuramoto-Sivashinsky Equation
Definition
$$ {\frac{ \partial u }{ \partial t }} + {\frac{ \partial }{ \partial x }} \left( {\frac{ 1 }{ 2 }} u^{2} \right) + {\frac{ \partial^{2} u }{ \partial x^{2} }} + {\frac{ \partial^{4} u }{ \partial x^{4} }} = 0 $$
The above partial differential equation is called the Kuramoto–Sivashinsky equation. Another form, obtained by expanding the nonlinear term, can be written as follows.
$$ {\frac{ \partial u }{ \partial t }} + u {\frac{ \partial u }{ \partial x }} + {\frac{ \partial^{2} u }{ \partial x^{2} }} + {\frac{ \partial^{4} u }{ \partial x^{4} }} = 0 $$
Description
From a physical viewpoint, including derivatives of $t$ and $x$ yields the Burgers equation; the second-derivative term in $x$ represents viscosity, while the fourth-derivative term in $x$ reflects reaction–diffusion.
The following shows the equation solved using a spectral method.
In $1$-dimensional space, $u = u \left( t ; x \right)$ that follows the above governing equation forms a dynamical system exhibiting characteristic patterns□ref.□.
It is not an exaggeration to say that, in the field of dynamical systems when chaotic systems are discussed, the Lorenz attractor plays that role for ordinary differential equations while the Kuramoto–Sivashinsky equation does so for partial differential equations. It is that fundamental and widely used a system.