دانلود کتاب What Is Integrability?

The idea of devoting a complete book to this topic was born at one of the
Workshops on Nonlinear and Turbulent Processes in Physics taking place
regularly in Kiev. With the exception of E. D. Siggia and N. Ercolani, all authors
of this volume were participants at the third of these workshops. All of them
were acquainted with each other and with each other's work. Yet it seemed to
be somewhat of a discovery that all of them were and are trying to understand
the same problem - the problem of integrability of dynamical systems, primarily
Hamiltonian ones with an infinite number of degrees of freedom. No doubt that
they (or to be more exact, we) were led to this by the logical process of scientific
evolution which often leads to independent, almost simultaneous discoveries.
Integrable, or, more accurately, exactly solvable equations are essential to
theoretical and mathematical physics. One could say that they constitute the
"mathematical nucleus" of theoretical physics whose goal is to describe real
classical or quantum systems. For example, the kinetic gas theory may be considered
to be a theory of a system which is trivially integrable: the system of classical
noninteracting particles. One of the main tasks of quantum electrodynamics is
the development of a theory of an integrable perturbed quantum system, namely,
noninteracting electromagnetic and electron-positron fields. Another well-known
example is that in solid-state physics where linear equations describe a system
of free oscillators representing atoms connected to each other by linear elastic
forces. On the other hand, nonlinear forces yield nonlinear equations for this
system.
Nonlinear integrable systems were discovered as early as the 18th century.
At that time only a few were known and with no real understanding of their
characteristics and solutions. Now, however, it is correct to state that it is impossible
to overestimate their importance in the development of all areas of science.
Among their applications is the integrable problem arising for the motion
of a particle in a central field, associated with atomic and nuclear physics. The
problem of a particle moving in the fields of two Coulomb centers is fundamental
to celestial mechanics and molecular physics. Also in molecular and nuclear
physics the integrability of the Euler problem for the motion of a heavy rigid body
is used. The development of the theory of gyroscopes would have been impossible
without the Lagrange solution of a symmetric top in a gravitational field. Only one
of the classical nonlinear integrable systems, namely, the Kovalewsky top, has
not yet found direct physical applications. But within mathematics this problem
the multiscale expansion method, the results, as derived for chosen models, are
found to be true on a more general level. This can be shown easily if the original
system possesses a Hamiltonian structure.
Several examples of this type may be found in the paper by V. E. Zakharov
and E. I. Schulman which is primarily devoted to a quite different question: how
can we determine whether a given system is integrable or not? This problem has
recently become more and more urgent, and is therefore thoroughly addressed
in this volume. There are essentially three approaches to solving it, all discussed
here. They originate from classical work initiated in the previous century. The
approach used in the paper by E. D. Siggia and N. Ercolani and in the contribution
by H. Flaschka, A. C. Newell and M. Tabor is essentially based on the classic
paper by S. Kovalewskaya discussing the integrability of a top in a gravitational
field.
Kovalewskaya observed that the majority of known integrable systems is
integrated in terms of elliptic and, consequently, meromorphic functions and thus
cannot have any movable critical points. This particular condition of the
nonexistence of movable critical points led subsequently to the integrable equation for
the Kovalewsky top. Kovalewskaya's idea was pursued further by Painleve. This
method of verifying the integrability of equations through an analysis of the
arrangement of critical points of their solutions in the complex plane is called the
Painleve test. In the contribution by Flaschka, Newell, and Tabor the Painleve
test is used on partial differential equations and is proved to be a powerful tool.
It allows not only to verify the integrability of systems but also, in the case of a
positive answer, it helps to find their Lax representation as a compatibility
condition (imposed on an overdetermined linear system), symmetries, and Hirota's
bilinear form.
One of the highlights of the third workshop in Kiev was the demonstration
(by A. C. Newell) of the power of the Painleve test as applied to the integrable
system found by A. V. Mikhailov and A. B. Shabat. It is worth noting that in spite
of all the advances of the Painleve test there is no reliable assurance for systems
not satisfying this test to be definitely nonintegrable. It should also be added
that further research is required to provide an even more solid mathematical
foundation for this quite useful and successful method.
The next paper in the volume is from A. V. Mikhailov, V. V. Sokolov, and
A. B. Shabat. They develop a symmetry approach originating from the famous
Sophus Lie. The question posed is under which conditions does a class of partial
differential equations admit a nontrivial group of local symmetry transformations
(depending on a finite number of derivatives). In the cases under consideration the
authors succeed in constructing a complete classification of systems possessing
symmetries. They also prove that when a few symmetries exist it follows that
there are actually an infinite number of them. It should be noted that in this paper
not only Hamiltonian but also dissipative systems are considered which cannot
be integrable in the classical sense but may be C-integrable, i.e., they may be
reduced to linear systems by changing variables.

The paper by V. E. Zakharov and E. I. Schulman is based on Poincar6's works.
Rather than choosing some differential equations and transforming them to their
Fourier representation where differential and pseudo-differential operators differ
only in coefficient functions, a Hamiltonian translationally invariant system is
taken as the starting point. The question posed is whether at least one additional
invariant motion for this system exists. It is shown that the existence of such an
integral implies rather important conclusions, discussed thoroughly in the paper.
They are formulated as restrictions on the perturbation series in the vicinity of
linearized (and trivially integrable) systems. In particular, the existence of an
additional invariant of motion implies the existence of an infinite number of
invariants. This result agrees with the paper by Mikhailov, Sokolov, and Shabat.
An extremely important result of this report is to make clear that the existence
of an infinite set of invariants of motion does not always mean integrability in
Liouville's sense. The set of integrals may be incomplete. Effective criteria for
identifying such cases are presented.
The contribution by A. P. Veselov is devoted to systems with discrete time and
thus has significant applications in physics. In this paper the particular concept
of integrability of systems of this type is defined. The contribution by V. A.
Marchenko devoted to the solution of the Cauchy problem of the KdV equation
(with nondecaying boundary conditions at infinity) lies to some degree outside the
general scope of this volume. It has been incorporated here, however, because
it seems to me that the inclusion of a classic paper of modern mathematical
physics can only increase the value and beauty of any presentation of associated
problems.
Moscow, August 1990
V.E. Zakharov