Interaction among a lump,periodic waves,and kink solutions to the fractional generalized CBS‐BK equation

The Hirota bilinear method is prepared for searching the diverse soliton solutions for the fractional generalized Calogero‐Bogoyavlenskii‐Schiff‐Bogoyavlensky‐Konopelchenko (CBS‐BK) equation. Also, the Hirota bilinear method is used to finding the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and multi‐kink soliton solutions will be investigated. Also, the solitary wave, periodic wave, and cross‐kink wave solutions will be examined for the fractional gCBS‐BK equation. The graphs for various fractional order α are plotted to contain 3D plot, contour plot, density plot, and 2D plot. We construct the exact lump and interaction among other types solutions, by solving the under‐determined nonlinear system of algebraic equations for the associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions and the interaction behaviors are revealed. The existence conditions are employed to discuss the available got solutions.     

Aspects of Generalized Calogero Model

A multispecies model of Calogero type in D 1 dimensions is constructed. The model includes harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we find the exact eigenenergies corresponding to a class of the exact global collective states. Analyzing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior.     

Shock wave solution of the variable coefficient nonlinear partial differential equations

In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)‐dimensional variable coefficient Kadomtsev–Petviashvili equation with time‐dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanhp function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley & Sons, Ltd.     

Symmetric and Non-Symmetric Bases of Quantum Integrable Particle Systems with Long-Range Interactions

We study in an algebraic manner the symmetric basis of the Calogero model and the non-symmetric basis of the corresponding Calogero model with distinguishable particles. The Rodrigues formulas are presented for the polynomial parts of both bases. The square norm of the non-symmetric basis is evaluated. Symmetrization of the non-symmetric basis reproduces the symmetric basis and enables us to calculate its square norm.     

Limit theorems for multivariate Bessel processes in the freezing regime

Multivariate Bessel processes describe the stochastic dynamics of interacting particle systems of Calogero–Moser–Sutherland type and are related with $\beta$-Hermite and Laguerre ensembles. It was shown by Andraus, Katori, and Miyashita that for fixed starting points, these processes admit interesting limit laws when the multiplicities $k$ tend to $\infty$, where in some cases the limits are described by the zeros of classical Hermite and Laguerre polynomials. In this paper we use SDEs to derive corresponding limit laws for starting points of the form $\sqrt{k}?x$ for $k\to \infty$ with $x$ in the interior of the corresponding Weyl chambers. Our limit results are a.s. locally uniform in time. Moreover, in some cases we present associated central limit theorems.     

Near-horizon states of black holes and Calogero models

We find self-adjoint extensions of the rational Calogero model in the presence of the harmonic interaction. The corresponding eigenfunctions may describe the near-horizon quantum states of certain types of black holes.     

Unification of Stieltjes‐Calogero type relations for the zeros of classical orthogonal polynomials

The classical orthogonal polynomials (COPs) satisfy a second‐order differential equation of the form σ(x)y^′′+τ(x)y^′+λy = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed‐form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.     

A Series of Exact Solutions for a New (2+1)-Dimensional Calogero KdV Equation

An algebraic method is proposed to solve a new (2 1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.     

Fun and frustration with Calogero model

We review some algebraical (oscillator) aspects of N-body single-species and multispecies Calogero models in one dimension. We treat them as a particular cases of deformed harmonic oscillators and discuss the corresponding Fock spaces. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.