scholarly journals A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants

2021 ◽  
Author(s):  
Sofia Tarricone ◽  
Keyword(s):  
Hankel Operators ◽  
Lax Pair ◽  
Airy Functions ◽  
Fredholm Determinants ◽  
The Matrix ◽  
Painlevé Ii

We consider Fredholm determinants of matrix Hankel operators associated to matrix versions of the n-th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix-valued version of the Lenard operators. In particular, the Riemann-Hilbert techniques used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitly written in terms of the matrix-valued Lenard operators and some solutions of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy Hankel operators.

scholarly journals Calculating the Energy Spectrum of Complex Low-Dimensional Heterostructures in the Electric Field

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Svetlana N. Khonina ◽  
Sergey G. Volotovsky ◽  
Sergey I. Kharitonov ◽  
Nikolay L. Kazanskiy
Keyword(s):  
Steady State ◽  
Electric Field ◽  
Ground State ◽  
State Energy ◽  
Airy Functions ◽  
Piecewise Constant ◽  
Matrix Rank ◽  
The Matrix ◽  
Constant Potential ◽  
Low Dimensional

An algorithm for solving the steady-state Schrödinger equation for a complex piecewise-constant potential in the presence of theE-field is developed and implemented. The algorithm is based on the consecutive matching of solutions given by the Airy functions at the band boundaries with the matrix rank increasing by no more than two orders, which enables the characteristic solution to be obtained in the convenient form for search of the roots. The algorithm developed allows valid solutions to be obtained for the electric field magnitudes larger than the ground-state energy level, that is, when the perturbation method is not suitable.

Aharony–Bergman–Jafferis–Maldacena Wilson Loops in the Fermi Gas Approach

2013 ◽  
Vol 68 (1-2) ◽  
pp. 178-209 ◽  
Author(s):  
Albrecht Klemm ◽  
Marcos Mariño ◽  
Masoud Soroush
Keyword(s):  
Vacuum Expectation ◽  
Fermi Gas ◽  
Wilson Loops ◽  
Winding Number ◽  
Airy Functions ◽  
Mechanical Problem ◽  
Expectation Values ◽  
The Matrix ◽  
Statistical Mechanical ◽  
Chern Simons

The matrix model of the Aharony-Bergman-Jafferis-Maldacena theory can be formulated in terms of an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show that, in this formalism, vacuum expectation values (vevs) of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the Wentzel- Kramer-Brillouin expansion.We present explicit results for the vevs of 1/6 and the 1/2 Bogomolnyi- Prasad-Sommerfield Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the ’t Hooft expansion.

scholarly journals ON A NOVEL CLASS OF INTEGRABLE ODEs RELATED TO THE PAINLEVÉ EQUATIONS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250211
Author(s):  
ATHANASSIOS S. FOKAS ◽  
DI YANG
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Lax Pair ◽  
Painlevé Equations ◽  
Painlevé Equation ◽  
Painleve Equations ◽  
Independent Variable ◽  
Painlevé Ii ◽  
Painleve Equation

One of the authors recently introduced the concept of conjugate Hamiltonian systems: the solution of the equation h = H(p, q, t), where H is a given Hamiltonian containing t explicitly, yields the function t = T(p, q, h), which defines a new Hamiltonian system with Hamiltonian T and independent variable h. By employing this construction and by using the fact that the classical Painlevé equations are Hamiltonian systems, it is straightforward to associate with each Painlevé equation two new integrable ODEs. Here, we investigate the conjugate Painlevé II equations. In particular, for these novel integrable ODEs, we present a Lax pair formulation, as well as a class of implicit solutions. We also construct conjugate equations associated with Painlevé I and Painlevé IV equations.

Electrical structures of interfaces: a Painlevé II model

2010 ◽  
Vol 466 (2119) ◽  
pp. 2117-2136 ◽  
Author(s):  
L. Bass ◽  
J. J. C. Nimmo ◽  
C. Rogers ◽  
W. K. Schief
Keyword(s):  
Electric Field ◽  
Boundary Value Problems ◽  
Field Distribution ◽  
Electric Field Distribution ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Physical Parameters ◽  
Airy Functions ◽  
Ion Concentrations ◽  
Painlevé Ii

A Painlevé II model derived out of the classical Nernst–Planck system is applied in the context of boundary value problems that describe the electric field distribution in a region x >0 occupied by an electrolyte. For privileged flux ratios of the ion concentrations, the auto-Bäcklund transformation admitted by the Painlevé II equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems. These representations involve, in turn, either Yablonskii–Vorob’ev polynomials or classical Airy functions. The requirement that the electric field distribution and ion concentrations in these representations be non-singular imposes constraints on the physical parameters. These are investigated in detail along with asymptotic properties.

scholarly journals Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

2019 ◽  
Author(s):  
Mattia Cafasso ◽  
Tom Claeys ◽  
Manuela Girotti
Keyword(s):  
Statistical Physics ◽  
Point Processes ◽  
Fredholm Determinant ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Contour Integrals ◽  
Painlevé Ii ◽  
Derivatives Of ◽  
Gap Probabilities ◽  
Logarithmic Derivatives

Abstract We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr [20]. In addition, we obtain asymptotics at $\pm \infty $ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

The N-soliton solutions to the Hirota and Maxwell–Bloch equation via the Riemann–Hilbert approach

2021 ◽  
pp. 2150153
Author(s):  
Jian Li ◽  
Tiecheng Xia ◽  
Hanyu Wei
Keyword(s):  
Physical Meaning ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Bloch Equation ◽  
Lax Pair ◽  
Integrable Equation ◽  
Motion Trajectory ◽  
The Matrix ◽  
Riemann Hilbert Problem

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

scholarly journals Multi-Soliton Solutions of the N-Component Nonlinear Schrodinger Equations via Riemann-Hilbert Approach*

2021 ◽  
Author(s):  
Yan Li ◽  
Jian Li ◽  
Ruiqi Wang
Keyword(s):  
Soliton Solutions ◽  
Lax Pair ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Spectral Parameters ◽  
Nonlinear Schrödinger ◽  
Jump Matrix ◽  
Nonlinear Schrodinger Equations ◽  
The Matrix

Abstract In this paper, we utilize the Riemann-Hilbert approach to discuss multi-soliton solutions of the N-component nonlinear Schrodinger equations. Firstly, by transformed Lax pair, we construct the matrix valued functions P1,2 that satisfy the analyticity and normalization and the corresponding jump matrix can be determined. Then, in the reflectionless case, we get the multi-soliton solutions ql(l = 1, ..., N) of the N-component nonlinear Schrodinger equations, which are related to spectral parameters. Particularly, the 2-soliton solutions q1, q2 and q3 of the three-component nonlinear Schrodinger equations are given and the corresponding 2-soliton diagrams are drawn.

scholarly journals Riemann-Hilbert Approach of The Complex Sharma-Tasso-Olver Equation and Its N-Soliton Solutions

2021 ◽  
Author(s):  
Sha Li ◽  
Tiecheng Xia ◽  
Jian Li
Keyword(s):  
Soliton Solution ◽  
Hilbert Problem ◽  
Three Dimensional ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Integrable Equation ◽  
The Matrix ◽  
Relationship Of ◽  
Parameter Values ◽  
The Relationship

Abstract In this paper, we use Riemann-Hilbert method to study the N-soliton solutions of the complex Sharma-Tasso-Olver(cSTO) equation. And then, based on analyzing the spectral problem of the Lax pair, the matrix Riemann-Hilbert problem for this integrable equation can be constructed, the N-soliton solutions about this system are given explicitly under the relationship of scattering matrix. At last, under the condition that some specifific parameter values are given, the three-dimensional diagram of the 2-soliton solution and the trajectory of the soliton solution will be simulated.

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