Family Of Second-order Differential Equations Research Articles

Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.

Group classification of a family of second-order differential equations

We find the group of equivalence transformations for equations of the form y ″ = A ( x ) y ′ + F ( y ) , where A and F are arbitrary functions. We then give a complete group classification of this family of equations using a direct method of analysis, together with the equivalence transformations.

The period function for second-order quadratic ODEs is monotone

Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [3] conjectured that all the centers encountered in the family of second-order differential equations $$\ddot x = V(x,\dot x)$$ , beingV a quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [8]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true whenV is a polynomial which nonlinear part is homogeneous of degreen>2.