To integrability

of nonlinear second-order ordinary differential equation

M.S.Boquien, S.É.Bouquet, P.G.L.Leach

Commissariat à l'Énergie Atomique, Département de Physique Théorique et Appliquée, DPTA-SPPE, BP12, F-91680 BRUYÈRES-LE-CHÂTEL

FRANCE

Abstract. The ordinary differential equation, , where љand љare two arbitrary constants is studied. This equation, describing the motion of an electron in the electromagnetic field of a magnetron device, was discovered seventy years ago. Yet, its integrability properties were not clearly known up to now. Kamke [Kamke E., Differentialgleichungen: Lösungmethoden und Lösungen, B.G.Teubner, Stuttgart (1983)] does not provide any analytical solution for this equation, but he points out that it can be integrated in a graphical way. A few years ago Cheb-Terrab and Roche [Cheb-Terrab E.S. and Roche A.D., Integrating Factors for Second Order ODEs, J. Sym. Comp. 27 (1999) 501-519] found an integrating factor and concluded that the equation was integrable. Very recently Leach and Bouquet [Leach P.G.L. and Bouquet S.É., Symmetries and Integrating Factors, J. Nonlin. Math. Phys. 9 (2002) 73-91] reached the opposite conclusion. They were not able to determine any symmetries for it, thus the reduction of its order was not possible. Considering this disagreement the equation is revisited by combining Lie group theory and Hamiltonian formalism. New solutions, playing a basic physical role with fundamental properties, are presented.

 

 

Médéric Boquien actually pursue his studies at l'EPF‑École d'Ingénieurs, in the option Energy and Environment.

S.É. Bouquet studied under the noted plasma physicist, M.R.Feix, at the Université d'Orléans for his first doctorate; since completing that degree he has been with the Commissariat Energie Atomique, firstly at Moron-Villiers and more recently at Saclay and neighbouring laboratories; a few years ago he completed his Habilation.

Peter Gavin Lawrence Leach, Dr., Graduate of the University of Melbourne and La Trobe University in Australia and the University of Natal in South Africa; Professor of Mathematics of the University of Natal, Durban; formerly Professor of Applied Mathematics, University of the Witwatersrand, Johannesburg; active in the applications of symmetry and singularity analyses to differential equations arising in Mechanics, Quantum Mechanics, Cosmology, Mathematical Economics and Mathematical Biosciences; also interested in problems of molecular structure.