On the complete integrability and linearization of nonlinear ordinary differential equations. V. Linearization of coupled second-order equations

Linearization of coupled second-order nonlinear ordinary differential equations (SNODEs) is one of the open and challenging problems in the theory of differential equations. In this paper, we describe a simple and straightforward method to derive linearizing transformations for a class of two coupled SNODEs. Our procedure gives several new types of linearizing transformations of both invertible and non-invertible kinds. In both cases, we provide algorithms to derive the general solution of the given SNODE. We illustrate the theory with potentially important examples.

[1]  M. Plank.  Hamiltonian structures for the n‐dimensional Lotka–Volterra equations , 1995 .

[2]  A. Qadir.  Geometric Linearization of Ordinary Differential Equations , 2007, 0711.0814.

[3]  A non-linear oscillator with quasi-harmonic behaviour: two- and n-dimensional oscillators , 2004, math-ph/0406002.

[4]  N. Ibragimov,et al.  Elementary Lie Group Analysis and Ordinary Differential Equations , 1999 .

[5]  V. K. Chandrasekar,et al.  On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[6]  P. Olver.  Equivalence, Invariants, and Symmetry: References , 1995 .

[7]  Joel Merker,et al.  Characterization of the Newtonian Free Particle System in $m\geqslant 2$ Dependent Variables , 2006 .

[8]  P. Enders,et al.  An embedding technique for the solution of coupled Riccati equations , 1992 .

[9]  F. Mahomed,et al.  THE LIE ALGEBRA sl(3, R) AND LINEARIZATION , 1989 .

[10]  M. Feix,et al.  Families of invariants of the motion for the Lotka–Volterra equations: The linear polynomials family , 1992 .

[11]  Léon Brenig,et al.  Complete factorisation and analytic solutions of generalized Lotka-Volterra equations , 1988 .

[12]  M. Euler,et al.  Sundman Symmetries of Nonlinear Second-Order and Third-Order Ordinary Differential Equations , 2004 .

[13]  L. G. S. Duarte,et al.  Linearization under nonpoint transformations , 1994 .

[14]  Vladimir V. Sokolov,et al.  Generalized Operator Yang-Baxter Equations, Integrable ODEs and Nonassociative Algebras , 2000 .

[15]  F. Schwarz,et al.  Symmetries and first integrals for dissipative systems , 1984 .

[16]  V. K. Chandrasekar,et al.  On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations , 2004, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  S. Meleshko,et al.  Conditions for linearization of a projectable system of two second-order ordinary differential equations , 2008 .

[18]  G. Bluman,et al.  Symmetry and Integration Methods for Differential Equations , 2002 .

[19]  Characterization of the Newtonian Free Particle System in Dependent Variables , 2006 .

[20]  Eduardo Martínez,et al.  Linear connections for systems of second-order ordinary differential equations , 1996 .

[21]  V. K. Chandrasekar,et al.  O ct 2 00 5 On the complete integrability and linearization of nonlinear ordinary differential equations-Part II : Third order equations , 2005 .

[22]  Alain Goriely,et al.  Algebraic degeneracy and partial integrability for systems of ordinary differential equations , 1990 .

[23]  T. Wolf,et al.  Linearisable Third-Order Ordinary Differential Equations and Generalised Sundman Transformations: The Case X′′′=0 , 2002 .

[24]  D. Grossman.  Torsion-free path geometries and integrable second order ODE systems , 2000 .

[25]  M Senthilvelan,et al.  On the complete integrability and linearization of nonlinear ordinary differential equations. II. Third-order equations , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[26]  F. Mahomed,et al.  Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations , 2007 .

[27]  M. Fels.  The Equivalence Problem for Systems of Second order Ordinary Differential Equations , 1995 .

[28]  Fazal M. Mahomed,et al.  Linearization criteria for a system of second-order ordinary differential equations , 2001 .