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Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains.pdf

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Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains.pdf

Commun Nonlinear Sci Numer Simulat 36 (2016) 303–318 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: /locate/cnsns Order preserving contact transformations and dynamical symmetries of scalar and coupled Riccati and Abel chains R. Gladwin Pradeep a, V.K. Chandrasekar b, R. Mohanasubha c, M. Senthilvelan c,?, M. Lakshmanan c a Department of Physics, KCG College of Technology, Karapakkam, Chennai - 600 097, India b Centre for Nonlinear Science and Engineering, School of Electrical and Electronics Engineering, SASTRA University, Thanjavur - 613 401, India c Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, India article info Article history: Received 23 April 2015 Revised 23 September 2015 Accepted 23 October 2015 Available online 30 October 2015 Keywords: Contact transformation Linearization Dynamical symmetries Riccati and Abel chains abstract We identify contact transformations which linearize the given equations in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The identi?ed contact transformations are not of Cole-Hopf type and are new to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the ?rst time. The contact transformations can be utilized to derive dynamical symmetries of the associated nonlinear ODEs. The wider applicability of identifying this type of contact transformations and the method of deriving dynamical symmetries by using them is illustrated through two dimensional generalizations of the Riccati and Abel chains as well. ? 2015 Elsevier B.V. All rights reserved. 1. Introduction Solving a nonlinear ordinary differential equation (ODE) by transforming it to a linear ODE is one of the classical methods of ?nding solutions in the theory of differential equations. Such a study is called linearization/equivalence problem [1]. The linearization can