General solutions of the two-dimensional system of Volterra equations which realize the Bäcklund transformation for the Toda lattice

A. N. Leznov   ; M. V. Saveliev   ; V. G. Smirnov

Teoretičeskaâ i matematičeskaâ fizika, Tome 47 (1981) no. 2, pp. 216-224

Résumé

It is shown that a two-dimensional system of Volterra equations (or difference Kortewegde Vries equations) is a Bäcklund transformation for the Toda lattice in two-dimensional space. This makes it possible to find the explicit form of the general selutions of these equations, which depend on the required number of arbitrary functions, on the basis of the known general solutions for the Toda lattice [1]. At the same time, the solutions of one-dimensional Volterra equations (and aIso numerous related nonlinear differentialdifference equations) are obtained as special cases of the general solutions.

Bibliographie

[1] Leznov A. N., Saveliev M. V., “Representation of zero curvature for the system of nonlinear partial differential equations $x_{\alpha,z\overline{z}}=\exp(Kx)_\alpha$ and its integrability”, Lett. Math. Phys., 3 (1979), 489–494 | DOI | MR | Zbl

[2] Volterra V., Leçons sur la theorie Mathematique de la Lutte pour la Vie, Gautheier-Villars, Paris, 1931 | MR | Zbl

[3] Breizman B. N., Zakharov V. E., Musher S. A., “O kinetike indutsirovannogo rasseyaniya lengmyurovskikh voln na ionakh plazmy”, ZhETF, 64:4 (1973), 1297–1313; Захаров В. Е., Мушер С. А., Рубенчик А. М., “О нелинейной стадии параметрического возбуждения волн в плазме”, Письма ЖЭТФ, 19:5 (1974), 249–253

[4] “Exact treatment of nonlinear lattice waves”, Suppl. Progr. Theor. Phys., 59 (1976), 1–161 | DOI

[5] Dubrovin V. A., Matveev V. B., Novikov S. P., “Nelineinye uravneniya tipa Kortevega–de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, Usp. matem. nauk, 31:1 (1976), 55–136 | MR | Zbl

[6] Manakov S. V., “O polnoi integriruemosti i stokhastizatsii v diskretnykh dinamicheskikh sistemakh”, ZhETF, 67:2 (1974), 543–555 | MR

[7] Kac M., van Moerbeke P., “On an explicitely soluble system of nonlinear differential equations related to certain Toda lattices”, Adv. Math., 16 (1975), 160–169 | DOI | MR | Zbl

[8] Toda M., “Studies of a non-linear lattice”, Phys. Reps., C18 (1975), 1–123 ; Henon M., “Integrals of the Toda lattice”, Phys. Rev., B9 (1974), 1921–1923 ; Flaschka H., “Toda lattice”, Phys. Rev., B9 (1974), 1924–1925 ; Bogoyavlensky O. I., “On perturbations of the periodic Toda lattice”, Commun. Math. Phys., 51 (1976), 201–209 ; Moser J., “Finitely many mass points on the line under the influence of an exponential potential - an integrable system”, Lecture notes in Physics, 38 (1976), 97–101 ; Olshanetsky M. A., Perelomov A. M., “Explicit Solutions of Classical Generalized Toda Models”, Invent. Math., 54 (1979), 261–269 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | MR | DOI | MR | Zbl

[9] Leznov A. N., Saveliev M. V., Cylindrically symmetric instantons for the arbitrary compact gauge groups, Preprint IHEP 78-177, Serpukhov, 1978 ; Leznov A. N., Saveliev M. V., “Exact monopole solutions in gauge theories for an arbitrary semisimple compact groups”, Lett. Math. Phys., 3 (1979), 207–211 ; Leznov A. N., Saveliev M. V., “Cylindrically symmetric instantons for the gauge groups of rank $2: SU(3)$, $O(5)$ and $G_2$”, Phys. Lett., 83B:3 (1979), 314–317 | MR | DOI | MR | DOI

[10] Leznov A. N., “O polnoi integriruemosti odnoi nelineinoi sistemy differentsialnykh uravnenii v chastnykh proizvodnykh v dvumernom prostranstve”, TMF, 42 (1980), 343–349 | MR | Zbl

[11] Bais F. A., Wilkinson D., “Exact $SU(N)$ monopole solutions with spherical symmetry”, Phys. Rev., D19 (1979), 2410–2422 | MR

[12] Miura R. M., “Korteweg-de Vries Equation and Generalization, I”, J. Math. Phys., 9:8 (1968), 1202–1204 | DOI | MR | Zbl

[13] Leznov A. N., Saveliev M. V., “Spherically summetric equations in gauge theories for an arbitrary semisimple compact Lie group”, Phys. Lett., 79B (1978), 294–297 | DOI | MR