Infopaket

Soliton theory

ОК18

Обов’язкова дисципліна для ОП

Second

2024/2025

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.P. G. Drazin, R. S. Johnson. Solitons: an introduction (Cambridge University Press, 1989). 2.M. J. Ablowitz, P. A. Clarkson. Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, 1991). 3.Boling Guo, Xiao-Feng Pang, Yu-Feng Wang, and Nan Liu. Solitons (Walter de Gruyter GmbH, Berlin/Boston, 2018). 4.M. J. Ablowitz, H. Segur. Solitons and the inverse scattering transform, (SIAM, 1981). 5.L.D. Faddeev, L.A. Takhtajan. Hamiltonian Methods in theTheory of Solitons (Springer-Verlag Berlin Heidelberg 2007). 6.G. Eilenberger. Solitons. Mathematical Methods for Physicists (Springer-Verlag Berlin Heidelberg, 1981). 7.A. Ludu. Nonlinear Waves and Solitons on Contours and Closed Surfaces (Springer, 2022). 8.A.M. Kosevich. The Crystal Lattice. Phonons, Solitons, Dislocations, Superlattices (Wiley-VCH, 2005). 9.E. Kengne, WuMing Liu. Nonlinear Waves. From Dissipative Solitons to Magnetic Solitons (Springer 2023).

The total amount of 12 hours, including: Lectures - 12 hours.

Control is carried out according to the modular rating system, which consists of 2 content modules. The knowledge assessment system includes current, modular and semester knowledge control. The results of students' educational activities are assessed on a 100-point scale. Forms of current control: evaluation of homework, written independent tasks, tests and tests completed by students during practical classes. A student can receive a maximum of 30 points for completing homework, independent tasks, oral answers, tests, additions to practical classes (15 points in each content module). Modular control: 2 modular tests. A student can receive a maximum of 30 points for modular tests (15 points for each work). Final semester control is carried out in the form of an exam (40 points) in the third semester. The examination card includes 2 theoretical questions (20 points each).

Ukrainian