I have read two possible definitions. A nonlinear field is
A field taking values on a manifold.
A field whose equation is nonlinear.
What do you understand by a nonlinear field or a nonlinear theory?
A nonlinear field or a nonlinear theory is, well, a field or a theory that is not linear. There are two obstructions to something being linear: a equation is said to be linear if, whenever $\phi$ and $\psi$ are solutions to the equation and $a,b$ are constant scalars, so is $a\phi + b\psi$. So for the definition to make sense, you need (a) a way to ¨add two solutions¨ and (b) the statement that the ¨sum of two rescaled solutions is again a solution¨.
In other words, in the first case we cannot even meaningfully define superpositions; in the second case superpositions can be defined, but the principle of superposition for decomposing solutions can no longer be used.
The distinction drawn is mainly epistemological. For practical purposes there is little point in distinguishing between the two: the difference between the two cases is minute compared with their difference to linear theories.
You mean a "nonlinear sigma model", which is a field which takes value on a manifold. I guess some people call that a "nonlinear field". The more common usage is "A field whose equation is nonlinear".
First, what is a difference between linear and nonlinear physical processes? If a deviation of a system from an equilibrium is small, then the system is said to be linear. Formally, in this case, the system is described by a linear equation. A simple example of a linear system is a pendulum that performs small oscillations near the equilibrium (vertical) position.
Another definition of a linear system is that it is a system that "responds" linearly to an external perturbation. "Linear responce" means that the system reaction (signal) is proportional to the strength of perturbation. Usual examples are Hooke's law F= k*X and Ohm's law U= R*I, where k and R do not depend on X and I, respectively.
Now, if the deviation of a system from an equilibrium is large then the system (or a process) is nonlinear. For example, oscillation of a pendulum with large (> ~30 degrees) deviation is a nonlinear process. Alternative view, if the parameters of a system depend on the preturbation, then the system responce is nonlinear. In the examples above, if the elasticity coefficient k depends on X, or resistance R on I, then the system is nonlinear.
We can say that all processes in nature are nonlinear. However, if the deviation is small then the process can be considered (treated mathematically) as linear. This is just a reflection of a simple fact that any function $f(x)$ can be represented as a linear function if $x$ is small, $f(x) \approx f(0)+f'(0) x$.
"Field" is used when you describe an extended system, and deviation from equlibrium is different in different points of the system. Ususal examples are an oscillating string, electromagnetic field etc.
Finally, if you consider an extended system with large deviation from the equilibrium, you need to use nonlinear field theory.
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