Solvability of Systems of Differential Equations by Complex Symmetry Analysis

Abstract

Complex symmetry analysis (CSA) establishes a connection between a class of n and 2n, dimensional complex and real systems of ordinary/partial differential equations (ODEs/PDEs), respectively. Similarly, another class of systems of PDEs is extractable from the base complex systems of ODEs by CSA. The equivalence of the base complex systems under an invertible transformation of the dependent and independent variables has been exploited to study the corresponding real systems. Of particular interest is the extension of Lie groups of transformations developed to solve scalar and systems of non-linear second order ODEs, to two and four dimensional systems of the same order, respectively. For scalar equations, they may be used to either reduce them to the free particle equation (linearize) or integrate them. The former requires an 8-dimensional algebra while a 2-dimensional solvable algebra is sufficient to apply the latter. In this thesis, those two and four dimensional systems are characterized that arise from linearizable or integrable scalar and two dimensional systems of second order ODEs, respectively. Further, invariance of systems of PDEs under equivalence transformations is studied with CSA by employing the invariance properties of the base complex PDEs. Three linearizable classes of two dimensional systems of cubically semi-linear (in the first derivative) second order ODEs has appeared so far in the literature. A comparison of two of the classes, obtainable from geometric methods and CSA, is presented. Both these classes are transformable to the system of free particle equations subject to certain linearization conditions, even though their general (cubic) semi-linear forms are proved to be inequivalent under point transformations. There are five equivalence classes of two dimensional linearizable systems of second order ODEs namely, those with 5, 6, 7, 8 and 15-dimensional Lie algebras. For those systems that arise from a scalar complex linearizable second order ODE, treated as a pair of real ODEs, a reduced optimal canonical form is established. Of the five only three equivalence classes with 6, 7 or 15-dimensional algebras are recovered by this procedure. Both the equations of these systems are found to satisfy Cauchy-Riemann (CR) equations with respect to the dependent variables. Therefore, here as elsewhere in this thesis, such systems are called CR-structured systems. A class of non-linearizable two dimensional CR-structured systems of second order ODEs is presented to show that the linearizability of the scalar complex equations is not sufficient to map the emerging systems to linear forms. A general system of n second order ODEs with 2n symmetry generators may not be amenable to quadratures by real symmetry analysis. However, it is shown that the CR-structured systems may be solvable by a procedure called complex-linearization even if they have fewer symmetries than required to linearize or integrate them. A symmetry generator of the base complex ODE associates a pair of Lie-like operators with the CR-structured systems. It is proved that all such operators are not necessarily real symmetries of the emerging system. A criterion has been developed which shows when and how the real symmetry generators of the CR-structured systems of two second order ODEs are extractable from the associated complex Lie symmetries of the base ODEs. The most general complex-linearizable form and the complex-linearization criteria for four dimensionalsystemsofsecondorderODEsarederivedbyextendingthegeometriclinearizationcriteria presented for two dimensional systems of cubically semi-linear second order ODEs. Two canonical forms of such systems have been derived by employing CSA on a system of dimension two once and a scalar equation twice. A specific form of the complex linearizing transformations associated with the base two dimensional systems is shown to furnish the reduction of the corresponding four dimensional complex-linearizable systems to the free particle Newtonian systems. Semi-invariants for a class of systems of two linear parabolic type PDEs in two independent variables under equivalence transformations of the dependent variables have been deduced. This class of systems of two linear parabolic type PDEs and the real transformations that map such systems into themselves with different coefficients in general, are shown to correspond to complex scalar linear parabolic equations and associated complex transformations, respectively. Moreover, the semi-invariants for such systems of PDEs also correspond to complex Ibragimov invariants of the complex scalar linear parabolic PDEs. Particular cases of systems of parabolic type equations, i.e., when they are uncoupled or coupled in a special manner, have been studied with CSA.