Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schrödinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrödinger-Hirota-Maxwell-Bloch equations, along with their Lax pairs.
Motion of curves and surfaces in R 3 lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that a more general situation in which the curves evolve in the presence of additional self consistent vector potentials can lead to interesting generalized spin systems with self consistent potentials or soliton equations with self consistent potentials. We obtain the general form of the evolution equations of underlying curves and report specific examples of generalized spin chains and soliton equations. These include principal chiral model and various Myrzakulov spin equations in (1+1) dimensions and their geometrically equivalent generalized nonlinear Schrödinger (NLS) family of equations, including Hirota-Maxwell-Bloch equations, all in the presence of self consistent potential fields. The associated gauge equivalent Lax pairs are also presented to confirm their integrability.
In this paper, we construct a Darboux transformation (DT) of the (2+1)-dimensional Schrödinger-Maxwell-Bloch equation (SMBE) which is integrable by the Inverse Scattering Method. Using this DT, the one-soliton solution and periodic solution are obtained from the "seed" solutions.
This article discusses the importance of hydraulic structures, particularly dams, in Kazakhstan’s economy due to water scarcity in many regions. The article highlights the need for a comprehensive approach to the location and design of dams considering climate change and land use scenarios. The article also emphasizes the need to update the design parameters and characteristics of dams to consider changes in land use and climate. The Samarkand reservoir hydrosystem is used as an example to discuss the potential hazards of the new spillway and the need for technical upgrading. The article suggests that the development of an integrated GIS-based approach to survey works, location, and design of dams could improve the quality of design and survey works and lead to the modernization of design and operation of reservoirs. The study proposes the use of modern research methods, including space methods, to provide more flexible and safe management of river flow with consideration of environmental and economic interests. The article concludes that the comprehensive GIS-based methodology for the location and design of dams, considering potential changes in river flow characteristics and the impact of changes in land use and climatic scenarios, can improve efficiency and optimize the work of hydraulic engineers.
The Myrzakulov-Lakshmanan-II (ML-II) equation is one of a (2+1)-dimensional generalizations of the Heisenberg ferromagnetic equation. It is integrable and has a non-isospectral Lax representation. In this paper, the Darboux transformation (DT) for the ML-II equation is constructed. Using the DT, the 1-soliton and 2-soliton solutions of the ML-II equation are presented.
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