Induced dynamics. (English) Zbl 1436.37081
Summary: Construction of new integrable systems and methods of their investigation is one of the main directions of development of the modern mathematical physics. Here we present an approach based on the study of behavior of roots of functions of canonical variables with respect to a parameter of simultaneous shift of space variables. Dynamics of singularities of the KdV and Sinh-Gordon equations, as well as rational cases of the Calogero-Moser and Ruijsenaars-Schneider models are shown to provide examples of such induced dynamics. Some other examples are given to demonstrates highly nontrivial collisions of particles and Liouville integrability of induced dynamical systems.
MSC:
37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
35Q51 | Soliton equations |
35Q53 | KdV equations (Korteweg-de Vries equations) |
37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
Keywords:
complete integrability; dynamics of singularities; Calogero-Moser system; Ruijsenaars-Schneider systemReferences:
[1] | Arkad’ev, V. A.; Pogrebkov, A. K.; Polivanov, M. K., Singular solutions of the KdV equation and the method of the inverse problem, Zap. Nauchn. Sem. LOMI, 133, 17-37 (1984) · Zbl 0571.35094 |
[2] | Bogdanov, L. V.; Zakharov, V. E., The Boussinesq equation revisited, Physica D, 165, 137-162 (2002) · Zbl 0997.35068 · doi:10.1016/S0167-2789(02)00380-9 |
[3] | Calogero, F., Exactly solvable one-dimensional many-body problems, Lett. Nuovo Cim, 13, 411-416 (1975) · doi:10.1007/BF02790495 |
[4] | Calogero, F., Motion of poles and zeros of special solutions of nonlinear and linear partial differential equations and related “solvable” many-body problems, Nuovo Cim, 43, B, 177-241 (1978) · doi:10.1007/BF02721013 |
[5] | Calogero, F., Classical Many-Body Problems Amenable to Exact Treatments (2001), Springer: Springer, New York · Zbl 1011.70001 |
[6] | Falkovich, G. E.; Spector, M. D.; Turitsyn, S. K., Destruction of stationary solitons and collapse in the nonlinear string equation, Phys. Lett. A, 99, 271-274 (1983) · doi:10.1016/0375-9601(83)90882-4 |
[7] | Moser, J., Three integrable Hamiltonian systems connected with isospectral deformations, Adv. Math, 16, 197-220 (1975) · Zbl 0303.34019 · doi:10.1016/0001-8708(75)90151-6 |
[8] | Novikov, S. P.; Manakov, S. V.; Pitaevskii, L. P.; Zakharov, V. E., Theory of Solitons: The Inverse Scattering Method (1984), Springer: Springer, Berlin · Zbl 0598.35002 |
[9] | Olshanetsky, M. A.; Perelomov, A. M., Explicit solution of the Calogero model in the classical case and geodesic flows on symmetric spaces of zero curvature, Lett. Nuovo Cim. (2), 16, 11, 333-339 (1976) · doi:10.1007/BF02750226 |
[10] | Orlov, A. Yu., Collapse of solitons in integrable model, Preprint IAIE, 221 |
[11] | Pogrebkov, A. K., Singular solitons: an example of a sinh-Gordon equation, Lett. Math. Phys, 5, 4, 277-285 (1981) · Zbl 0471.35057 · doi:10.1007/BF00401475 |
[12] | Pogrebkov, A. K.; Polivanov, M. K., Interaction of particles and fields in classical theory, Soviet J. Particles and Nuclei, 14, 5, 450-457 (1983) |
[13] | Pogrebkov, A. K.; Todorov, I. T., Relativistic Hamiltonian dynamics of singularities of the Liouville equation, Ann. Inst. H. Poincaré Sect. A (N.S.), 38, 1, 81-92 (1983) · Zbl 0539.35068 |
[14] | Pogrebkov, A. K.; Polivanov, M. K., The Liouville and sinh-Gordon equations. Singular solutions, dynamics of singularities and the inverse problem method, Soviet Sci. Rev., Sect. C, Math. Phys. Rev., 5, 197-271 · Zbl 0604.70032 |
[15] | Pogrebkov, A.K., Induced dynamics, arXiv: 1904.09469v1 (2019). |
[16] | Ruijsenaars, S. N.M.; Schneider, H., A new class of integrable systems and its relation to solitons, Annals of Physics (NY), 170, 2, 370-405 (1986) · Zbl 0608.35071 · doi:10.1016/0003-4916(86)90097-7 |
[17] | Ruijsenaars, S. N.M., Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Commun. Math. Phys, 110, 2, 191-213 (1987) · Zbl 0673.58024 · doi:10.1007/BF01207363 |
[18] | Ruijsenaars, S. N.M., Action-angle maps and scattering theory for some finite-dimensional integrable systems. I: The pure soliton case, Commun. Math. Phys, 115, 127-165 (1988) · Zbl 0667.58016 · doi:10.1007/BF01238855 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.