On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M.

doi:10.1098/rspa.2008.0240

Cited by 13 publications

(34 citation statements)

References 27 publications

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“…To begin with, let us consider a system of two coupled SNODEs, R{t, x} (eqn (2.1) in part IV (Chandrasekar et al 2009)), t, x, y,ẋ,ẏ).…”

Section: Linearizing Transformations (A) Methods Of Deriving Linearizimentioning

confidence: 99%

“…by using the generalized modified PrelleSinger (PS) method formulated in part IV (Chandrasekar et al 2009). Then, the following theorem ensures that the transformation can be deduced from I i , i = 1, 2.…”

Section: Linearizing Transformations (A) Methods Of Deriving Linearizimentioning

confidence: 99%

“…The first two integrals associated with equation (3.1), which can be obtained using the formulation given in §2 of part IV (Chandrasekar et al 2009), can be written as…”

Section: Applicationsmentioning

confidence: 99%

“…Let us focus our attention on the two-dimensional Mathews-Lakshmanan oscillator system of the form (Cariñena et al 2004;Chandrasekar et al 2009) …”

Section: (Ii) Example 2: Generalized Sundman Transformation Of Type Imentioning

confidence: 99%

“…This study arises not only for the completeness of part IV (Chandrasekar et al 2009), but also to show the importance of unfinished tasks that exist in the theory of linearization of two coupled SNODEs. As far as the first point is concerned, we show that one can also solve a class of coupled SNODEs by transforming them into two second-order free particle equations and, from the solutions of the latter, one can construct the solution of the former, even though this is a non-trivial problem in many situations (one can also transform coupled nonlinear ODEs into uncoupled nonlinear ones, which has already been pointed out by us in the previous paper, i.e.…”

Section: Introductionmentioning

confidence: 99%

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

2009

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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.

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“…To begin with, let us consider a system of two coupled SNODEs, R{t, x} (eqn (2.1) in part IV (Chandrasekar et al 2009)), t, x, y,ẋ,ẏ).…”

Section: Linearizing Transformations (A) Methods Of Deriving Linearizimentioning

confidence: 99%

Section: Linearizing Transformations (A) Methods Of Deriving Linearizimentioning

confidence: 99%

“…The first two integrals associated with equation (3.1), which can be obtained using the formulation given in §2 of part IV (Chandrasekar et al 2009), can be written as…”

Section: Applicationsmentioning

confidence: 99%

“…Let us focus our attention on the two-dimensional Mathews-Lakshmanan oscillator system of the form (Cariñena et al 2004;Chandrasekar et al 2009) …”

Section: (Ii) Example 2: Generalized Sundman Transformation Of Type Imentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2009

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Factorization technique and isochronous condition for coupled quadratic and mixed Liénard-type nonlinear systems

Tiwari

Pandey

Chandrasekar

et al. 2015

Applied Mathematics and Computation

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In this paper, we discuss a systematic and self consistent procedure to factorize a rather general class of coupled nonlinear ordinary differential equations (ODEs), namely coupled quadratic and mixed Liénard type equations, which include various physical and mathematical models. The procedure is broadly divided into two parts. In the first part, we consider a general factorized form for the equation under consideration in terms of some unknown functions and identify the determining equations for them. In the second part, we systematically solve the determining equations and identify the compatible factorizing form for this class of equations. In addition, we also discuss the problem of identification of isochronous dynamical systems belonging to the above class of equations. In particular, we deduce an isochronicity condition for the coupled quadratic Liénard equation. We also present specific examples of physical interest.

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Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

Sakkaravarthi

Johnpillai²,

Devi

et al. 2018

Applied Mathematics and Computation

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We consider the nonlinear Helmholtz (NLH) equation describing the beam propagation in a planar waveguide with Kerr-like nonlinearity under non-paraxial approximation. By applying the Lie symmetry analysis, we determine the Lie point symmetries and the corresponding symmetry reductions in the form of ordinary differential equations (ODEs) with the help of the optimal systems of one-dimensional subalgebras. Our investigation reveals an important fact that in spite of the original NLH equation being non-integrable, its symmetry reductions are of Painlevé integrable. We study the resulting sets of nonlinear ODEs analytically either by constructing the integrals of motion using the modified Prelle-Singer method or by obtaining explicit travelling wave-like solutions including solitary and symbiotic solitary wave solutions. Also, we carry out a detailed numerical analysis of the reduced equations and obtain multi-peak nonlinear wave trains. As a special case of the NLH equation, we also make a comparison between the symmetries of the present NLH system and that of the standard nonlinear Schrödinger equation for which symmetries are long available in the literature.

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On the complete integrability and linearization of nonlinear ordinary differential equations. IV. Coupled second-order equations

Cited by 13 publications

References 27 publications

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

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

Factorization technique and isochronous condition for coupled quadratic and mixed Liénard-type nonlinear systems

Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

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