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EQUATION OF MATHEMATICAL PHYSICS AND OTHER AREAS OF APPLICATION: A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION OF SOLUTIONS OF THE HIGHER ORDER KAUP-NEWELL EQUATION
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作者 李忠定 《Acta Mathematica Scientia》 SCIE CSCD 1993年第4期449-456,共8页
Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be... Under the constrained condition induced by the eigenfunction expresson of the potential (u, v)T = (-[A2q, q], [A2p, p])T = f (q, p), the spatial part of the Lax pair of the Kaup-Newell equation is non linearized to be a completely integrable system (R2N, Adp AND dq, H = H-1) with the Hamiltonian H-1 = -[A3q, p]-1/2[A2p, p][A2q, q]. while the nonlinearization of the time part leads to its N-involutive system {H(m)}. The involutive solution of the compatible fsystem (H-1), (H(m)) is mapped by into the solution of the higher order Kaup-Newell equation. 展开更多
关键词 equation of mathematical physics AND OTHER AREAS of APPLICATION A CLASSICAL INTEGRABLE SYSTEM AND THE INVOLUTIVE REPRESENTATION of SOLUTIONS of THE HIGHER ORDER KAUP-NEWELL equation CTAM AZ
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Hidden Properties of Mathematical Physics Equations. Double Solutions. The Realization of Integrable Structures. Emergence of Physical Structures and Observable Formations
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作者 L. I. Petrova 《Journal of Applied Mathematics and Physics》 2020年第7期1255-1262,共8页
With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various... With the help of skew-symmetric differential forms the hidden properties of the mathematical physics equations are revealed. It is shown that the equations of mathematical physics can describe the emergence of various structures and formations such as waves, vortices, turbulent pulsations and others. Such properties of the mathematical physics equations, which are hidden (they appear only in the process of solving these equations), depend on the consistency of derivatives in partial differential equations and on the consistency of equations, if the equations of mathematical physics are a set of equations. This is due to the integrability of mathematical physics equations. It is shown that the equations of mathematical physics can have double solutions, namely, the solutions on the original coordinate space and the solutions on integrable structures that are realized discretely (due to any degrees of freedom). The transition from the solutions of the first type to one of the second type describes discrete transitions and the processes of origin of various structures and observable formations. Only mathematical physics equations, on what no additional conditions such as the integrability conditions are imposed, can possess such properties. The results of the present paper were obtained with the help of skew-symmetric differential forms. 展开更多
关键词 Integrability of mathematical physics equations Double Solutions Integrable Structures Discrete Transitions Skew-Symmetric Differential Forms
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New Approaches of Theoretical Astrophysics for Application to Some Astronomical Objects: I. An Application of Non-Classical Equation Mathematical Physics to the Magneto-Hydrodynamic Equilibrium (in Case of Mixed Magnetic Field) of Magnetic Stars
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作者 Mahammad A. Nurmammadov 《Open Journal of Applied Sciences》 2024年第11期3165-3182,共18页
In this paper, we consider the application of the equation of non-classical mathematical physics to magneto-hydrodynamic equilibrium (in the case of a mixed magnetic field) for magnetic stars. First, we give the neces... In this paper, we consider the application of the equation of non-classical mathematical physics to magneto-hydrodynamic equilibrium (in the case of a mixed magnetic field) for magnetic stars. First, we give the necessary concepts about the equation of non-classical mathematical physics and the possibility of their applicability to astrophysical problems. The conditions of magneto-hydrodynamic equilibrium are determinate, and self-consistence provides the means to derive the corresponding partial differential equations describing this equilibrium in a magnetosphere magnetic star. Namely, this process is to the non-classical equations of mathematical physics in cases of types. Keldysh-Tricomi, a common case equation of non-classical type, is at first introduced by the author. Using the two main physical efficiencies of MHD. A mathematical model of a poloidal-toroidal mixed magnetic field for magnetic stars is constructed, and this model is classified with respect to degenerating case equations. According to Hopf’s theorem, Maxwell’s equation and the magnetic force balance equation constructed equilibrium conditions of the poloidal-toroidal of the magnetic field for a magnetic star. At the same time, the taken example, which is the self-consistency of this model by observation dates, is investigated. At first, in an application, the method of straight lines for recurrent formulas of calculation of magnetic flux and stream functions is used. The physical means, the corresponding singular point of the sonic line, cutoff, and resonance phenomena are considered. In this case, a general solution equation is found, which is interpreted by this phenomenon as a cutoff, resonance. Finally, this obtained solution gives the conditions of magneto-hydrodynamic equilibrium on the magnetosphere of magnetic stars. Methodology and obtained equations are new approaches that are at first considered. 展开更多
关键词 Magnetic Star MHD Equilibrium Keldysh and Tricomi Type Plasma Non-Classical equations of mathematical physics
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