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.

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 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.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

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