Infopaket

Soliton theory

ОК18

Обов’язкова дисципліна для ОП

Second

2021/2022

3 Semester

3

The learning outcomes are knowledge of the mechanism of soliton formation, mastery of the methods of the inverse scattering problem and soliton perturbation theory, knowledge of solutions of basic soliton equations (Cortewega de Vries, Schrödinger nonlinear equation, Gordon sine) and their properties. Also, students will understand the physical meaning of soliton formation in a wide range of physical systems, such as superconducting tunnel contacts, ferromagnets, nonlinear optical waveguides, and surface waves in hydrodynamics. Students will also be able to make a continuous approximation for an arbitrary discrete equation of motion, write a nonlinear equation in Hamiltonian form, determine whether this functional is an integral of motion, using perturbation theory to obtain effective equations of motion for soliton parameters.

Full-time form

Know the basic laws of hydrodynamics, classical mechanics, quantum mechanics, mathematical analysis, optics, ordinary differential equations, mathematical physics. In particular, to know the Hamilton equation, the properties of the Sturm-Liouville problem solutions, the Green's function for differential equations, the basic properties of the stationary Schrödinger equation, its eigenfunctions and eigenvalues. Know the basic concepts of wave theory, in particular such as phase and group velocities, the law of variance. Be able to apply previous knowledge from the courses of mathematical analysis, mathematical physics, theory of function of a complex variable, in particular the basic properties of analytical functions, basics of vector analysis and differential equations. Have basic skills in calculating derivatives, integrals, operations on operations with vectors, calculate surpluses, define and decompose functions into a series and a Fourier integral.

The discipline "Soliton Theory" is a discipline that is part of the cycle of professional training of specialists of educational and qualification level "Master of Physics". Within the course there is a study of the nature and properties of nonlinear solitary excitations (solitons), acquaintance with integrated systems of mathematical physics and methods of their analysis. The inverse scattering problem method, which is an essential component of the proposed course, aims to teach students to construct exact solutions of soliton equations, to understand the nature of the integrals of motion of these equations and to explain the concept of full integrability of dynamical systems.

1. V.E. Zakharov, S.V. Manakov, SP Novikov and LP Pitaevsky, Theory of solitons: a method of the inverse problem (Nauka, Moscow, 1980). 2. A. Newell, Solitons in Mathematics and Physics (Mir, Moscow, 1989). 3. J. Wisham, Linear and nonlinear waves (Mir, Moscow, 1977) 4. M. Ablovitz and H. Sigur, Solitons and the method of the inverse problem (Nauka, Moscow, 1983). 5. Solitons in action, ed. K. Lindgren and A. Scott, (Mir, Moscow, 1981). 6. L.A. Takhtadjian and L.D. Faddeev, Hamilton's campaign in the theory of solitons (Nauka, Moscow, 1986). 7. В.В. Schmidt, Introduction to the Physics of Superconductors, (Nauka, Moscow, 1982). 8. А.М. Kosevych, B.А. Ivanov and AS Kovalev, Nonlinear waves of obsession. Dynamic and topological solitons (Naukova Dumka, Kyiv, 1983).

The total amount of 90 hours, including: Lectures - 30 hours. Independent work - 60 hours.

The control is carried out by a modular rating system, which consists of 3 content modules. The knowledge assessment system includes current, modular and semester control knowledge. Forms of current control: assessment of homework, written independent assignments, tests and tests performed by students during practical classes. The student can receive a maximum of 30 points for homework, independent assignments, oral answers, tests, additions to practical classes (10 points in each content module). Modular control: 3 modular control works. The student can receive a maximum of 30 points for modular tests (10 points for each work). The final semester control is conducted in the form of an exam (40 points) in the third semester. The exam ticket includes 2 theoretical questions (20 points each).

Ukrainian