Integrability of Nonlinear Systems

32 B. Grammaticos and A. Ramani
useful unless one defines more precisely what is meant by “function”. We shall
come back to this point in the next section, but let us point out here that the
most important feature that characterizes a function is its singlevaluedness.
Integrability is a rare phenomenon. The typical dynamical system is nonintegrable.
How does one study such generic systems? Until very recently this was
practically impossible. Except for some very particular cases, the only way to
study a generic dynamical system was through the use of computers: “without
computers you cannot visualize randomness in real systems” [3]. On the other
hand, integrable systems can be studied in much greater detail than generic, nonintegrable
ones. Algebraic and analytic methods are operative here. However,
an arbitrarily small change in an integrable equation can destroy its integrability.
Still, some structure of the integrable system persists under (not too large)
perturbations. Near-integrable systems can be studied through special analytical
techniques [4] which allow one to dicover the qualitative behaviour of the
system.
Given that integrability is structurally unstable, one may worry as to the
pertinence of integrable systems in the description of physical phenomena. Segur
points out [5] that “if a given problem can be approximated by an integrable
model then it is likely that it can also be approximated to the same accuracy
by a model that is not integrable”. Thus the worry is that results that depend
fundamentally on integrability cannot be very important. Still, this is not the
feeling shared by the integrability community. Calogero offers a basis for this optimistic
attitude [6]. He has pointed out that some integrable partial differential
equations (PDE) are both “universal” and “widely applicable”. His argument is
that a limiting (usually asymptotic) procedure applied to a large class of nonlinear
PDE’s leads, to the same limit, to a universal equation which is integrable.
If this limiting procedure is physically reasonable this guarantees the wide applicability
of the integrable equation. More recently, Fokas has shown that the
use of nonlinear transforms allows one to extend the class of universal integrable
equations [7]. Thus one expects integrable equations to play a non negligible role
in the description of realistic physical systems, even though they are expected
to describe some limiting, asymptotic situation.
Novikov [3] takes this argument one step further: “Physicists and mathematical
philosophers of science for the most part do not believe that the laws
of nature are to be expressed by arbitrarily chosen, general equations. Most of
them somehow believe de facto in a higher reason”. Indeed one is amazed at
the simple mathematical form of physical laws. What is still more amazing is
that the values of the fundamental physical constants are so finelly tuned as to
make the appearance of sentient life in the universe possible. This means not
only that the laws are simple but that the initial conditions of the universe are
appropriate.
At this point we should be able to answer the question of the title of this section.
However, in order to make things even clearer, let us present some definitions,
proposed by Segur [5], that will help our argumentation. According to Segur: