Abstract:
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.
