Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms

Abstract

Abstract. Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a unique formal "correction" of the nonlinear part so that the "corrected" system is formally linearizable.Normal forms of these systems are found, providing also their classification.

A computational method for converter analysis using point symmetries

“…The illustration of a symmetry method here is based on the observation that a reversible one-parameter transformation maps the generally nonlinear system to a linear and equivalent one (see [2]). …”

A class of matrix-valued polynomials generalizing Jacobi polynomials

“…(3) has an analytic solution if and only if the first coefficient in the Jacobi series in P ((n−1)a−1,(n−1)b−1) k k=0,1,2,... of g is zero (details can be found in [4,5]). …”

“…Numerical expansions in Jacobi series yield (what appear to be) rapidly convergent series. 1 In the multi-dimensional case the problem was also solved using expansions in special vectorvalued polynomials [4]. The natural question is whether the polynomial structure found in the one-dimensional case survives in the multi-dimensional case.…”

Matrix-valued polynomials generated by the scalar-type Rodrigues’ formulas

“…(i) The case deg Q = 1, say Q(x) = x, corresponds in the one-dimensional case to the Laguerre polynomials and L (α) n have the representation (5) and (6) with A k = −x +(k +α)+x∂ x . 1 However, a special instance was used by the author in [7].…”

“…Matrix-valued polynomials satisfying classical Rodrigues' formulas have appeared naturally in classification problems of differential equations [6], and the present paper investigates the properties of these polynomials; there are no assumptions on the "weight".…”