On Representation of the P-Q Pair Solution at the Singular Point Neighborhood

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived. The inverse monodromic (or isomonodromic) transformation (IMT) method [1, 2] is a powerful tool for studying the class of nonlinear ordinary differential equations (ODE’s) representable as the compatibility condition of the overdetermined linear system (P–Q pair). The IMT method reduces the initial value problem for the system of nonlinear ODE’s to solving the inverse problem for associated isomonodromic linear equation. This inverse problem is formulated in terms of the monodromy data, which are constructed using the asymptotic expansions of solution of the P–Q pair at neighborhood of the singular points. It was shown for particular cases that the matrix coefficient of the isomonodromic equation can be uniquely specified from the monodromy properties of its global solution [3]. The existence of the global solution can be investigated in the frameworks of the theory of the Riemann–Hilbert problem [2]. This work is devoted to obtaining in closed form of the expansion in series of solution of P–Q pair at neighborhood of the singular points. The problem considered here is important for developing the IMT method on P–Q pairs of arbitrary matrix dimension with different types of singularities of both the equations forming the pair. In particular, the compatibility of the equations, that are imposed on the remainder term of the asymptotic expansion of the P–Q pair solution, is suggested in implementing the direct problem of the IMT method for the monodromy data to be determined. The irregular and regular singularities of the second equation of P–Q pair (Q–equation) are studied in Section 3 and Section 4 respectively. We require no the additional conditions on the coefficient of Q–equation such as the inequality (on modulo integers for the regular singularity) of the eigenvalues of the leading coefficients of the expansions in singular points (cf. [3]). Besides, the independent variable of the first equation of P–Q pair is not supposed to be immediately connected with the subset of the “deformation parameters” of the monodromy data (see Ref. [3]). The theorems establishing the existence of the compatible expansion in series at the singular point neighborhood of solutions of the P–Q pair equations are proven. The conservation laws for system of nonlinear ODE’s admitting the compatibility condition representation are derived from the expansions in series of solution of corresponding P–Q pair.