Brief History of NEEDS

The NEEDS Series of international meetings began in 1980. The original motivation was twofold: on one hand the ambition to contribute to the major development in theoretical and mathematical physics, in applied mathematics, in pure mathematics and in several applicative domains, associated with the discovery in the 70s of many integrable systems (classical and quantum) and of various techniques for their study and solution (the Soliton revolution); on the other, the intention to foster exchanges among scientists from the “West” and the “East” (including, in the latter case, the former Soviet Union). The second goal appeared particularly crucial, since the contributions to this research field by scientists in the former Soviet Union were absolutely outstanding, yet many of those who had provided and were producing such contributions had enormous difficulties in making contact with their colleagues living elsewhere who were working in the same area - and, of course, viceversa.

This series of meetings was quite successful in fulfilling both goals: the participation of scientists from the former Soviet Union, East Europe, China – together with that of scientists from Western Europe, the USA, Japan, Canada, Australia,... - has been substantial, already in the early years when travelling to the West was very difficult for many scientists, especially from the former Soviet Union; in several cases, the NEEDS meetings have provided the first opportunity for outstanding researchers from the Eastern countries to travel abroad, beyond the Iron Curtain. Through the 80s and the 90s, when the field of nonlinear integrable equations has continued to witness a major development - possibly the most interesting and vital development in theoretical and mathematical physics and applied mathematics of the second half of the 20th century - these international meetings have significantly fostered the blooming of this research area.

The interdisciplinary character of the theory of integrable equations can be traced back to its earliest days. Historically, there exist different and independent beginnings in fields that are as far apart as hydrodynamics (Russell's great wave of translation, 1834; Korteweg - de Vries equation, 1895), differential geometry (pseudo-spherical surfaces, 1839), mechanics (Kowalevski top, 1889) solid state physics (Frenkel-Kontorova model, 1938) and numerics (Fermi-Pasta-Ulam experiment, 1955). The breakthrough occurred in 1967, thanks to the discovery of the appropriate technique (a kind of nonlinear generalization of the Fourier transform) to treat certain classes of evolution partial differential equations and thereby to fully understand the role of a new nonlinear coherent structure, the soliton. From then on the discipline combined all formerly independent roots and became the interdisciplinary subject that it is today. The NEEDS meetings, from the beginning of the 1980s, played a crucial role in this process: they provided the global opportunity of discussion and interaction for a scientific community who was developing a new field of research. A turning point in the theory of integrable equations was the exciting discovery, in the late sixties, that several partial differential equations possess a variety of nontrivial exact solutions. These include not only solitary waves, known since the nineteenth century, but also solutions involving an arbitrary number of solitary waves of varying speeds and amplitudes undergoing collective collisions. The solitary waves are now called solitons to emphasize their particle-like character, since they leave the interaction region of space-time with the same shape they had upon entry. These phenomena are not just mathematical curiosities, bearing no relation to reality; quite on the contrary, the most significant soliton systems arise in the context of outstanding problems in applied science.

After these discoveries a large amount of research was devoted to the study of nonlinear processes; in this context many important nonlinear equations turned out to have a universal character, thereby explaining the remarkable fact that they were both integrable and widely applicable. The NEEDS meetings provided not only the opportunity for the world’s leading scientists and young researchers to meet in an informal setting for discussions at the highest level on recent developments in the field, they acted moreover as a catalyst for creating new synergistic contacts throughout Europe and the rest of the world. Through the 80s and the 90s one of the main lines of research was devoted to discover new integrable systems and to find appropriate methods to solve them (Inverse Scattering Transform, Direct Linearization Method, Dressing Method, Hirota Bilinear Technique,...). Soon afterwards it became important to classify known equations and to develop techniques to recognise whether a given nonlinear system is integrable or not (symmetry approach, tests of integrability such as the Painleve’- Kowalevski test, necessary conditions of integrability,...). The mathematical structures underlying integrability were investigated using analytical, algebraic and geometrical approaches. In this context, new mathematical objects such as recursion operators, bi-Hamiltonian structures, reduction groups, master symmetries were introduced. These outstanding scientific results have been discussed at the NEEDS workshops and some of them have indeed been obtained during these workshops themselves.

Despite the major progress, much remains to be done. Nonlinearity constitutes a central theme of current research in theoretical and mathematical physics, as well as in applied and pure mathematics; and it plays a crucial role in several applied fields.

Besides their main scientific goal, the NEEDS meetings face nowadays a new organizational challenge: to increase the participation of outstanding researchers from developing countries.