2010-10-27
10:15 at HCI J 574

Solitons (Classical and Quantum Theory)

Maksym Osmanov

National Technical University of Ukraine, Kiev

A soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant velocity. Solitons arise as the solutions of a widespread class of nonlinear partial differential equations describing physical systems (hydrodynamics, Langmuir films, optical solitons in fibers, polymeric chain, nonlinear magnetization dynamics in ferromagnets with the easy-axis type anisotropy, transmissions lines made out of arrays of Josephson junctions of superconductors, proteins and DNA etc.). In this talk I will discuss the classical and quantum solitons theory. From the classical theory following issues will be represented: Hamiltonian structure of solitons equations [1], method of inverse scattering (Lax representation [2] and zero curvature representation [3]), Hirotaís method [4, 5]. From the quantum theory: soliton quantization (WKB method and path integral) [6], number state method [7] and quantum inverse scattering [8]. To illustrate the theory, following equations will be solved: Korteveg-de Vries, sine-Gordon equation, Nonlinear Schrodinger equation, Kadomtsev-Petviashvili equation, Ablowitz-Ladik equation, Heisenberg ferromagnet etc.

1. L. A. Dickey. Soliton Equations and Hamiltonian Systems, 2003, 420 p. 2. S. Novikov, S. V. Manakov, L. P. Pitaevslij, V. E. Zakharov. Theory of Solitons: The Inverse Scattering Method, 1984, 292 p. 3. L. D. Faddeev, L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons, 1987, 592 p. 4. R. Hirota, A. Nagai, J. Nimmo, C. Gilson. The Direct Method in Soliton Theory, 2004, 212 p. 5. T. Miwa, M. Jimbo, E. Date. M. Reid. Solitons: Differential equations, symmetries and infinite dimensional algebras, 2000, 118 p. 6. R. Rajaraman. An Introduction to Solitons and Instantons in Quantum Field Theory, 1987, 418 p. 7. A. Scott. Nonlinear Science: Emergence and Dynamics of Coherent Structures. 2003, 504 p. 8. V. E. Korepin, Bogoliubov N. M, Izergin A. G. Quantum Inverse Scattering Method and Correlation Functions, 1997, 576.