Relevance of the Project. The project is devoted to the study of the integrability of nonlocal nonlinear Schrödinger-type equations with PT symmetry. The main objectives include the classification of such equations, the proof of their integrability, and the construction of exact solutions using the inverse scattering transform, as well as Darboux and Bäcklund transformations.
The expected results involve obtaining and analyzing exact solutions, which will contribute to quantum mechanics, nonlinear optics, and the dynamics of complex systems. The project will strengthen the scientific and technological potential of the Republic of Kazakhstan, enhance the competitiveness of Kazakhstani research institutions, and may find applications in telecommunications and laser technologies. It is planned to publish two scientific articles in international peer-reviewed journals, thereby reinforcing Kazakhstan’s position in the global scientific community.
Aim of the Project. To investigate the integrability and classify nonlocal Schrödinger–Maxwell–Bloch (SMB) systems, to obtain their exact solutions using classical methods of mathematical physics, and to perform modeling of the obtained results.
Project Objectives. To achieve the project goals during 2025–2027, the following tasks will be carried out:
- Analysis of Schrödinger-type equations (STE):
Investigation of the mathematical and physical properties of STE, including symmetries, nonlinear effects, and types of solutions. This task is fundamental for understanding the properties of STE and is necessary for the correct derivation of nonlocal analogues and the construction of solutions. It serves as the basis for subsequent studies. - Classification of nonlocal PT-symmetric STE systems:
Derivation and classification of new classes of nonlocal PT-symmetric equations. This is important for identifying systems with special symmetries and remarkable properties, which will help determine promising equations for further analysis. - Study of integrability of nonlocal systems:
Proof of integrability using methods such as the Lax pair formalism. This study is crucial for obtaining exact analytical solutions and identifying special classes of nonlocal equations with physical applications. - Construction of exact solutions:
Solving nonlocal systems using the Darboux, Hirota, and Bäcklund transformation methods. These solutions constitute the main result, enabling the study of the physical behavior of nonlocal systems and their stability in various applications. - Visualization of solutions:
Construction of graphical representations of exact solutions of nonlocal PT-symmetric STEs to analyze their dynamical properties. Visualization improves understanding of the processes described by nonlocal systems and facilitates presentation of results in scientific publications.
These tasks form a logical sequence, leading from theoretical analysis and derivation of new equations to their visualization and practical interpretation.
Expected Results. Upon completion of the project, it is expected that two (2) articles will be published in journals belonging to the first three quartiles according to the Web of Science impact factor or having a CiteScore percentile of at least 50 in the Scopus database.
Publication of monographs/books/book chapters/reviews: not planned.
Obtaining patents in foreign or national patent offices: not planned.
Development of scientific and technical or design documentation: not planned.
Dissemination of Results
The project results are of a theoretical nature and will be made available to potential users, the scientific community, and the general public through publication in prestigious international journals with an impact factor. In addition, the results will be disseminated through presentations at conferences, symposia, and scientific seminars.
Scientific and Social Impact
The scientific impact of the project lies in advancing classical-level results toward the next stage of studying soliton equations at the quantum level. The social impact includes contributing to the training of highly qualified and competitive scientific personnel, thereby supporting the long-term development of science in Kazakhstan.
Application Area and Target Audience
The target audience includes researchers working on related problems in mathematical physics and the differential geometry of nonlinear media. The results may also be introduced as specialized courses for master’s and PhD students at universities and used in further scientific research.
Since the project is fundamental in nature, the results are not intended for commercialization.
Achieved Results. An analysis of Schrödinger-type equations was conducted to further investigate their mathematical and physical properties. Various forms of nonlinear Schrödinger equations, including classical and generalized versions, were examined. Special attention was paid to the structure of nonlinear terms governing wave packet behavior and stability. Relationships between model parameters and types of solutions—solitonic, elliptic, and rational—were analyzed. Physical interpretations relevant to nonlinear optics, quantum mechanics, and excitation dynamics in complex media were also investigated. Symmetries and transformations preserving integrability were identified. These results refine approaches to constructing exact solutions and outline directions for further research in nonlocal models.
The computation and classification of nonlocal PT-symmetric Schrödinger-type systems were performed. Based on symmetry principles and reduction techniques, nonlocal equations exhibiting PT invariance were derived. Detailed calculations confirmed the correctness of the proposed transformations and their compliance with integrability conditions. Each system was classified according to the type of PT reduction, parameter signs, and the nature of nonlocal interactions. Both focusing and defocusing nonlinearities and their physical analogues were considered. Particular attention was paid to cases where nonlocal interactions lead to new types of stable solutions. As a result, a set of fundamental models of interest for further analytical and numerical study was identified.
Nonlocal PT-symmetric Schrödinger-type systems were formulated and defined. The constructed systems are based on symmetric relations between the field and its mirror-conjugate components, ensuring PT invariance. The equations include parameters responsible for gain–loss balance and the nature of nonlocal coupling. These models generalize the well-known results of Ablowitz and Musslimani and extend the class of integrable Schrödinger–Maxwell–Bloch systems. At the current stage, compatibility analysis and construction of corresponding Lax pairs are underway. Conservation laws and integrals of motion are being investigated for each reduction. These results form the basis for applying Darboux and Hirota methods to obtain exact solutions and analyze their stability.
A classification of the obtained systems according to their mathematical and physical properties was carried out, and a database was created. All derived equations were systematized based on structure, type of nonlinearity, and form of reduction. For each model, PT symmetry conditions, interaction characteristics, and integrable structure were specified. Comparison with known analogues from the literature made it possible to identify new, previously unexplored systems. The database includes analytical expressions, model parameters, and numerical results, serving as a tool for further analysis, visualization, and verification of physical interpretations.
Research Team
Zakariyeva Zaruyet – Project Leader, Master of Pedagogical Sciences
H-index: 1
Web of Science Researcher ID: LPQ-0230-2024
ORCID: https://orcid.org/0009-0008-6054-0160
Email: This email address is being protected from spambots. You need JavaScript enabled to view it.
Myrzakulova Zhaidary – Scientific Consultant, PhD
H-index: 6
Web of Science Researcher ID: AAR-9826-2020
ORCID: https://orcid.org/0000-0002-4047-4484
Email: This email address is being protected from spambots. You need JavaScript enabled to view it.
Publications
Myrzakulova Z.; Zakariyeva Z.; Zhumakhanova A.; Yesmakhanova K. Exact solution of the nonlocal PT-symmetric (2 +1)-dimensional Hirota–Maxwell Bloch system. Mathematics 2025, 13, 1101. https://doi.org/10.3390/ math13071101