Linear vs. Nonlinear Equations

The key feature that makes a PDE nonlinear is self-reference — the dependent variable appears multiplied or composed with itself.

If we unpack the advection equation

$$ \frac{\partial u}{\partial t} = -(\mathbf{u} \cdot \nabla)\mathbf{u}, $$

we see that the rate of change of $\mathbf{u}$ with respect to time depends on $\mathbf{u}$ itself and its gradient.
This creates a feedback loop: the velocity field modifies itself over time.

In contrast, consider the diffusion term

$$ \frac{\partial u}{\partial t} = \nu \nabla^2 u. $$

If we scale $u$ by a constant factor $a$, the entire equation still holds with the same scaling:

$$ L(a u) = a L(u), $$

which means the relationship is linear.

Superposition Principle

For linear differential equations, superposition holds.
If $u_1$ and $u_2$ are both solutions, then any linear combination

$$ \alpha u_1 + \beta u_2 $$

is also a valid solution.

Linear operators — such as gradient ($\nabla$), divergence ($\nabla \cdot$), Laplacian ($\nabla^2$), or constant multiplication — all satisfy this property:

$$ L(au_1 + bu_2) = aL(u_1) + bL(u_2) $$

For example:

$$ L(u) = \frac{\partial u}{\partial x} \Rightarrow L(au_1 + bu_2) = a\frac{\partial u_1}{\partial x} + b\frac{\partial u_2}{\partial x}. $$

Nonlinear Operator Example

Now let’s take a nonlinear operator:

$$ L(u) = u \frac{du}{dx}. $$

If this were linear, we should have

$$ L(au_1 + bu_2) = aL(u_1) + bL(u_2), $$

but let’s check:

$$ L(au_1 + bu_2) = (au_1 + bu_2) \frac{d(au_1 + bu_2)}{dx} = (au_1 + bu_2)(a\frac{du_1}{dx} + b\frac{du_2}{dx}). $$

Expanding:

$$ a^2u_1\frac{du_1}{dx} + ab\,u_1\frac{du_2}{dx} + ab\,u_2\frac{du_1}{dx} + b^2u_2\frac{du_2}{dx}. $$

Compare that to:

$$ aL(u_1) + bL(u_2) = a,u_1\frac{du_1}{dx} + b,u_2\frac{du_2}{dx}. $$

They are not equal — extra cross terms appear.
This violates superposition, confirming that $L(u) = u \frac{du}{dx}$ is a nonlinear operator.

TODO explain further stuff