Nurshuak-Tolkynay-Myrzakulov system: integrability, geometry and solutions

In this paper, we study an integrable system with the self-consistent potentials called the Nurshuak-Tolkynay-Myrzakulov (NTM) system. This system is of great importance in the theory of integrable nonlinear equations, since this system describes the dynamics of nonlinear wave processes in various fields of physics, such as hydrodynamics, optics, quantum mechanics, and plasma dynamics. Various integrable reductions of this system are also given and their Lax pairs are found. It is shown that the NTM system, being integrable, has some deep geometric roots, and that its geometric interpretation can lead to an understanding of more complex geometric structures. Thus, it is shown that the NTM system describes the dynamics of waves and allows us to understand how those waves interact with the geometry of space, which is an important aspect of many physical processes. Solitonic solutions of the NTM system are found. These solutions exhibit various signs of the periodicity, exponentiality, and rationality of soliton structures, including the elliptic Jacobi function. The results are visualized using three-dimentional (3D) and contour plots to clearly illustrate the response of the behavior to momentum propagation and to find appropriate values for the system's parameters. This visualization provides valuable insights into the characteristics and dynamics of the soliton solutions obtained from the integrable NTM equation.