Integrability of Nonlinear Systems

40 B. Grammaticos and A. Ramani
this was indeed possible [25]. The lattice with exponential interactions between
nearest neighbours that bears his name,
d 2 x n
dt 2 = e xn+1−xn + e xn−xn−1 , (2.21)
is indeed integrable, has an infinite number of conservation laws, possesses a Lax
pair and satisfies every condition for integrability.
The KdV equation and its integrability might have been an exception, but
this soon turned out not to be the case. Zakharov and Shabat [26] discovered
another integrable nonlinear PDE, the nonlinear Schrödinger equation: iu t +
u xx + κ|u| 2 u = 0. Ablowitz, Kaup, Newell and Segur [27], motivated by important
observations of Kruskal, solved the Sine-Gordon equation: u xt = sinu. Soon
the domain of integrable PDE’s was blossoming. It was not astonishing that the
Painlevé equations started making their appearance in connection with integrable
evolution equations. Thus, Ablowitz and Segur [28] showed that the IST techniques
could be used to linearize the Painlevé equations. The interesting point
is that this linearization was in terms of integrodifferential equations. This may
explain why this solution was not obtained earlier although the ‘Lax pairs’ for
the Painlevé equations were known since the works of Garnier and Schlesinger.
The appearance of the Painlevé equations as well as of other equations belonging
to the Painlevé-Gambier classification, as reductions of integrable PDE’s, was
not fortuitous. Ablowitz and Segur realized that integrability was the key word
and soon the ARS conjecture was proposed [29] (in collaboration with one of us,
A.R.): “Every ODE which arises as a reduction of a completely integrable PDE is
of Painlevé type (perhaps after a transformation of variables)”. This conjecture
provided a most useful integrability detector. In the years that followed the ARS
approach, which is very close in spirit to that of Kovalevskaya, turned out to be
a most powerful tool for the investigation of integrability. Several new integrable
systems were discovered through the singularity analysis approach.
Improvements to this approach were proposed. Weiss and collaborators managed
to treat PDE’s directly without the constraint of considering reductions
[30]. This was significant progress because, according to the ARS conjecture,
one had to treat every reduction before being able to assert anything about the
given PDE (and, of course, it is very difficult to make sure that every reduction
has been considered). Kruskal extended the singularity-analysis approach
in a nonlocal way through his poly-Painlevé method [31]. While, in the traditional
approach, one is concerned whether the solutions are multivalued, in
the poly-Painlevé approach the distinction is made between nondense and dense
multivaluedness. The former is considered to be compatible with integrability
while the latter is not. Apart from these innovative approaches, considerable
progress has been made in the ‘mainstream’ singularity-analysis domain but, a
major open question still remains: “what are the acceptable transformations?”
This, and the absence of a certain rigor, reduce the singularity analysis approach
to a good heuristic tool for the study of integrability. It is undoubtedly powerful