About

The term non-linear or nonlinear has several definitions but is generally used to describe a system that cannot be approximated by a superposition principle or perturbative approach.

Background

In many cases, a linear system is one in which one can assume that things will add following the superposition principle. One of the consequences of a system which obeys such a principle is that one can use a first order perturbation to approximate the fluctuations in the system as $Q \approx Q_{o} + \delta Q$, where $Q_{o}$ is a quasi-static term and $\delta Q$ a fluctuation term. An example of a nonlinear system is one where $Q \neq Q_{o} + \delta Q$.

However, there are additional complications. For instance, a linear plane wave can be described by the relationship $$ Q\left( \mathbf{x}, t \right) = Q_{o} \ e^{ i \left( \mathbf{k} \cdot \mathbf{x} - \omega t \right) } $$ where $Q_{o}$ is a constant amplitude, $\mathbf{k}$ is the wavenumber, $\omega$ is the angular frequency, $\mathbf{x}$ is the spatial position, and $t$ is the time. If the wave's amplitude was no longer constant, e.g., $Q_{o} \rightarrow Q_{o}\left( \mathbf{k}, \omega \right)$, the wave would now be considered to be a nonlinear wave regardless of the magnitude of $Q_{o}$.

Examples of nonlinear systems

Below are some examples of nonlinear systems, but they are certainly not all-encompassing.

For an example, let $Q_{o}$ represent the wave height (as in a water wave). If the magnitude of $Q_{o}$ becomes large enough that it exceeds the magnitude of the depth of the water in which it propagates, it would no longer be considered a linear wave (the phenomena of a wave dispersion induced by propagating into shallow water is called wave shoaling).

In the Navier-Stokes eqautions, a system where the $\mathbf{u} \cdot \nabla \mathbf{u}$ term is non-negligible is also considered nonlinear. Note that the $\mathbf{u} \cdot \nabla \mathbf{u}$ term is responsible for wave steepening in shock waves and breaking water waves.

In a plasma, a nonlinear wave is one that satisfies either of the following: $$ \frac{ \delta B }{ B_{o} } \gtrsim 0.1 \\ \text{ or } \\ \frac{ \varepsilon_{o} \delta E^{2} }{ 2 \ n_{e} \ k_{B} \ T_{e} } \gtrsim 0.1 $$ where $B$($E$) is the magnetic(electric) field (where $Q_{o}$ is a quasi-static term and $\delta Q$ a fluctuation term as above), $\varepsilon_{o}$ is the permittivity of free space, $n_{e}$ is the electron number density, $k_{B}$ is the Boltzmann constant, and $T_{e}$ is the average electron temperature. The reason these limits are considered nonlinear is that in both cases the fluctuation would have the potential to significantly alter the medium in which it exists.

When to use

Nonlinear systems are many but some obvious cases apply like , , wave breaking phenomena, , non-perturbative theories, etc.

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