Multi-particle dynamical systems and polynomials

Demina, Maria V.; Kudryashov, Nikolay A.

doi:10.1134/s1560354716030072

Cited by 8 publications

(6 citation statements)

References 44 publications

(85 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Exceptional polynomials appear in a number of applications in mathematical physics, mostly as solutions to exactly solvable quantum mechanical problems describing bound states [25,45,49]. They appear also in connection with super-integrable systems [38,48], exact solutions to Dirac's equation [50], diffusion equations and random processes [32], finite-gap potentials [31] and point vortex models [34].…”

Section: Application To Exceptional Hermite Polynomialsmentioning

confidence: 99%

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

2018

View full text Add to dashboard Cite

We derive identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize an equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, and describe the solution using an elegant graphical interpretation. The results are applied to provide a more efficient representation for exceptional Hermite polynomials and for rational solutions of the Painlevé IV equation. The latter are expressed in terms of the Okamoto and generalized Hermite polynomials.

show abstract

Section: Application To Exceptional Hermite Polynomialsmentioning

confidence: 99%

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

2018

View full text Add to dashboard Cite

show abstract

“…Moreover, comparing (2.17) with (2.5) and (2.16) we conclude that the zeros of consecutive Adler-Moser polynomials are stationary configurations (or equivalently, μ = (μ 1 , μ 2 ) defined in (2.13) is a critical vector measure) for the vector energy E ϕ,a ( μ) defined in (2.14), with ϕ ≡ (0, 0) and a = − 1 2 , fact that was already observed in [17]. Further generalizations of these ideas, and in particular, their relation to rational solutions of Painlevé equations, can be found in [15,16,30,34,35,37,38,57,62,78], to cite a few.…”

Section: Definition 22mentioning

confidence: 73%

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Martínez–Finkelshtein

Orive

Sánchez-Lara

2022

Constr Approx

View full text Add to dashboard Cite

For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect to w allows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasi-orthogonal polynomials, we recover some of the known results and generalize others. Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems. We also discuss the discrete-to-continuous transition of these models in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.

show abstract

“…Note that if r = 0 then the partition λ(0) = ∅ which has only 0 parts, and when r = m the last multi-index in the lexicographic order, λ(m) = λ. Note that the index sets (03) and ( 12) are incomparable although both (03) ( 13), (12) (13). The partitions λ(r) corresponding to r = (r 1 r 2 ) are given by λ((01)) = (0, 0) = ∅, λ((02)) = (0, 1), λ((03)) = (0, 2), λ(( 12)) = (1, 1), λ(( 13)) = (1, 2), and λ(( 23)) = (2, 2).…”

Section: Definition 31 (Partition)mentioning

confidence: 99%

Classification of KPI lumps

Chakravarty¹,

Zowada²

2022

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

A large family of nonsingular rational solutions of the Kadomtsev-Petviashvili (KP) I equation are investigated. These solutions are constructed via the Gramian method and are identiﬁed as points in a complex Grassmannian. Each solution is a traveling wave moving with a uniform background velocity but have multiple peaks which evolve at a slower time scale in the co-moving frame. For large times, these peaks separate and form well-deﬁned wave patterns in the xy-plane. The pattern formation are described by the roots of well-known polynomials arising in the study of rational solutions of Painlev´e II and IV equations. This family of solutions are shown to be described by the classical Schur functions associated with partitions of integers and irreducible representations of the symmetric group of N objects. It is then shown that there exists a one-to-one correspondence between the KPI rational solutions considered in this article and partitions of a positive integer N.

show abstract

Multi-particle dynamical systems and polynomials

Cited by 8 publications

References 44 publications

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials

Classification of KPI lumps

Contact Info

Product

Resources

About