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).

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.

Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Edyta Hetmaniok ◽  
Iwona Nowak ◽  
Damian Słota ◽  
Roman Wituła
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Linear Integral ◽  
Special Case ◽  
Linear Integral Equations

AbstractIn this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

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 On Solvability Conditions for a Certain Conjugation Problem

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 234
Author(s):  
Vladimir Vasilyev ◽  
Nikolai Eberlein
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Integral Equations ◽  
Mellin Transform ◽  
Algebraic Equations ◽  
Conjugation Problem ◽  
Solvability Conditions ◽  
Linear Algebraic Equations ◽  
Linear Integral ◽  
Linear Integral Equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

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