arXiv subjects

Xu-Dan Luo

Publications and source records attributed to Xu-Dan Luo.

9 recordsLinked to original sources

arXiv2024

Breather gas fission from elliptic potentials in self-focusing media

We present an analytical model of integrable turbulence in the focusing nonlinear Schrödinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter m of the potential. We then demonstrate that, upon augmenting the potential by a small random noise (which is inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon we call breather gas fission. We show that the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential. We analytically compute the kurtosis of the breather gas as a function of the elliptic parameter m, and we show that it is greater than 2 for all non-zero m, implying non-Gaussian statistics. Finally, we verify the theoretical predictions by comparison with direct numerical simulations of the fNLS equation. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases.

nlin.SI↗

arXiv2023

Elliptic finite-band potentials of a non-self-adjoint Dirac operator

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials for a non-self-adjoint Dirac operator which connects two previously known limiting cases in which the elliptic parameter is zero or one. A full characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space and to appropriate connection problems for Heun's equation. In turn, these problems are related to four non-self-adjoint unbounded tridiagonal operators, all of which nonetheless have only real eigenvalues. For certain parameter values, the corresponding elliptic potentials generate finite-genus solutions for all the positive and negative flows of the focusing nonlinear Schrödinger hierarchy.

math.SP↗

arXiv2020

Instability intervals of the Whittaker-Hill operator

The Hill operator admits a band gap structure. As a special case, like the Mathieu operator, one has only open gaps, however, the instability intervals of the Whittaker-Hill operator may be open or closed. In 2007, P. Djakov and B. Mityagin gave the asymptotics of band gaps for a special Whittaker-Hill operator [P. Djakov and B. Mityagin, J. Funct. Anal., 242, 157-194 (2007).]. In this paper, a more general Whittaker-Hill operator is considered and the asymptotics of the instability intervals are studied.

math.SP↗

arXiv2018

Solitons, the Korteweg-de Vries equation with step boundary values and pseudo-embedded eigenvalues

The Korteweg-deVries (KdV) equation with step boundary conditions is considered, with an emphasis on soliton dynamics. When one or more initial solitons are of sufficient size they can propagate through the step; in this case the phase shift is calculated via the inverse scattering transform. On the other hand, when the amplitude is too small they become trapped. In the trapped case the transmission coefficient of the associated associated linear Schrödinger equation can become large at a point exponentially close to the continuous spectrum. This point is referred to as a {\it pseudo-embedded eigenvalue}. Employing the inverse problem it is shown that the continuous spectrum associated with a branch cut in the neighborhood of the pseudo-embedded eigenvalue plays the role of discrete spectra, which in turn leads to a trapped soliton in the KdV equation.

nlin.SI↗

arXiv2017

General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

General soliton solutions to a nonlocal nonlinear Schrödinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions {are considered} via the combination of Hirota's bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. First, general $N$-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.

nlin.SI↗

arXiv2017

Inverse scattering transform for the nonlocal reverse space-time Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations with nonzero boundary conditions

The reverse space-time (RST) Sine-Gordon, Sinh-Gordon and nonlinear Schrödinger equations were recently introduced and shown to be integrable infinite-dimensional dynamical systems. The inverse scattering transform (IST) for rapidly decaying data was also constructed. In this paper, IST for these equations with nonzero boundary conditions (NZBCs) at infinity is presented. The NZBC problem is more complicated due to the associated branching structure of the associated linear eigenfunctions. With constant amplitude at infinity, four cases are analyzed; they correspond to two different signs of nonlinearity and two different values of the phase at infinity. Special soliton solutions are discussed and explicit 1-soliton and 2-soliton solutions are found. In terms of IST, the difference between the RST Sine-Gordon/Sinh-Gordon equations and the RST NLS equation is the time dependence of the scattering data. Spatially dependent boundary conditions are also briefly considered.

math-ph↗

arXiv2016

Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions

In 2013 a new nonlocal symmetry reduction of the well-known AKNS scattering problem was found; it was shown to give rise to a new nonlocal $PT$ symmetric and integrable Hamiltonian nonlinear Schrödinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one soliton solution were found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in the four cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem is developed via a left-right Riemann-Hilbert problem in terms of a suitable uniformization variable and the time dependence of the scattering data is obtained. This leads to a method to linearize/solve the Cauchy problem. Pure soliton solutions are discussed and explicit 1-soliton solution and two 2-soliton solutions are provided for three of the four different cases corresponding to two different signs of nonlinearity and two different values of the phase difference between plus and minus infinity. In the one other case there are no solitons.

nlin.SI↗

arXiv2015

On the zeros of Confluent Hypergeometric Functions

In this paper, we study the zero sets of the confluent hypergeometric function $_{1}F_{1}(α;γ;z):=\sum_{n=0}^{\infty}\frac{(α)_{n}}{n!(γ)_{n}}z^{n}$, where $α, γ, γ-α\not\in \mathbb{Z}_{\leq 0}$, and show that if $\{z_n\}_{n=1}^{\infty}$ is the zero set of $_{1}F_{1}(α;γ;z)$ with multiple zeros repeated and modulus in increasing order, then there exists a constant $M>0$ such that $|z_n|\geq M n$ for all $n\geq 1$.

math.CA↗

arXiv2015

Some new properties of Confluent Hypergeometric Functions

The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(α;γ;z):=\sum_{n=0}^{\infty}\frac{(α)_{n}}{n!(γ)_{n}}z^{n}\ (γ\neq 0,-1,-2,\cdots)$, which are of many properties and great applications in statistics, mathematical physics, engineering and so on, have been given. In this paper, we investigate some new properties of ${}_{1}F_{1}(α;γ;z)$ from the perspective of value distribution theory. Specifically, two different growth orders are obtained for $α\in \mathbb{Z}_{\leq 0}$ and $α\not\in \mathbb{Z}_{\leq 0}$, which are corresponding to the reduced case and non-degenerated case of ${}_{1}F_{1}(α;γ;z)$. Moreover, we get an asymptotic estimation of characteristic function $T(r,{}_{1}F_{1}(α;γ;z))$ and a more precise result of $m\left(r, \frac{{}_{1}F_{1}'(α;γ;z)}{{}_{1}F_{1}(α;γ;z)}\right)$, compared with the Logarithmic Derivative Lemma. Besides, the distribution of zeros of the confluent hypergeometric functions is discussed. Finally, we show how a confluent hypergeometric function and an entire function are uniquely determined by their $c$-values.

math.CV↗