arXiv subjects

Guo-Fu Yu

Publications and source records attributed to Guo-Fu Yu.

27 records · Page 2Linked to original sources

arXiv2018

Galoisian approach to complex oscillation theory of some Hill equations

We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That is, solutions are obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential has four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions in $\mathbf{C}$ that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.

math.CA↗

arXiv2018

Bright-dark soliton solutions to the multi-component AB system

In this paper we investigate the multi-component AB system that comes from the geophysical fluid dynamics. We construct bright-dark soliton solutions through Hirota's bilinear method. For the two-component AB system, asymptotic behaviours of two-soliton solution are obtained and interactions between two bright and two dark solitons are proved to be elastic. Under different parameter conditions, the oblique interactions, bound states of solitons are analyzed in details. Meanwhile, by use of the Pfaffian technique, we present $N$-bright and $N-$dark soliton solutions to the two- and multi-component AB system. The results will be meaningful for the study of vector multi-dark solitons in many physical systems such as nonlinear optics and fluid dynamics.

nlin.SI↗

arXiv2017

Symmetric abstract hypergeometric polynomials

Consider an abstract operator $L$ which acts on monomials $x^n$ according to $L x^n= λ_n x^n + ν_n x^{n-2}$ for $λ_n$ and $ν_n$ some coefficients. Let $P_n(x)$ be eigenpolynomials of degree $n$ of $L$: $L P_n(x) = λ_n P_n(x)$. A classification of all the cases for which the polynomials $P_n(x)$ are orthogonal is provided. A general derivation of the algebras explaining the bispectrality of the polynomials is given. The resulting algebras prove to be central extensions of the Askey-Wilson algebra and its degenerate cases.

math.CA↗

arXiv2016

On complex oscillation theory, quasi-exact solvability and Fredholm Integral Equations

Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere $\hat{\mathbb{C}}$. Motivated by a Nevanlinna theory (complex oscillation theory) approach, we have established a theory of \textit{periodic} BHE (PBHE) in parallel with the Lamé equation verses the Heun equation, and the Mathieu equation verses the confluent Heun equation. We have established condition that lead to explicit construction of eigen-solutions of PBHE, and their single and double orthogonality, and a related first-order Fredholm-type integral equation for which the corresponding eigen-solutions must satisfy. We have also established a Bessel polynomials analogue at the BHE level which is based on the observation that both the Bessel equation and the BHE have a regular singular point at the origin and an irregular singular point at infinity on the Riemann sphere $\hat{\mathbb{C}}$, and that the former equation has orthogonal polynomial solutions with respect to a complex weight. Finally, we relate our results to an equation considered by Turbiner, Bender and Dunne, etc concerning a quasi-exact solvable Schrödinger equation generated by first order operators such that the second order operators possess a finite-dimensional invariant subspace in a Lie algebra of $SL_2(\mathbb{C})$

math.CA↗

arXiv2016

Supersymmetry of the quantum rotor

The quantum rotor is shown to be supersymmetric. The supercharge $Q$, whose square equals the Hamiltonian, is constructed with reflection operators. The conserved quantities that commute with $Q$ form the algebra $so(3)_{-1}$, an anticommutator version of $so(3)$. The subduced representation of $so(3)_{-1}$ on the space of spherical harmonics with total angular momentum $j$ is constructed and found to decompose into two irreducible components. Two natural bases for the irreducible representation spaces of $so(3)_{-1}$ are introduced and their overlap coefficients prove expressible in terms of orthogonal polynomials of a discrete variable called anti-Krawtchouk polynomials.

math-ph↗

arXiv2015

Two integrable differential-difference equations derived from NLS-type equation

Two integrable differential-difference equations are derived from a (2+1)-dimensional modified Heisenberg ferromagnetic equation and a resonant nonlinear Schröinger equation respectively. Multi-soliton solutions of the resulted semi-discrete systems are given through Hirota's bilinear method. Elastic and inelastic interaction behavior between two solitons are studied through the asymptotic analysis. Dynamics of two-soliton solutions are shown with graphs.

nlin.SI↗

arXiv2015

-1 Krall-Jacobi Polynomials

We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists in the continuous measure of the little -1 Jacobi polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type polynomials with a point mass inside the orthogonality interval. These polynomials can be obtained by a Geronimus transform of the little $q$-Jacobi polynomials in the limit $q=-1$.

math.CA↗

arXiv2015

On an integrable system related to the relativistic Toda lattice -Bäcklund transformation and integrable discretization

We study an integrable system related to the relativistic Toda lattice. The bilinear representation of this lattice is given and the Bäckulund transformation obtained. A fully discrete version is also introduced with its bilinear Bäcklund transformation and Lax pair. One-soliton solution of the discrete system is presented by use of Bäcklund transformation.

nlin.SI↗

arXiv2014

$q$-Rotations and Krawtchouk polynomials

An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown to arise as matrix elements of unitary "$q$-rotation" operators expressed as $q$-exponentials in the $\mathcal{U}_{q}(sl_{2})$ generators. The properties of the polynomials (orthogonality relation, generating function, structure relations, recurrence relation, difference equation) are derived by exploiting the algebraic setting. The results are extended to another family of polynomials, the affine $q$-Krawtchouk polynomials, through a duality relation.

math-ph↗