国际流体力学研究期刊
DOI: 10.1615/InterJFluidMechRes.v22.i56.40
pages 134154
Nonlinearity. New Problems, New Opportunities
G. G. Malinetskii
A. B. Potapov
ABSTRACT
Work done on mathematical modeling and applied mathematics makes it currently possible to solve a large number of applied problems. However, it is at times useful to look around and ask several general questions. What development in the discipline under study changes the investigators' view of nature and affects their world outlook? Which new ideas have recently appeared? What would be interesting to tell scientists working in related fields, from a bird's eye view of one's own field?
This is particularly important for approaches that arose relatively recently and are currently developing at a high rate. Such a discussion will assist in understanding which hopes that were initially pinned on the new approach have justified themselves, and what can be expected in the future.
Nonlinear science is precisely such an approach. One of its purposes is to find universal relationships governing the behavior of nonlinear systems. These relationships manifest themselves in the general nature of mathematical description of a very large number of objects in physics, biology, technology, chemistry and most likely also in social sciences. Nonlinear science attempts to see a new, higher level of the unity of nature behind the enormous number of equations, models and problems.
Such a program of study is highly attractive. If general relationships indeed exist in the nonlinear world, then they can be found and investigated employing elementary models.
The hopes that were pinned on nonlinear science were discussed almost ten years ago by one of the present authors (G. M.) with his coworkers [1]. Many of the problems listed in that article have found their way from scientific journals into books and are on the way to textbooks. A survey of many relatively recent results, can be found, among others, in [2, 3]. It is hence sensible to single out ever new ideas and discuss the future of nonlinear science.
