The Interaction of Solitary Waves in Two-Fluid Magnetohydrodynamics in a Longitudinal Magnetic Field
Authors: Gavrikov M.B., Savelyev V.V. | Published: 14.02.2017 |
Published in issue: #1(70)/2017 | |
DOI: 10.18698/1812-3368-2017-1-59-77 | |
Category: Mathematics and Mechanics | Chapter: Mechanics of Liquid, Gas and Plasma | |
Keywords: two-fluid magnгtohydrodynamics, solitary waves, plasma, solitons, a longitudinal magnгtic field, wave dispersion, Lax - Wendroff difference scheme |
The article examines analytically and numerically the interactions of solitary waves in two-fluid magnetohydrodynamics (MHD). We consider the most general case of waves in a cold plasma in a longitudinal magnetic field. The main feature of this work is the use of "exact" equations, rather than an approximate approach (the model equations). We have numerically studied the solutions of a system of 8 partial differential equations. Findings of the research show that the solitary waves interact with great precision as the solitons, i. e. solitary waves being the solutions to various model equations. The considered solitary waves transfer dense, strongly magnetized plasmoids with velocities of the Alfven velocity order. As the main difference method for solving the system of equations we used the natural generalization of the classical two-step Lax - Wendroff difference scheme for hyperbolic equation.
References
[1] Novikov S.P., Manakov S.V., Pitaevsky L.P., Zakharov V.E. Theory of solitons: The inverse scattering method. N.Y., Plenum, 1984.
[2] Faddeev L.D., Takhtajan L.A. Hamiltonian methods in the theory of solitons. Berlin, Springer-Verlag, 1987.
[3] Lonngren K., Scott A., eds. Solitons in action. N.Y., Academic Press, 1978.
[4] Zakharov V.E. Collapse of langmuir waves. 1БТР, 1972, vol. 35, no. 5, pp. 908-916.
[5] Kadomtsev B.B., Petviashvili V.I. On the stability of solitary waves in weakly dispersing media. Soviet Physics DoPlady, 1970, vol. 15, p. 539.
[6] Mio K., Ogino T., Minamy K., Takeda S. Modified nonlinear Schrodinger equation for Alfven waves propagating along the magnetic field in cold plasma. J. Phys. Soc. Japan, 1976, vol. 41, pp. 265-271.
[7] Gavrikov M.B. Cold plasma aperiodic oscillations. Preprint no. 33. Moscow, Keldysh Inst. Prihl. Mat. of the Academy of Sciences of the USSR, 1991. 28 p.
[8] Braginskiy S.I. Yavleniya perenosa v plazme. Voprosy teorii plazmy. Vyp. 1 [Transport phenomena in plasma. The problems of the plasma theory. Vol. 1]. Moscow, Atomizdat Publ., 1963, pp. 183-272.
[9] Gavrikov M.B., Sorokin R.V. Homogeneous deformation of a two-fluid plasma with allowance for electron inertia. Fluid Dynamics, 2008, vol. 43, iss. 6, pp. 977-989. DOI: 10.1134/S0015462808060197
[10] Kulikovskiy A.G., Lyubimov G.A. Magnetohydrodynamics. Addison-Wesley, Reading, Mass, 1965.
[11] Morozov A.I., Solov’ev L.S. Statsionarnye techeniya plazmy v magnitnom pole. Voprosy teorii plazmy. Iss. 8 [Steady plasma flows in a magnetic field. Problems in the plasma theory]. Moscow, Atomizdat Publ., 1974, pp. 3-86.
[12] Saffman P.G. Propagating of a solitary wave along a magnetic field in a cold collision-free plasma. J. Fluid Mech., 1961, vol. 11, pp. 16-20.
[13] Gavrikov M.B., Savel’ev V.V., Tayurskiy A.A. Solitons in two-fluid magnetohydrodynamics with non-zero electron inertia. Izvestiya vuzov. PriPladnaya nelineynaya dinamika [Izvestiya VUZ. Applied Nonlinear Dynamics], 2010, vol. 18, no. 4, pp. 132-147 (in Russ.).
[14] Roach P. Computational fluid dynamics. Hermosa Publishers Albuquerque, 1976.
[15] Gavrikov M.B., Kudryashov N.A., Petrov B.A., Savelyev V.V., Sinelshchikov D.I. Solitary and periodic waves in two-fluid magnetohydrodynamics. Communications of Nonlinear Science and Numerical Simulation, 2016, vol. 38, pp. 1-7.