L.N. Gumilyov Eurasian National University, a dissertation defense for the degree of Doctor of Philosophy (PhD) by Найзағараева Ақгүл Аманжолқызы on the topic «Modeling and analysis of solutions of integrable spin systems» by specialty «6D070500 – Математикалық және компьютерлік модельдеу».
The dissertation was carried out at the «Mathematical and computer modeling education department» of L.N. Gumilyov Eurasian National University.
The language of defense is kazakh
Official reviewers:
Talgat Zhakupovich Mazakov, Professor of the Software Engineering Department, Doctor of Physical and Mathematical Sciences, International Engineering and Technology University (Almaty);
Baltabek Esmatovich Kanguzhin - Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematics at Al-Farabi Kazakh National University (Almaty).
Temporary members of the Dissertation Council:
Bakhytzhan Zhamaladinovich Zhanmoldaev, Doctor of Technical Sciences, Professor of the Mathematics and Applied Mechanics Department, Korkyt Ata Kyzylorda University (Kyzylorda);
Murat Ibraevich Ramazanov, Doctor of Physical and Mathematical Sciences, Honored Professor of the Department of Mathematical Analysis and Differential Equations, E.A. Buketov Karaganda University (Karaganda);
Elmira Aynabekovna Bakirova, Candidate of Physical and Mathematical Sciences, Professor at the Institute of Mathematics and Mathematical Modeling (Almaty).
Scientific advisors:
Kuralay Ratbaykyzy Esmakhanova, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical and Computer Modeling, L.N. Gumilyov Eurasian National University (Astana);
Tihomir Valchev, Professor, PhD, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Republic of Bulgaria (Sofia).
The defense will take place on January 23, 2026, at 03:00 PM in the Dissertation Council for the training direction «8D061 – Information and communication technologies» in the specialty «6D070500 – Математикалық және компьютерлік модельдеу» of L.N. Gumilyov Eurasian National University. The defense meeting is planned to be held online.
Link: https://clck.ru/3Qqw6Z
Address: Astana, st. Kazhymukan, 13, auditorium No. 205 of educational building No. 3.
Abstract (English): Relevance of the research topic Modern nonlinear dynamical systems require the development of accurate analytical and numerical models that enable the analysis of stability and wave structures in order to describe processes in solid-state physics, optics, and spintronics. Integrable spin systems—such as the Akbota and Zhanbota-IIA equations—play a significant role in describing the evolution of the magnetization vector and the propagation of spin waves in ferromagnetic media. The relevance of this research is determined by the need to develop new approaches for finding and visualizing solutions of complex nonlinear models that combine analytical and computational methods. An interactive 2D/3D software package played a crucial role in visualizing the dynamics of solutions, analyzing parametric sensitivity, refining physical interpretation, and ensuring the reproducibility of the results. Degree of research development The theory of integrable systems has been actively developing since the mid-20th century. Fundamental solutions of nonlinear equations possessing integrability properties were obtained in the works of Korteweg–de Vries, Zakharov and Shabat, Lax, Hirota, and Darboux. Subsequently, this theory was extended through the use of determinant structures, variational iteration methods, and Lie symmetries. Modern research is focused on the discovery of new integrable models, the investigation of their stability, and the derivation of analytical solution forms. At the same time, issues related to the numerical implementation of solutions and their visualization—especially for multicomponent spin systems—remain insufficiently explored. Research objectives and tasks The objective of the dissertation is to model solutions of integrable spin systems, investigate their analytical and numerical properties, and obtain new physical effects based on computational experiments carried out in a programming environment. To achieve this objective, the following tasks were set: 1. To apply the φ⁶ expansion method and obtain new classes of bright, dark, and mixed solutions; 2. To use the extended ShGEE (sinh–Gordon Expansion Equation) and JEFE (Jacobi Elliptic Function Expansion) methods to derive analytical solutions of spin systems and to construct software-based models of the obtained solutions; 3. To implement numerical algorithms such as the Crank–Nicolson and Runge–Kutta methods and visualize the evolution of solutions in 2D and 3D formats; 4. To analyze the modulational instability and stability conditions of the obtained solutions; 5. To develop a software package for computer modeling of the dynamics of soliton solutions. Object and subject of the research The object of the research is integrable nonlinear spin systems describing the dynamics of the magnetization vector in a ferromagnetic medium. The subject of the research is the derivation of analytical, numerical, and graphical solutions of the Akbota and Zhanbota-IIA equations, as well as the investigation of their properties. Research methods The study employs methods of the theory of nonlinear differential equations. The φ⁶ expansion method, as well as the extended ShGEE and JEFE methods, are applied for the first time to the considered spin systems. Numerical methods such as the Crank–Nicolson and Runge–Kutta schemes are used, along with Python, Matplotlib, Plotly, and HTML/JavaScript software tools. Scientific novelty of the research The φ⁶ expansion method is applied for the first time to integrable spin systems, resulting in 17 new families of bright, dark, and mixed-type solutions; The ShGEE and JEFE methods are applied for the first time, yielding elliptic and hyperbolic solutions; Numerical algorithms such as the Crank–Nicolson and Runge–Kutta schemes are implemented, and the evolution of solutions is analyzed; Modulation instability is investigated, and parametric regions of stable wave solutions are identified; The developed software package automates the study of solutions of integrable systems and introduces a new level of computer modeling for the first time. Theoretical significance of the research The theoretical significance of the research is determined by the first-time application of new methods—namely the φ⁶ expansion method, ShGEE, and JEFE—for obtaining solutions of integrable spin systems. Using these methods, 17 new analytical solutions of bright, dark, and mixed types were derived. In addition, a comprehensive analysis of modulation instability made it possible to determine the stability conditions and parametric regions of stationary waves. The new solutions and their properties contribute to the further development of soliton and wave theory and open new directions in mathematical physics. Practical significance of the research The practical significance of the research lies in the development of a software tool that includes graphical models of analytical and numerical solutions. This tool allows the simulation of wave evolution by varying different parameters. The results of such modeling can be applied in physical and engineering fields, such as optical waves, plasma oscillations, and Bose–Einstein condensates. In addition, the visual demonstrations can be effectively used for educational purposes. Approbation of research results The research results were discussed at scientific seminars of the Department of Mathematical and Computer Modeling at L.N. Gumilyov Eurasian National University, presented at international conferences, and published in international high–impact factor journals. An interactive software package for computer modeling of solutions of integrable systems was developed, and a copyright certificate was obtained (No. 55704, March 12, 2025). Publications on the research topic The main results of the dissertation have been published in journals with an impact factor according to JCR and indexed in the Web of Science Core Collection (Q2 quartile), as well as in journals with a high CiteScore in the Scopus database. In total, 3 scientific articles and 2 articles in the materials of international scientific conferences were published. 1. Articles in International peer-reviewed journals (indexed in Web of Science and Scopus): 1. Optical wave structures and stability analysis of integrable Zhanbota equation. Modern Physics Letters B, 2024. CiteScore (2024) – 4; Journal Impact Factor (2024) – 1.8; field – physics, mathematical; quartile – Q2. 2. Dynamical visualization and propagation of soliton solutions of Akbota equation arising in surface geometry. Modern Physics Letters B, 2024. CiteScore (2024) – 4; Journal Impact Factor (2024) – 1.8; field – statistical and nonlinear physics; quartile – Q2. 3. The generalized soliton wave structures and propagation visualization for Akbota equation. Zeitschrift für Naturforschung A – Journal of Physical Sciences, 2024. CiteScore (2024) – 3.0; Journal Impact Factor (2024) – 1.8; field – mathematical physics; quartile – Q2. 2. Materials of international conferences: 1. Spin systems associated with integrable nonlinear Schrödinger equations. AIP Conference Proceedings, Vol. 2872, No. 1, 2023, 11th International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE). CiteScore (2022) – 0.7; Percentile (Physics & Astronomy) – 10. 2. Nonlocal Schrödinger–Maxwell–Bloch Equations. Journal of Physics: Conference Series, Vol. 2090, No. 1, 2021, 10th International Conference on Mathematical Modeling in Physical Sciences (IC-MSQUARE). CiteScore (2020) – 0.7; Percentile (Physics & Astronomy) – 27. 3. Presentation of research results The research results were presented at the international conference “Topical Issues of Modern Mathematics and Computer Science,” held at L.N. Gumilyov Eurasian National University (Astana, 2024, dedicated to the 10th anniversary of the department). All published works fully correspond to the research topic and describe analytical solutions of integrable spin systems, the construction of their computer models, and methods for graphical visualization. Reliability and validity of the research results The reliability and validity of the scientific results are confirmed by publications in international peer-reviewed journals indexed in the Web of Science and Scopus databases, as well as by citations of the research results by foreign scientists. The article published in Modern Physics Letters B has been cited 10 times. Scientometric indicators: • h-index in the Web of Science database – 1; • h-index in the Scopus database – 6. These indicators clearly demonstrate the scientific novelty and practical significance of the work. Structure of the dissertation The dissertation consists of an introduction, four chapters, a conclusion, and a list of references. The total volume of the work is 82 pages, including 30 figures, 3 tables, an appendix, and 97 references. In the introduction, the relevance of the research topic, objectives and tasks of the study, scientific novelty, research object and subject, applied methods, and the practical significance of the obtained results are presented. The first chapter examines the theoretical foundations of integrable spin systems. The physical meaning and historical development of the Heisenberg model are analyzed, and the connection of integrable systems with spin equations and nonlinear wave models is demonstrated. The second chapter presents the analytical solutions of the Akbota equation and their physical interpretation. The structure of the solutions obtained by the φ⁶-expansion method is described, and numerical modeling is carried out using the Crank–Nicolson and Runge–Kutta methods. Additionally, the modulation instability of the Akbota equation is investigated, and its physical significance is analyzed. The third chapter provides analytical solutions of the Zhanbota-IIA equation and analyzes their stability. The mathematical formulation of the model is presented, and bright, dark, and elliptic soliton solutions obtained using the ShGEE and JEFE methods are studied. The conditions of modulation instability and the results of numerical visualization are provided. The fourth chapter demonstrates the practical significance of the research results. The structure and functionality of the developed interactive web interface are described, and the computational complexity is evaluated. In the conclusion, the main scientific results and conclusions of the dissertation are summarized. Appendix A. Parametric analysis of the Zhanbota-IIA equation solutions for the first choice and first case. Internal consistency of the dissertation The internal consistency of the dissertation is ensured through the logical interconnection of the research objectives, tasks, and obtained results. The research stages are organized sequentially, covering the complete scientific cycle — from theoretical modeling to numerical analysis and visualization. This approach allows for a comprehensive characterization of the properties of integrable spin systems and a deep understanding of the physical meaning of the obtained results. All sections of the work are subordinated to a single scientific idea — the formation of an integrated methodology for analyzing solutions of integrable spin models, studying their stability, and implementing their visualization. Theoretical analyses, numerical calculations, and software implementation complement each other, ensuring the coherence of the research and the consistency of the results obtained.
Conclusion of the Research Ethics Committee
Defense of the dissertation: https://www.youtube.com/watch?v=QWe-Ezoq3Ko
