scholarly journals Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems

2006 ◽  
Vol 47 (9) ◽  
pp. 092701 ◽  
Author(s):  
Błażej M. Szablikowski ◽  
Maciej Błaszak
Keyword(s):  
Dispersionless Systems

scholarly journals Integrable discrete systems on R and related dispersionless systems

2008 ◽  
Vol 49 (7) ◽  
pp. 072702 ◽  
Author(s):  
Maciej Błaszak ◽  
Metin Gürses ◽  
Burcu Silindir ◽  
Błażej M. Szablikowski
Keyword(s):  
Discrete Systems ◽  
Dispersionless Systems

scholarly journals QUASICLASSICAL LIMIT OF THE SCHRÖDINGER-MAXWELL- BLOCH EQUATIONS

2021 ◽  
Vol 2 (336) ◽  
pp. 179-184
Author(s):  
Zh. B. Umurzhakhova ◽  
M. D. Koshanova ◽  
Zh. Pashen ◽  
K. R. Yesmakhanova
Keyword(s):  
Differential Equations ◽  
Applied Mathematics ◽  
Nonlinear Differential Equations ◽  
Soliton Solutions ◽  
Optical Solitons ◽  
Theoretical Physics ◽  
Physical Processes ◽  
Integrable Equations ◽  
Quasiclassical Limit ◽  
Dispersionless Systems

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.

scholarly journals Twistor theory at fifty: from contour integrals to twistor strings

2017 ◽  
Vol 473 (2206) ◽  
pp. 20170530 ◽  
Author(s):  
Michael Atiyah ◽  
Maciej Dunajski ◽  
Lionel J. Mason
Keyword(s):  
Holomorphic Curves ◽  
Twistor Theory ◽  
Gravitational Instantons ◽  
Yang Mills ◽  
Conformal Curvature ◽  
Twistor Strings ◽  
Equations Of Mathematical Physics ◽  
Elementary Construction ◽  
Dispersionless Systems

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space–time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold—the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics—anti-self-duality equations on Yang–Mills or conformal curvature—can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang–Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang–Mills equations, and Einstein–Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose’s proposal for a role of gravity in quantum collapse of a wave function.

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