QUASICLASSICAL LIMIT OF THE SCHRÖDINGER-MAXWELLBLOCH EQUATIONS
Ключевые слова:
Dispersionless integrable system, quasiclassical limit, Schrödinger-Maxwell-Bloch equations, Lax pair.Аннотация
The study of integrable equations is one of the most important aspects of modern mathematical and
theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical
application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a
nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a
soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion.
Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the
equation compensates for the dispersion term.
In addition to integrable nonlinear differential equations, there is also an important class of integrable partial
differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless
(quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of
known integrable equations. In recent years, the study of dispersionless systems has become of great importance,
since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied
mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex
plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations.
In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell-
Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of
nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with
resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB
using the obtained Lax representation.
