×

Generalized conditional symmetries, related solutions and conservation laws of the Grad-Shafranov equation with arbitrary flow. (English) Zbl 1420.35229

Summary: The generalized conditional symmetry (GCS) method is applied to the case of a generalized Grad-Shafranov equation (GGSE) with incompressible flow of arbitrary direction. We investigate the conditions which yield the GGSE that admits a special class of second-order GCSs. Three GCS generators and the associated families of invariant solutions are pointed out. Several plots of the level sets or flux surfaces of the new solutions are displayed. These results extend the recent solutions with 5 parameters recently obtained on the basis of Lie-point symmetries. They could be useful in the study of plasma equilibrium, of transport phenomena, and of magnetohydrodynamic stability. Futher, by making use of the multiplier’s method, three nontrivial conservation laws that are admitted by the concerned equation and which involve arbitrary functions, are highlighted.

MSC:

35Q35 PDEs in connection with fluid mechanics
58J70 Invariance and symmetry properties for PDEs on manifolds
34C14 Symmetries, invariants of ordinary differential equations
35L65 Hyperbolic conservation laws
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anco, S. C.; Bluman, G. W., Direct construction method for conservation laws of partial differential equations, Part II: General treatment, Eur. J. Appl. Math., 9, 567-585 (2002) · Zbl 1034.35071
[2] Bluman, G.; Cheviakov, A. F.; Ivanova, N. M., Framework for nonlocally related partial differential equation systems and nonlocal symmetries: Extension, simplification, and examples, J. Math. Phys., 47 (2006) · Zbl 1112.35010 · doi:10.1063/1.2349488
[3] Bluman, G. W.; Cole, J. D., The general similarity solution of the heat equation, J. Math. Mech., 18, 1025-1042 (1969) · Zbl 0187.03502
[4] Cerfon, A. J.; Freidberg, J. P., “One size fits all” analytic solutions to the Grad-Shafranov equation, Plasma Phys., 17, 3 (2010) · doi:10.1063/1.3328818
[5] Cimpoiasu, R., Generalized conditional symmetries and related solutions of the Grad-Shafranov equation, Phys. Plasmas, 21 (2014) · doi:10.1063/1.4871857
[6] Clarkson, P. A.; Kruskal, M. D., New similarity reductions of the Boussinesq equation, J. Math. Phys., 30, 2201-2213 (1989) · Zbl 0698.35137 · doi:10.1063/1.528613
[7] Constantinescu, R., Generalized conditional symmetries and related solutions of the Klein Gordon Fock equation with central symmetry, Rom. J. Phys., 61, 77-88 (2016)
[8] Hydon, P. E., How to construct the discrete symmetries of partial differential equations, Eur. J. Appl. Math., 11, 515-527 (2000) · Zbl 1035.35005 · doi:10.1017/S0956792500004204
[9] Ibragimov, N. H.; Kolsrud, T., Lagrangian approach to evolution equations: Symmetries and conservation laws, Nonlinear Dynam., 36, 29-40 (2004) · Zbl 1106.70012 · doi:10.1023/B:NODY.0000034644.82259.1f
[10] Kuiroukidis, Ap.; Throumoulopoulos, G. N., Equilibria with incompressible flows from symmetry analysis, Phys. Plasmas, 22, 8 (2015) · Zbl 1492.76149
[11] Ibragimov, N. H., A new conservation theorem, J. Math. Anal. Appl., 333, 311-428 (2007) · Zbl 1160.35008 · doi:10.1016/j.jmaa.2006.10.078
[12] Kaltsas, D.; Throumoulopoulos, G. N., Generalized Solovev equilibrium with sheared flow of arbitrary direction and stability consideration, Phys. Plasmas, 21, 8 (2014) · doi:10.1063/1.4892380
[13] Kara, A. H.; Mahomed, F. M., Relationship between symmetries and conservation laws, Int. J. Theor. Phys., 39, 23-40 (2000) · Zbl 0962.35009 · doi:10.1023/A:1003686831523
[14] Kuiroukidis, A.; Throumoulopoulos, G. N., New classes of exact solutions to the Grad-Shafranov equation with arbitrary flow using Lie-point symmetries, Phys. Plasmas, 23, 11 (2016)
[15] Kunzinger, M.; Popovych, R. O., Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl., 379, 444-460 (2016) · Zbl 1211.35021 · doi:10.1016/j.jmaa.2011.01.027
[16] Qu, C., Group classification and generalized conditional symmetry reduction of the nonlinear diffusionconvection euation with a nonlinear source, Stud. Appl. Math., 92, 2, 107-136 (1997) · Zbl 0879.35031 · doi:10.1111/1467-9590.00058
[17] Qu, C., Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math., 62, 3, 283-302 (1999) · Zbl 0936.35039 · doi:10.1093/imamat/62.3.283
[18] Leveque, R. J., Numerical Methods for Conservation Laws, Lectures in Mathematics (ETH Zurich, Birkhauser Verlag, 1992). · Zbl 0847.65053
[19] Lina, J., Conditional Lie-Backlund symmetries and diferential constraints for inhomogeneous nonlinear difusion equations due to linear determining equations, J. Math. Anal. Appl., 440, 286-299 (2016) · Zbl 1381.35005 · doi:10.1016/j.jmaa.2016.03.047
[20] Liu, Q. M.; Fokas, A. S., Exact interaction of solitary waves for certain nonintegrable equations, J. Math. Phys., 37, 324-345 (1996) · Zbl 0861.35095 · doi:10.1063/1.531393
[21] Maschke, E. K., Exact solutions of the MHD equilibrium equation for a toroidal plasma, Plasma Phys., 15, 535-543 (1973) · doi:10.1088/0032-1028/15/6/006
[22] Negrea, M.; Petrisor, I.; Weyssow, B., Role of stochastic anisotropy and shear on magnetic field lines diffusion, Plasma Phys. Contr. Fusion, 49, 1767-1781 (2007) · doi:10.1088/0741-3335/49/11/002
[23] Negrea, M.; Petrisor, I.; Weyssow, B., Influence of magnetic shear and stochastic electrostatic field on the electron diffusion, J. Optoelectron. Adv., 10, 1942-1945 (2008)
[24] Negrea, M.; Petrisor, I.; Constantinescu, D., Aspects of the diffusion of electrons and ions in tokamak plasma, Rom. J. Phys., 55, 1013-1023 (2010)
[25] Olver, P. J., Applications of Lie groups to differential equations (1993), New York: Springer-Verlag, New York · Zbl 0785.58003
[26] Shalchi, A.; Negrea, M.; Petrisor, I., Stochastic field-line wandering in magnetic turbulence with shear. I. Quasi-linear theory, Phys. Plasmas, 23 (2016) · doi:10.1063/1.4958809
[27] Simintzis, Ch.; Throumoulopoulos, G. N.; Pantis, G.; Tasso, H., Analytic magnetohydrodynamic equilibria of a magnetically confined plasma with sheared flows, Phys. Plasmas, 8, 6, 2641-2648 (2001) · doi:10.1063/1.1371768
[28] Solovev, S., Sov. Theory of hydromagnetic stability of toroidal plasma configurations, Phys. JETP, 53, 626-643 (1967) · Zbl 0172.57401
[29] Srinivarsan, R.; Lao, L.; Chu, M. S., Analytical description of poloidally diverted tokamak equilibrium with linear stream functions, Plasma Phys. Controlled Fusion, 52, 3 (2010)
[30] Steinbrecher, G.; Negrea, M.; Pometescu, N.; Misguich, J. H., On non-Markovian relative diffusion of stochastic magnetic lines, Plasma Phys. Contr. Fusion, 49, 12, 2039-2049 (1997) · doi:10.1088/0741-3335/39/12/007
[31] Tasso, H.; Throumoulopoulos, G. N., Axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows, Phys. Plasmas, 5, 6, 2378-2383 (1998) · doi:10.1063/1.872912
[32] Throumoulopoulos, G. N.; Tasso, H., International thermonuclear experimental reactor-like extended Solovév equilibria with parallel flow, Phys. Plasmas, 19, 1 (2012) · doi:10.1063/1.3672509
[33] Wang, J.; Lina, J., Conditional Lie Backlund symmetry, second-order diferential constraint and direct reduction of difusion systems, J. Math. Anal. Appl., 427, 1101-1118 (2015) · Zbl 1370.37132 · doi:10.1016/j.jmaa.2015.02.059
[34] Zhdanov, R. Z., Conditional Lie-Backlund symmetry and reduction of evolution equations, J. Phys. A, 28, 3841-3853 (1995) · Zbl 0859.35115 · doi:10.1088/0305-4470/28/13/027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.