Nonlinear physics…

One of the most widely used concepts of physics is that of linearity or superposition. From a mathematical point of view, a linear system is described by equations—algebraic, differential, or integral—that are linear in the dependent variables. From a physical point of view, a system is linear or possesses a superposition principle if various modes of behavior of the system that arise from various causes or initial conditions can be added together algebraically to produce the same behavior that would result if the causes or initial conditions were added algebraically and applied to the system. For example, water waves of small amplitude superpose linearly on the surface of a pond, so that the ripples produced by two disturbances at the same time are just the sum of those produced by the separate disturbances. In most cases, the assumption of linearity is believed to be justified either as an exact representation of the situation or as a close approximation to the truth. Thus the water waves referred to above form a linear system only so long as the amplitude is small; otherwise, the shape of the resultant wave is different from that of the component waves, as is apparent when the wave is about to “break”. There are some systems, however, that are not even approximately linear, and yet are commonly described by linear equations for want of more effective mathematical methods.