If there would be God ~ then what a staff-captain am I? ~ said one of the characters in a vel by Dostoevskii. In a similar way we can exclaim: If there would be nlinearity ~ than what physics would that be'? . Really, the most interesting and exciting effects are described by n- linear equations, and vanish in the linear approximation. For example, the general theory of relativity by A.Einstein comes to mind first - one of the most beautiful physical theories, which is in fact essentially nlinear. Next, the phase transitions crystal ~ liquid and liquid ~ gas are due to the anhar- monicity of inter-particle interactions, to dissociation and infinite motion. Similarly, transitions into the superconducting state or the superftuid would be impossible with purely harmonic interaction potentials. Ather bril- liant achievement in nlinear physics was the construction of a laser and the subsequent development of nlinear optics. The latter describes the in- teraction of the matter with light of super-high intensity, when multi-quanta intra-molecular transitions become essential. Last, we should te here the very beautiful mathematical theory ~ the theory of catastrophes. Its subject is the study of invariant general properties of multi-dimensional surfaces in the vicinity of bifurcation points with respect to continuous transformations.
Alexandr A. Ovchinnikov, K. A. Pronin, Nikolai S. Erikhman