## Integrable Systems

### Integrable systems in 19th century

Euler and Lagrange established a mathematically satisfactory foundation of Newtonian mechanics. Hamilton developed analogous formulation of optics. Jacobi imported Hamilton's idea in mechanics, and eventually arrived at a new formulation (now referred to as the Hamilton-Jacobi formalism).

The Hamilton-Jacobi formalism was a crucial step towards Liouville's rigorous definition of the notion of ``integrability''. Moreover, Jacobi himself is also famous for having discovered (1839) that the geodesic motion on an ellipsoid is an integrable system (and solvable by hyperelliptic functions) .

 C.G. Jacobi Vorlesungen über Dynamik, Königsberg University 1842 - 1843 (edited by Clebsch and published from Reimer, Berlin, 1884)

Jacobi's research on dynamical started in 1837 and deeply motivated by Hamilton's formulation of optics based on the least action. One may say that Hamilton's work played the same role as Abel's work in Jacobi's researches of elliptic functions (click here for the history of elliptic functions).

 J. Liouville Note sur les équations de la dynamique, J. Math. Pures Appl. 20 (1855), 137-138.

Liouville's definition of integrability is based on the notion of ``first integrals'' (conserved quantities). In his definition, a (Hamiltonian) system is said to be integrable if it has sufficiently many first integrals in involution. The same idea has ever been inherited in many variants of the notion of integrability.

Liouville's definition of integrable Hamiltonian systems naturally covered many classical examples. Among them are the Kepler motion solved by Newton, harmonic oscillators solvable by trigonometric functions, the rigid bodies (``spinning tops'') of the Euler-Poinsot type and the Lagrange type, and Jacobi's example of geodesic motion on an ellipsoid.

The spinning tops and Jacobi's example were significant because they were known to be solvable by elliptic functions. Soon after the work of Liouville, C. Neumann discovered a new integrable Hamiltonian system, and pointed out that this system can be solved by hyperelliptic functions. That was the beginning of subsequent discoveries of many integrable systems.

 C. Neumann De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur, J. Reine Angew. Math. 56 (1859), 46-63. Neumann's Hamiltonian system was a prototype of integrable Hamiltonian systems discovered in the 19th century. They all are more or less connected with hyperelliptic functions. G. Kirchhoff Über die Bewegung eines Rotationskörper in einer Flüssigkeit, J. reine. angew. Math. 71 (1870), 237-262. A. Clebsch Über die Bewegung eines festen Körper in einer Flüssigkeit, Math. Ann. 3? (1871), 238-262. V. Steklov (Stekloff) Über die Bewegung eines festen Körper in einer Flüssigkeit, Math. Ann. 42 (1893), 273-374. Kirchhoff derived the equation of motion of a rigid body in an ideal fluid. This is a generalization of Langrange's rigid body (spinning top). Clebsch and Steklov studied some other cases in a similar setup. H. Weber Anwendung der Thetafunctionen zweiter Veränderlichen auf die Theorie der Bewegung eines festen Körper in einer Flüssigkeit, Math. Ann. 14 (1878), 143-206. Weber solved the motion of a rigid body in an ideal fluid in terms of genus-two hyperelliptic functions. S. Kowalevski Sur le problème de la rotation d'un corps solide autour d'un point fixe, Acta Math. 12 (1889), 177-232. Sur une propriété du système d'équations différentielles qui défunit la rotation d'un corps solide autour d'un point fixe, Acta Mat. 14 (1889), 81-93. Sophie Kowalevski (Sofia Kowalevskaya) discovered her famous rigid body as the third (and final) example of solvable rigid bodies, and further solved it in terms of genus-two hyperelliptic functions. P. Stäckel Über die Integralen der Hamilton-Jacobischen Differential Gleichung mittelst Separation der Variable (Habilitationsschrift, Halle, 1891) Stäckel presented a systematic classification of Hamiltoninan systems that can be solved by ``separation of variables''.

### Solitary waves and surface geometry

The aforementioned researches form the main stream of studies on integrable systems in the 19th century. They are, however, not the direct origin of the breakthrough in the seventies of the 20th century. The breakthrough originates in quite different sources in 19th century --- solitary waves and surface geometry. Studies on solitary waves (or ``solitons'' in the modern language) were initiated by J. Scott Russell when he observed a solitary wave in a canal in 1834. His report invoked a controversy. Sir G.B. Airy was suspicious, but Lord Rayleigh advocated Scott Russel's observation by calculation (1876). Finally, in 1895, D.J. Korteweg and G. de Vries proposed a nonlinear partial differential equation (the KdV equation) that well fits in the result of Lord Rayleigh. Note that, unlike the equation of motion of point particles and rigid bodies, the KdV equation is a partial differential equation. In other words, it has an infinite degrees of freedom. This is a common feature of many ``soliton equations'' discovered in the breakthrough in the seventies.

Surface geometry is also a subject started and developed in the 19th century. The sine-Gordon equation, which is also an important soliton equation, a partial differential equation that characterizes a special family of surfaces. Furthermore, members of this family are connected by the so called Bäcklund transformations. A good classical overview of this subject is provided in G. Darboux's book, Leçons sur la théorie générale des surfaces (Gauthier-Villars, Paris, 1895), in which one can even find the Toda lattice!

### Precursors early in the 20th century

The brilliant success of the search for integrable Hamiltonian systems was rapidly fading on the turn to the century. It was the beginning of a long blank that continued until the sixties. Routes towards the breakthrough after that blank, however, were already prepared in these days.

A route was opened by Paul Painlevé and the contemporaries. Actually, the idea of Painlevé was conceptually very close to Kovalevskaya's work on integrable rigid bodies. Kovalevskaya's method was to search for the cases where the solutions of the equation of motion, analytically continued to the complex plane, have no singularity other than poles. Painlevé did the same for second order nonlinear ordinary differential equations, slightly relaxing the conditions on possible singularities of solutions. (click here for the history of the Painlevé equations).

It should be noted that the celebrated six Painlevé equations (and presumably their various generalizations ) are NOT integrable in the sense that the aforementioned meaning. Rather, apart from special cases, they are not solvable by any abelian function, nor reducible to a linear ordinary differential functions; this is known as the ``irreducibility'' of solutions of the Painlevé equations.

Nevertheless Garnier, one of the successors of Painlevé's researches, pointed out a link with integrable systems:

 R. Garnier Sur une classe de systè:mes différentiels abéliens déduits de la théorie des équations linéaires, Rend. Circ. Mat. Palermo 43 (1918-19), 155-191. In this paper Garnier discovered an integrable system solvable by hyperelliptic functions as a byproduct of his researches on a generalization of Painlevés equations. This system was derived by taking a special ``limit' of his generalized Painlevé equations. Remarkably, his calculations implicitly uses a ``Lax equation'', which is a clue in the present approach to integrable systems.

Another route to the revival of integrable systems was discovered in a quite different direction by Drach, Burchnall and Chaundy:

 J. Drach Détermination des cas de réduction de l'équation différentielle d2/dx2 = phi(x) + h ]y, C.R. Acad. Sci. Paris 168 (1919), 47-50. Sur l'intégration par quadratures de l'équation d2/dx2 = [phi(x) + h ]y, C.R. Acad. Sci. Paris 168 (1919), 337-340. Drach considered the second order linear ordinary equation from a Galois-theoretic point of view, and discovered the case where the coefficient phi(x) and the solution y(x) are both somehow related to hyperelliptic abelian functions. J.L. Burchnall and T.W. Chaundy Commutative ordinary differential operators, Proc. London Math. Soc. (2) 21 (1922), 420-440. Burchnall and Chaundy considered the case where the same linear ordinary operator L as Drach's commutes with another ordinary differential operator M, namely satisfies the commutation relation [L,M] = 0. Remarkably, this leads to the same situation as Drach's work.

What Drach, Burchnall and Chaundy discovered are now called ``finite band'' (or ``finite gap'') operators, and known to be related to hyperelliptic solutions of the KdV equations. Their work is thus a precursor of algebro-geometric studies of soliton equations in the seventies.

### Integrable systems revived as soliton theory

In 1965, M. Zabusky and M.D. Kruskal reported the celebrated numerical computation of solutions of the KdV equation. Their research was motivated by the work of E. Fermi, J. Pasta and S. Ulam done in the fifties. Zabusky was re-examining this work, approximating the nonlinear lattice of Fermi et al. by a continuous system. The continuous system turned out to be described by the KdV equation. He and Kruskal thus attempted to solve the KdV equation numerically, and discovered numerical solutions in which many solitary waves coexisted. The numerical solutions revealed remarkable stability of the solitary waves, each of which behaved like a ``particle''. Because of this behavior, they called the solitary waves ``solitons''.

This observation stimulated theoretical researches, and soon led to the discovery, by Gardner, Greene, Kruskal and Miura, of exact ``multi-soliton'' solutions and the ``inverse scattering method'' that produces those solutions:

 C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095-1097. The clue of their discovery was a relation between the KdV equation and a second order linear ordinary differential equation (equivalently, the stationary Schrödinger equation in one spatial dimension). Using this relation, they could derive the multi-soliton solutions, an finite number of conserved quantities, etc. systematically.

Peter Lax soon proposed a more convenient and universal reformulation of the work of Gardner et al. This formulation is now called the Lax formalism:

 P.D. Lax Integrals of nonlinear equations of evolutions and solitary waves, Comm. Pure and Appl. Math. 21 (1968), 467-490.

The Toda lattice was also discovered in these days. This discovery, however, was done independently:

 M. Toda Vibration of a chain with a non-linear interaction, J. Phys. Soc. Japan 22 (1967), 431-436. Wave propagation in anharmonic lattice, J. Phys. Soc. Japan 23 (1967), 501-596. Being unaware of the work of Femi, Pasta and Ulam, Toda was looking for an exact model of heat conduction, and eventually arrived at his exponential lattice. His method was to go back the usual way ``from equations to solutions'': He started from a nonlinear wave given by an elliptic function, attempted to derive a nonlinear equation to be satisfied by the nonlinear wave, and eventually arrived at his exponential lattice.

Within less than a decade, the method of Gardner et al. and Lax was extended to many other ``soliton equations'':

 M.J. Ablowitz, D.J. Kaup, A.C. Newell, and H. Segur The inverse scattering transform --- Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 249. Ablowitz et al. extended the inverse scattering method to a matrix system. A wide range of new equations, such as the modified KdV equation, the nonlinear Schrödinger equation, and the classical sine-Gordon equations, thus turned out to be solvable by the inverse scattering method. H. Flaschka The Toda lattice, Phys. Rev. B9 (1974), 1924; The Toda lattice II, Progr. Theor. Phys. 51 (1974), 703-716. S.V. Manakov Complete integrability and stochastization of discrete dynamical systems, Soviet Phys. JETP 40 (1974), 269-274. Flaschka and Manakov independently developed a Lax formalism of the Toda lattice. The inverse scattering method was thus further extended to a spatially discrete system. V.E. Zakharov and A.B. Shabat A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem, I., Funct. Anal. Appl. 8 (1974), 226-235. Zakharov and Shabat extended the inverse scattering method to systems with two spatial-dimensions. A typical example was the Kadomtsev-Petviashvili (KP) equation.

While the inverse scattering method was thus refined, several new techniques were also invented in the seventies. In particular, the direct method (bilinearization) of R. Hirota, the algebro-geometric method of B.A. Durovin, V.B. Matveev, S.P. Novikov and I.M. Krichever, and the group-theoretical (or Lie-algebraic) method due to M. Adler, B. Kostant, W.W. Symes, A.G. Reyman and M.A. Semenov-Tian-Shansky emerged in the second half of the seventies, and grew up to the mainstream of the progress through the beginning of the eighties.

### Yet another route --- Calogero systems

The world of integrable Hamiltonian systems of the 19th century, too, came back soon after the birth of the soliton theory, with a novel family of integrable systems --- the Calogero systems. The Calogero systems are Hamiltonian systems of interacting particle on a line. Calogero discovered the simplest case of these systems as a quantum integrable system:

 F. Calogero Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971), 419-436. This paper deals with the quantum integrability of the system with two-body potential 1/x2.

Sutherland soon proposed another version of Calogero's systems:

 B. Sutherland Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev. A5 (1972), 1372-1376. Sutherland considered a variant of Calogero's system with two-body potential 1/sin2x. This system is nowadays called the Sutherland system.

Calogero conjectured the integrability of classical analogues of these systems, and presented a partial answer. Moser solved this probelm by constructing a Lax formalism of these systems.

 F. Calogero and C. Marchioro Exact solution of a one-dimensional three-body scattering problem with two-body and/or three body inverse square potential, J. Math. Phys. 15 (1974), 1425-1430. Calogero and Marchioro proved the classical integrability of the three-body case by explicit calculation. J. Moser Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197-220. Moser constructed a Lax representation for both the Calogero and Sutherland systems, and proved the classical integrability (at least for the Calogero system). In fact, this paper is also known to be one of the papers in which Moser presented his method for sovling the non-periodic finite Toda (and Kac-van Moerbeke) system using Stiektjes' theory of continued fraction.

Moser's work suggested that the Lax formalism, originally developed in soliton theory, would be also useful for finite-dimensional integrable systems. Moser's idea was further extended by Olshanetsky and Perelomov:

 M.A. Olshanetsky and A.M. Perelomov Completely Integrable Hamiltonian Systems Connected with Semisimple Lie Algebras, Inventions Math. 37 (1976), 93-108.

This work is also one of the earliest attempts of the Lie algebraic methods for constructing (and solving) integrable systems. A similar work was done by Bogoyavlensky for the (finite) Toda lattice (O. I. Bogoyavlensky, On perturbations of the Toda lattice, Commun. Math. Phys. 51 (1976), 201-209).

After the aforementioned work, Moser turned to classical integrable systems of the 19th century, i.e., Jacobi's geodesic flows on an ellipsoid, C. Neumann's system, etc., and demonstrated that these systems, too, can be treated in the Lax formalism:

 J. Moser Geometry of quadrics and spectral theory, in ``Chern Symposium, Berkeley, 1979'', Springer-Verlag, 1980, pp. 147-188)