The Thermodynamic Bethe Ansatz and a Connection with Painlevé Equations

1997 ◽  
Vol 11 (01n02) ◽  
pp. 69-74
Author(s):  
Craig A. Tracy ◽  
Harold Widom
Keyword(s):  
Integral Equations ◽  
Bethe Ansatz ◽  
Painlevé Equations ◽  
Nonlinear Integral Equations ◽  
Thermodynamic Bethe Ansatz ◽  
Painleve Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We summarize some recent connections between a class of nonlinear integral equations related to the thermodynamic Bethe Ansatz and a class of linear integral equations related to the Painlevé equations.

scholarly journals Solution of Baxter equation for the $q$-Toda and Toda$_2$ chains by NLIE

2018 ◽  
Vol 5 (4) ◽  
Author(s):  
Olivier Babelon ◽  
Karol Kozlowski ◽  
Vincent Pasquier
Keyword(s):  
Integral Equations ◽  
Bethe Ansatz ◽  
Baxter Equation ◽  
Thermodynamic Bethe Ansatz ◽  
Non Linear ◽  
Linear Integral ◽  
Linear Integral Equations

We construct a basis of solutions of the scalar t-Q equation describing the spectrum of the q-Toda and Toda_22 chains by using auxiliary non-linear integral equations. Our construction allows us to provide quantisation conditions for the spectra of these models in the form of thermodynamic Bethe Ansatz-like equations.

scholarly journals Linearization method for solving nonlinear integral equations

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
P. Darania ◽  
A. Ebadian ◽  
A. V. Oskoi
Keyword(s):  
Integral Equations ◽  
Piecewise Linear ◽  
Linearization Method ◽  
Nonlinear Integral Equations ◽  
Numerical Examples ◽  
Linear Integral ◽  
Linear Integral Equations ◽  
Methods Numerical

The objective of this paper is to assess both the applicability and the accuracy of linearization method in several problems of general nonlinear integral equations. This method provides piecewise linear integral equations which can be easily integrated. It is shown that the accuracy of linearization method can be substantially improved by employing variable steps which adjust themselves to the solution. This approach can reveal that, under this method, the nonlinear integral equations can be transformed into the linear integral equations which may be integrated using classical methods. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.

Rational Type Inequality with Applications to Voltera–Hammerstein Nonlinear Integral Equations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Muhammad Sarwar ◽  
Mian Bahadur Zada ◽  
Stojan Radenović
Keyword(s):  
Integral Equations ◽  
Existence Result ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Type Inequality ◽  
Nonlinear Integral Equations ◽  
Linear Integral ◽  
Complex Valued ◽  
Rational Type ◽  
Linear Integral Equations

AbstractThe aim of this work is to establish fixed point theorems under rational type contractions in the framework of complex-valued metric spaces. These theorems extend and generalize some prominent results in the present literature. Furthermore, as an application the existence result is given for the system of Volterra–Hammerstein non-linear integral equations.

scholarly journals Reciprocity and Self-Tuning Relations without Wrapping

2015 ◽  
Vol 2015 ◽  
pp. 1-21
Author(s):  
Davide Fioravanti ◽  
Gabriele Infusino ◽  
Marco Rossi
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Bethe Ansatz ◽  
High Spin ◽  
Nonlinear Integral Equation ◽  
Anomalous Dimension ◽  
Hooft Coupling ◽  
Self Tuning ◽  
Linear Integral ◽  
Linear Integral Equations

We consider scalar Wilson operators ofN= 4 SYM at high spin,s, and generic twist in the multicolor limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain “reciprocity” and functional “self-tuning” relations up to all terms1/s(ln s)n(inclusive) at any fixed ’t Hooft couplingλ. Of course, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order ins. On this basis we give some evidence that wrapping corrections should enter the nonlinear integral equation and anomalous dimension expansions at the next order(ln s)2/s2, at fixed ’t Hooft coupling, in such a way to reestablish the aforementioned relation (which fails otherwise).

scholarly journals QQ-system and non-linear integral equations for scattering amplitudes at strong coupling

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Davide Fioravanti ◽  
Marco Rossi ◽  
Hongfei Shu
Keyword(s):  
Scattering Amplitudes ◽  
Integral Equations ◽  
Bethe Ansatz ◽  
Strong Coupling ◽  
Strong Coupling Limit ◽  
Wilson Loops ◽  
Functional Relations ◽  
Non Linear ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract We provide the two fundamental sets of functional relations which describe the strong coupling limit in AdS3 of scattering amplitudes in $$ \mathcal{N} $$ N = 4 SYM dual to Wilson loops (possibly extended by a non-zero twist l): the basic QQ-system and the derived TQ-system. We use the TQ relations and the knowledge of the main properties of the Q-function (eigenvalue of some Q-operator) to write the Bethe Ansatz equations, viz. a set of (‘complex’) non-linear-integral equations, whose solutions give exact values to the strong coupling amplitudes/Wilson loops. Moreover, they have some advantages with respect to the (‘real’) non-linear-integral equations of Thermodynamic Bethe Ansatz and still reproduce, both analytically and numerically, the findings coming from the latter. In any case, these new functional and integral equations give a larger perspective on the topic also applicable to the realm of $$ \mathcal{N} $$ N = 2 SYM BPS spectra.

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