Elements of the theory of solitons
- familiarization with solutions of nonlinear partial differential equations of the soliton type, the study of their characteristic properties and typical dynamics, in contrast to linear and nonlinear dispersive waves
- formation of the basic knowledge and skills applied to the research of properties of soliton-type waves
- awareness of the Inverse Scattering Transform as a nonlinear spectral method for studying wave dynamics
- familiarization with actual scientific problems related to the dynamics of soliton-type waves
- Aware of the basic properties of the KdV equation, can derive the first integrals of the equation, familiar with the properties of the solitons and their collisions, have a broader view on the KdV-type integrable equations and their soliton-type solutions
- Aware of the basic properties of the NLS equation, of the exact solutions bright/gray/dark solutions and breathers, rogue waves, familiar with the concept of the complex amplitude.
- Aware of the main equations of the wave type, able to recognize nonlinear equations and linear equations with dispersion, can write the dispersion relation for a given linear equation with dispersion, can derive the soliton solution as the localized stationary solution
- Has general understanding of the Inverse Scattering Transform, its similarity and difference from the Fourier method, understands the relation between isolated solutions, discrete eigenspectrum of the associated scattering problem and reflectionless potentials.
- Wave equations: linear, nonlinear, with soliton solutions
- Korteweg-de Vries equation and its generalizations
- Nonlinear Schrödinger equation
- Nonlinear spectral analysis
- Generalization and repetition
- 2022/2023 4th module0.2 * Home Works and Short Classworks + 0.4 * Written examination + 0.1 * Activity during the seminars
- Guo, B., Pang, X.-F., Wang, Y.-F., & Liu, N. (2018). Solitons. Berlin: De Gruyter. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1746464
- Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L.O. Methods Of Qualitative Theory In Nonlinear Dynamics (Part II). World Sci //Singapore, New Jersey, London, Hong Kong. – 2001.
- Sheng ZHANG, & Caihong YOU. (2019). Inverse Scattering Transform for a Supersymmetric Korteweg-De Vries Equation. Thermal Science, 23, S677–S684. https://doi.org/10.2298/TSCI180512081Z