The symmetries of solitons

Palais, Richard

doi:10.1090/S0273-0979-97-00732-5

The symmetries of solitons
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by Richard S. Palais PDF

Bull. Amer. Math. Soc. 34 (1997), 339-403

Abstract:

In this article we will retrace one of the great mathematical adventures of this century—the discovery of the soliton and the gradual explanation of its remarkable properties in terms of hidden symmetries. We will take an historical approach, starting with a famous numerical experiment carried out by Fermi, Pasta, and Ulam on one of the first electronic computers, and with Zabusky and Kruskal’s insightful explanation of the surprising results of that experiment (and of a follow-up experiment of their own) in terms of a new concept they called “solitons”. Solitons however raised even more questions than they answered. In particular, the evolution equations that govern solitons were found to be Hamiltonian and have infinitely many conserved quantities, pointing to the existence of many non-obvious symmetries. We will cover next the elegant approach to solitons in terms of the Inverse Scattering Transform and Lax Pairs, and finally explain how those ideas led step-by-step to the discovery that Loop Groups, acting by “Dressing Transformations”, give a conceptually satisfying explanation of the secret soliton symmetries.

References

M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, vol. 149, Cambridge University Press, Cambridge, 1991. MR 1149378, DOI 10.1017/CBO9780511623998
M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Method for solving the sine-Gordon equation, Phys. Rev. Lett. 30 (1973), 1262–1264. MR 406175, DOI 10.1103/PhysRevLett.30.1262
Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), no. 4, 249–315. MR 450815, DOI 10.1002/sapm1974534249
Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141
M. Adler, On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations, Invent. Math. 50 (1978/79), no. 3, 219–248. MR 520927, DOI 10.1007/BF01410079
M. Adler and P. van Moerbeke, Completely integrable systems, Euclidean Lie algebras, and curves, Adv. in Math. 38 (1980), no. 3, 267–317. MR 597729, DOI 10.1016/0001-8708(80)90007-9
S. Bergmann and J. Marcinkiewicz, Sur les fonctions analytiques de deux variables complexes, Fund. Math. 33 (1939), 75–94 (French). MR 57, DOI 10.4064/fm-33-1-75-94
V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR 0232910
Michèle Audin, Spinning tops, Cambridge Studies in Advanced Mathematics, vol. 51, Cambridge University Press, Cambridge, 1996. A course on integrable systems. MR 1409362
R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math. 37 (1984), no. 1, 39–90. MR 728266, DOI 10.1002/cpa.3160370105
R. Beals and R. R. Coifman, Inverse scattering and evolution equations, Comm. Pure Appl. Math. 38 (1985), no. 1, 29–42. MR 768103, DOI 10.1002/cpa.3160380103
Richard Beals and R. R. Coifman, Linear spectral problems, nonlinear equations and the $\overline \partial$-method, Inverse Problems 5 (1989), no. 2, 87–130. MR 991913, DOI 10.1088/0266-5611/5/2/002
Richard Beals and D. H. Sattinger, On the complete integrability of completely integrable systems, Comm. Math. Phys. 138 (1991), no. 3, 409–436. MR 1110449, DOI 10.1007/BF02102035
R. Beals, P. Deift, and X. Zhou, The inverse scattering transform on the line, Important developments in soliton theory, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1993, pp. 7–32. MR 1280467
Birkhoff, G.D., Proof of the Ergodic Theorem, Proc. Nat. Acad. Sci. USA 17 (1931), 650–660.
J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A 278 (1975), no. 1287, 555–601. MR 385355, DOI 10.1098/rsta.1975.0035
A. S. Budagov, A completely integrable model of classical field theory with nontrivial particle interaction in two-dimensional space-time, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 77 (1978), 24–56, 229 (Russian). Questions in quantum field theory and statistical physics. MR 541693
Robin K. Bullough and Philip J. Caudrey (eds.), Solitons, Topics in Current Physics, vol. 17, Springer-Verlag, Berlin-New York, 1980. MR 625877
Gaston Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. Troisième partie, Chelsea Publishing Co., Bronx, N.Y., 1972. Lignes géodésiques et courbure géodésique. Paramètres différentiels. Déformation des surfaces; Réimpression de la première édition de 1894. MR 0396213
Da Rios, Rend. Circ. Mat. Palermo 22 (1906), 117-135.
P. G. Drazin and R. S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1989. MR 985322, DOI 10.1017/CBO9781139172059
V. G. Drinfel′d, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 285–287 (Russian). MR 688240
V. G. Drinfel′d and V. V. Sokolov, Equations of Korteweg-de Vries type, and simple Lie algebras, Dokl. Akad. Nauk SSSR 258 (1981), no. 1, 11–16 (Russian). MR 615463
Fermi, E., Beweis dass ein mechanisches Normalsysteme im Allgemeinen quasi-ergodisch ist, Phys, Zeit. 24 (1923), 261–265.
L. D. Faddeev and L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. Translated from the Russian by A. G. Reyman [A. G. Reĭman]. MR 905674, DOI 10.1007/978-3-540-69969-9
Alan C. Newell (ed.), Nonlinear wave motion, Lectures in Applied Mathematics, Vol. 15, American Mathematical Society, Providence, R.I., 1974. MR 0336014
H. Flaschka, A. C. Newell, and T. Ratiu, Kac-Moody Lie algebras and soliton equations. II. Lax equations associated with $A_{1}^{(1)}$, Phys. D 9 (1983), no. 3, 300–323. MR 732574, DOI 10.1016/0167-2789(83)90274-9
Flaschka, H., Newell, A.C., Ratiu, T., Kac-Moody Lie algebras and soliton equations, IV. Lax equations associated with $A^{(1)}_{1}$, Physica 9D (1983), 333–345.
A. G. Reĭman, M. A. Semenov-Tjan-Šanskiĭ, and I. E. Frenkel′, Graded Lie algebras and completely integrable dynamical systems, Dokl. Akad. Nauk SSSR 247 (1979), no. 4, 802–805 (Russian). MR 553832
Clifford S. Gardner, Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system, J. Mathematical Phys. 12 (1971), 1548–1551. MR 286402, DOI 10.1063/1.1665772
Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M., Method for solving the Korteweg-de Vries equation, Physics Rev. Lett. 19 (1967), 1095–1097.
I. M. Gel′fand and L. A. Dikiĭ, Fractional powers of operators, and Hamiltonian systems, Funkcional. Anal. i Priložen. 10 (1976), no. 4, 13–29 (Russian). MR 0433508
I. M. Gel′fand and I. Ja. Dorfman, Hamiltonian operators and algebraic structures associated with them, Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 13–30, 96 (Russian). MR 554407
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Hasimoto, H., Motion of a vortex filament and its relation to elastic, J. Phys. Soc. Japan 31 (1971), 293–295.
Hasimoto, H., Kodama, Y., Solitons in Optical Communications, Clarendon Press, Oxford, 1995.
Tosio Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, Adv. Math. Suppl. Stud., vol. 8, Academic Press, New York, 1983, pp. 93–128. MR 759907
Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
Korteweg, D.J., de Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. Ser. 5 39 (1895), 422–443.
Bertram Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), no. 3, 195–338. MR 550790, DOI 10.1016/0001-8708(79)90057-4
Kay, B., Moses, H.E., The determination of the scattering potential from the spectral measure function, III, Nuovo Cim. 3 (1956), 276–304.
Klein, F., Sommerfeld A., Theorie des Kreisels, Teubner, Liepzig, 1897.
George L. Lamb Jr., Elements of soliton theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1980. MR 591458
Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 235310, DOI 10.1002/cpa.3160210503
Peter D. Lax, Periodic solutions of the KdV equations, Nonlinear wave motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972) Lectures in Appl. Math., Vol. 15, Amer. Math. Soc., Providence, R.I., 1974, pp. 85–96. MR 0344645
Lax, P.D., Outline of a theory of the KdV equation, in Recent Mathematical Methods in Nonlinear Wave Propogation, Lecture Notes in Math., vol. 1640, Springer-Verlag, Berlin and New York, 1996, pp. 70–102.
Luther, G.G., Alber, M.S., Nonlinear Waves, Nonlinear Optics, and Your Communications Future, in Nonlinear Science Today, Springer-Verlag New York, Inc., 1997.
Marchenko,V.A., On the reconstruction of the potential energy from phases of the scattered waves, Dokl. Akad. Nauk SSSR 104 (1955), 695–698.
Alan C. Newell, Solitons in mathematics and physics, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 48, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 847245, DOI 10.1137/1.9781611970227
S. Novikov, S. V. Manakov, L. P. Pitaevskiĭ, and V. E. Zakharov, Theory of solitons, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984. The inverse scattering method; Translated from the Russian. MR 779467
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Richard S. Palais and Chuu-Lian Terng, Critical point theory and submanifold geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, Berlin, 1988. MR 972503, DOI 10.1007/BFb0087442
A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. Vol. I, Birkhäuser Verlag, Basel, 1990. Translated from the Russian by A. G. Reyman [A. G. Reĭman]. MR 1048350, DOI 10.1007/978-3-0348-9257-5
Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
Reyman, A.G., Semenov-Tian-Shansky, M.A., Current algebras and non-linear partial differential equations, Sov. Math., Dokl. 21 (1980), 630–634.
Rica, R.L., Rediscovery of the Da Rios Equation, Nature 352 (1991), 561–562.
Russell, J.S., Report on Waves, 14th Mtg. of the British Assoc. for the Advance. of Science, John Murray, London, pp. 311–390 + 57 plates, 1844.
D. H. Sattinger, Hamiltonian hierarchies on semisimple Lie algebras, Stud. Appl. Math. 72 (1985), no. 1, 65–86. MR 773831, DOI 10.1002/sapm198572165
Graeme Segal and George Wilson, Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61 (1985), 5–65. MR 783348, DOI 10.1007/BF02698802
Michael A. Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), no. 6, 1237–1260. MR 842417, DOI 10.2977/prims/1195178514
M. A. Semenov-Tian-Shansky, Classical $r$-matrices, Lax equations, Poisson Lie groups and dressing transformations, Field theory, quantum gravity and strings, II (Meudon/Paris, 1985/1986) Lecture Notes in Phys., vol. 280, Springer, Berlin, 1987, pp. 174–214. MR 905898, DOI 10.1007/3-540-17925-9_{3}8
A. B. Šabat, An inverse scattering problem, Differentsial′nye Uravneniya 15 (1979), no. 10, 1824–1834, 1918 (Russian). MR 553630
Gilbert Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal. 5 (1968), 506–517. MR 235754, DOI 10.1137/0705041
W. W. Symes, Systems of Toda type, inverse spectral problems, and representation theory, Invent. Math. 59 (1980), no. 1, 13–51. MR 575079, DOI 10.1007/BF01390312
Alan C. Newell (ed.), Nonlinear wave motion, Lectures in Applied Mathematics, Vol. 15, American Mathematical Society, Providence, R.I., 1974. MR 0336014
Chuu Lian Terng, A higher dimension generalization of the sine-Gordon equation and its soliton theory, Ann. of Math. (2) 111 (1980), no. 3, 491–510. MR 577134, DOI 10.2307/1971106
Terng, C.L., Soliton equations and differential geometry, J. Differential Geometry 45 (1997), 407–445.
Terng, C.L., Uhlenbeck, K., Poisson Actions and Scattering Theory for Integrable Systems, dg-ga/9707004 (to appear).
Terng, C.L., Uhlenbeck, K., Bäcklund transformations and loop group actions (to appear).
Karen Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, J. Differential Geom. 30 (1989), no. 1, 1–50. MR 1001271
Karen Uhlenbeck, On the connection between harmonic maps and the self-dual Yang-Mills and the sine-Gordon equations, J. Geom. Phys. 8 (1992), no. 1-4, 283–316. MR 1165884, DOI 10.1016/0393-0440(92)90053-4
S. M. Ulam, Adventures of a mathematician, Charles Scribner’s Sons, New York, 1976. MR 0485098
Miki Wadati, The modified Korteweg-de Vries equation, J. Phys. Soc. Japan 34 (1973), 1289–1296. MR 371251, DOI 10.1143/JPSJ.34.1289
George Wilson, The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras, Ergodic Theory Dynam. Systems 1 (1981), no. 3, 361–380 (1982). MR 662474, DOI 10.1017/s0143385700001292
Zabusky, N.J., Kruskal, M.D., Interaction of solitons in a collisionless plasma and the recurrence of initial states, Physics Rev. Lett. 15 (1965), 240–243.
V. E. Zaharov and L. D. Faddeev, The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funkcional. Anal. i Priložen. 5 (1971), no. 4, 18–27 (Russian). MR 0303132
Zakharov, V.E., Manakov, S.V., On resonant interaction of wave packets in non-linear media, JETP Letters 18 (1973), 243–247.
V. E. Zakharov and S. V. Manakov, The theory of resonance interaction of wave packets in nonlinear media, Ž. Èksper. Teoret. Fiz. 69 (1975), no. 5, 1654–1673 (Russian, with English summary); English transl., Soviet Physics JETP 42 (1975), no. 5, 842–850. MR 0426678
Zakharov, V.E., Mikhailov, A.V., Example of nontrivial interaction of solitons in two-dimensional classical field theory, JETP Letters 27 (1978), 42–46.
V. E. Zakharov and A. V. Mikhaĭlov, Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method; Russian transl., Soviet Phys. JETP 74 (1978), no. 6, 1953–1973. MR 524247
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Ž. Èksper. Teoret. Fiz. 61 (1971), no. 1, 118–134 (Russian, with English summary); English transl., Soviet Physics JETP 34 (1972), no. 1, 62–69. MR 0406174