How is it possible that linear theories, for example maxwells equations or the schroedinger equation, produce nonlinear physics?
One way of getting non-linear results from linear equations is to have interactions that are spread over time and/or space. This will require integration of the linear interaction, which will result in non-linear behavior. For example, if you integrate dx/dt = kx, you get $x=exp(kt)$, which is clearly non-linear in both k and t.
In another example, take Hooke's law $F = -k x$ (F = Force, k = constant, x = displacement from equilibrium). You can generalize it to continuous media with stress = constant X strain. This is clearly linear. However, if you calculate the deflection of a beam of a material of uniform composition and cross-section, you get non-linear effects, for example with deflection with a given force depending non-linearly on the length of the beam. This is because deformation at one point in the beam influences deformation at another point as static equilibrium is achieved. This coupling between different parts of the beam causes the non-linear term to arise.