Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

Abstract

Abstract. In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the "invariance" of the objects of the "Darboux theory of integrability". In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. Finally, as illustration of these results, some examples of planar vector fields coming from supersymmetric quantum mechanics are studied.

Algebraic and qualitative remarks about the family yy′ = (αxm+k–1 + βxm–k–1)y + γx2m–2k–1

“…The associated system of the Polyanin-Zaitsev vector eld, with a, b, c, m, k ∈ R, is given by: The next proposition can illustrate the cases in which the Polyanin-Saitsev vector eld is formed by non trivial polynomial functions. x = ẏ y = −cx s+p+ (9) x = ẏ y = a s+p+ yx s − a r+ x s+ − cx r (10) x = ẏ y = +b(m + p + )yx p − b mx p+ (11) x = ẏ y = a(m + s + )yx s − amx s+ (12) with s, p, r ∈ Z de ned in the proof.…”

“…On another hand, E. Picard and E. Vessiot introduced an algebraic approach to study linear di erential equations based on the Galois theory for polynomials, see [8][9][10][11][12][13]. Combination of dynamical systems with di erential Galois theory is a recent topic which started with the works of J.J Morales-Ruiz (see [12] and references therein) and with the works of J.-A.…”

“…We call as Polyanin-Zaitsev vector eld to the vector eld associated to this system of di erential equations that comes from the corrigendum of the Exercise 11 in [18, §1.3.3]. Moreover, we study integrability aspects using di erential Galois theory, following [9,19] as well qualitative aspects due to the foliation associated to Polyanin-Zaitzev vector eld is a Liénard equation, which is closely related to a Van Der Pol equation.…”

Dynamical and Algebraic Analysis of Planar Polynomial Vector Fields Linked to Orthogonal Polynomials

“…Also, G. Darboux introduced an algebraic theory to analyze the integrability of polynomial vector fields, which is known as Darboux theory of integrability [15]. The final ingredient of this paper corresponds to orthogonal polynomials [16], [17], which are very important in both theoretical and applied mathematics: they contribute to random matrices, approximation theory, trigonometric series, and especially differential equations, among others.…”

Differential Galois Groups and Representation of Quivers for Seismic Models With Constant Hessian of Square of Slowness

“…One of these applications corresponds to Morales-Ramis theory (see [32]), that is, the differential Galois theory linked with the non-integrability of dynamical systems, being the starting point to prove non-integrability of Hamiltonian and non-Hamiltonian systems, see [1][2][3][4][5]11]. Similarly, differential Galois theory has been used to study integrability and nonintegrability of polynomial vector fields (see [7,14]), integrability in quantum mechanics (see [6][7][8][9][10]12]) and integrability in quantum optics (see [13]). …”