Interconnections between various analytic approaches applicable to third-order nonlinear differential equations

Abstract

We unearth the interconnection between various analytical methods which are widely used in the current literature to identify integrable nonlinear dynamical systems described by third-order nonlinear ODEs. We establish an important interconnection between the extended Prelle-Singer procedure and λ-symmetries approach applicable to third-order ODEs to bring out the various linkages associated with these different techniques. By establishing this interconnection we demonstrate that given any one of the quantities as a starting point in the family consisting of Jacobi last multipliers, Darboux polynomials, Lie point symmetries, adjointsymmetries, λ-symmetries, integrating factors and null forms one can derive the rest of the quantities in this family in a straightforward and unambiguous manner. We also illustrate our findings with three specific examples.

Interplay of symmetries and other integrability quantifiers in finite-dimensional integrable nonlinear dynamical systems

“…The interconnection between λ-symmetries and the Prelle-Singer method for the second-and third-order ODEs was discussed elaborately in [1][2][3]. Here, we generalize the interlink to nthorder ODEs.…”

“…To demonstrate the interconnection in third-order ODEs, we again consider the same example as we considered in our earlier work [2]; that is,…”

“…Substituting the null forms (3.2) and the integrating factors (3.3) in (2.52), we obtain (ζ (1) = −RS) 2 and ζ…”

“…The corresponding generalized λ-symmetries can be written as (telescopic vector fields) (2) and γ (2) .…”

“…In two of our earlier works [1,2], we have interconnected six widely used analytical methods for solving ordinary differential equations (ODEs), namely the extended Prelle-Singer method, Lie symmetry analysis, the Jacobi last multiplier method, the Darboux method, the adjointsymmetries method and the λ-symmetries approach by considering second-and third-order ODEs. Progressing further, in this paper, we unearth the interconnection between the extended Prelle-Singer procedure with two procedures, namely (i) contact symmetries and (ii) generalized λ-symmetries.…”