Auswahl der wissenschaftlichen Literatur zum Thema „Sinh-Gordon model“

AbstractThis work proposes the new extended rational sinh-Gordon equation expansion technique (SGEEM). The computational approach is formulated based on the well-known sinh-Gordon equation. The proposed technique generalizes the sine-Gordon/sinh-Gordon expansion methods in a rational format. The efficiency of the suggested technique is tested on the (2+1)imensional Kunduukherjeeaskar (KMN) model. Various of optical soliton solutions have been obtained using this new method. The conditions which guarantee the existence of valid solitons are given.

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Using the new regularization and renormalization scheme recently proposed by Yang and used by Ni et al., we analyze the sine–Gordon and sinh–Gordon models within the framework of Gaussian effective potential in D+1 dimensions. Our analysis suffers no divergence and so does not suffer from the manipulational obscurities in the conventional analysis of divergent integrals. Our main conclusions agree exactly with those of Ingermanson for D=1,2 but disagree for D=3: the D=3 sinh(sine)–Gordon model is nontrivial. Furthermore, our analysis shows that for D=1,2, the running coupling constant (RCC) has poles for sine–Gordon model (γ2<0) and the sinh–Gordon model (γ2>0) has a possible critical point [Formula: see text] while for D=3, the RCC has poles for both γ2>0 and γ2<0.

The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, the program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model.

The most general solution to the form factor problem in the sinh–Gordon model is presented in an explicit way. The linearly independent classes of solutions correspond to powers of the elementary field. We show how the form factors of exponential operators can be obtained from the general solution by adjusting free parameters. The general formula obtained in the sinh–Gordon case reproduces the form factors of the scaling Lee–Yang model in a certain limit of the coupling constant.

We review the Pohlmeyer reduction of the superstring sigma model on AdSn × Sn leading to a gauged WZW model with an integrable potential. In particular, we consider the case of GS superstrings on AdS3 × S3 supported by RR flux. The bosonic part of the reduced Lagrangian is given by the sum of the complex sine-Gordon Lagrangian and its sinh-Gordon counterpart. We determine the corresponding fermionic part and discuss possible 2d supersymmetry of the reduced action.

Three different aspects of the sinh-Gordon model are explored in this thesis. We begin, in chapter one, with a summary of the model and the necessary background. Chapter two studies the model with two boundary conditions. Two approaches are presented to investigate the reflection factors off the boundaries and the energy of the theory. In chapter three, perturbation theory is developed to study the theory with one general boundary condition. A contribution to the quantum reflection factor is obtained and compared with the result obtained for the special boundary condition. Chapters four and five investigate the supersymmetric extension of the model in the presence of a single boundary. Firstly, the classical limits of the supersymmetric reflection matrices are checked. The exact reflection factors are studied perturbatively up to the second order of the coupling constant. Secondly, the perturbation theory and the path integral formalism are employed in the supersymmetric model to study the quantum reflection factors. We conclude with a brief sixth chapter describing the outlook for further investigations.

This thesis studies the quantum reflection factor of the sinh-Gordon model under boundary conditions consistent with integrability. First, we review the affine Toda field theory in Chapter One. In particular, the classical and quantum integrability of the theory are reviewed on the whole line and on the half-line as well, that is, in the presence of a boundary. We next consider the sinh-Gordon model which is restricted to a half-line by boundary conditions maintaining integrability in Chapter Two. A perturbative calculation of the reflection factor is given to one loop order in the bulk coupling and to first order in the difference of the two parameters introduced at the boundary. The result provides a further verification of Ghoshal's formula. The calculation is consistent with a conjecture for the general dependence of the reflection factor on the boundary parameters and the bulk coupling. In Chapter Three, quantum corrections to the classical reflection factor of the sinh-Gordon model are studied up to second order in the difference of boundary data and to one loop order in the bulk coupling. Chapter Four deals with the quantum reflection factor for the sinh-Gordon model with general boundary conditions. The model is studied under boundary conditions which are compatible with integrability and in the framework of the conventional perturbation theory generalised to the affine Toda field theory. It is found that the general form of a subset of the related quantum corrections are hypergeometric functions. Finally, we sum up this thesis in Chapter Five along with some conclusions and suggestions for further future studies.

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Apresentamos a formulação da hierarquia mKdV e sinh-Gordon supersimétrica com N = 1 baseada na superálgebra de Kac-Moody sl(2,1). Obtivemos soluções para até quatro vértices para ambos os modelos de dressing e também discutimos a transformação de Bäcklund para o modelo sinh-Gordon supersimétrico utilizando supercampos
We introduce the formulation of supersymmetric mKdV and sinh-Gordon hierarchy with N = 1 based on Kac-Moody sl(2,1) superalgebra. We ontained solutions to up to four vertex operator for both models using the dressing method and we also discuss the Bäcklund transformation for the supersymmetric sinh-Gordon model using superfields

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A partir dos geradores da superágebra de Kac-Moody 's TIL'l(2,1) escrevemos o par de Lax para o modelo sinh-Gordon supersimétrico. Em adição, veri?camos que através da equação de curvatura nula o Lax obtido, reproduz as equações de movimento do modelo. Para a análise de teorias com defeito calculamos, por meio de uma transformação de Gauge, a matriz de defeito para o modelo sinh-Gordon supersimétrico, e apresentamos o resultado similar obtido para o modelo sine-Gordon. Notamos ainda que na região do defeito, os campos da teoria obedecem equações do tipo das tranformações de Backlund. Foram obtidas também a energia e o momento para o modelo super sinh-Gordon, a partir de uma expressão geral, que nos permite obter o conjunto in?nito de cargas conservadas para modelo. Além disso, as contribuições do defeito dessas quantidades foram também encontradas
From the generators of the Kac-Moody superálgebra 's TIL'l(2,1) write the Lax pair for the supersymmetric sinh-Gordon model. In addition, we ?nd that through the zero curvature obtained the Lax equation reproduces the equations of motion of the model. For the analysis of defective theories calculated by means of a gauge transformation, the default matrix for the model sinh supersymetric Gordon, and present the result similar to the sine-Gordon model. We also note that in the defect region, the ?eld equations of the theory obey the type of Backlund transformations. The energy and momentum for the super sinh-Gordon model were also obtained from a general expression that allows us to obtain the in?nite set of conserved charges for style. In addition, the contributions of the defect these quantities have also been found

Orientador: Abraham Hirsz Zimerman
Co-orientador: José Francisco Gomes
Banca: Marco Aurélio Cattacin Kneipp
Banca: Leandro Hayato Ymai
Resumo: A partir dos geradores da superágebra de Kac-Moody 's TIL'l(2,1) escrevemos o par de Lax para o modelo sinh-Gordon supersimétrico. Em adição, verificamos que através da equação de curvatura nula o Lax obtido, reproduz as equações de movimento do modelo. Para a análise de teorias com defeito calculamos, por meio de uma transformação de Gauge, a matriz de defeito para o modelo sinh-Gordon supersimétrico, e apresentamos o resultado similar obtido para o modelo sine-Gordon. Notamos ainda que na região do defeito, os campos da teoria obedecem equações do tipo das tranformações de Backlund. Foram obtidas também a energia e o momento para o modelo super sinh-Gordon, a partir de uma expressão geral, que nos permite obter o conjunto infinito de cargas conservadas para modelo. Além disso, as contribuições do defeito dessas quantidades foram também encontradas
Abstract: From the generators of the Kac-Moody superálgebra 's TIL'l(2,1) write the Lax pair for the supersymmetric sinh-Gordon model. In addition, we find that through the zero curvature obtained the Lax equation reproduces the equations of motion of the model. For the analysis of defective theories calculated by means of a gauge transformation, the default matrix for the model sinh supersymetric Gordon, and present the result similar to the sine-Gordon model. We also note that in the defect region, the field equations of the theory obey the type of Backlund transformations. The energy and momentum for the super sinh-Gordon model were also obtained from a general expression that allows us to obtain the infinite set of conserved charges for style. In addition, the contributions of the defect these quantities have also been found
Mestre

Cette thèse vise à étendre les techniques mathématiques qui permettent d’extraire le comportement asymptotique de certaines intégrales multiples quand le nombre d’intégrales tend vers l’infini. Un cas très bien compris est celui de la fonction de partition des beta-ensembles classiques. Les techniques probabilistes de grandes déviations et d’analyse des équations de boucles forment l’arsenal classique pour son étude et permettent de comprendre son comportement asymptotique dans une large généralité. Des généralisations non-triviales de cette intégrale multiple sont étudiés dans ce manuscrit: le régime haute température des beta-ensembles et le modèle sinh. Dans ce premier modèle, la température proportionnelle au nombre de particules permet de rendre l’entropie du même ordre que le potentiel confinant et l’interaction à deux corps. Cela a de multiples conséquences: un support non-borné pour la mesure d’équilibre contrairement au régime classique des beta-ensembles et un opérateur master beaucoup plus délicat à gérer. Une étude fine de son comportement permet de démontrer un théorème central limite et le comportement asymptotique à toute du logarithme de sa fonction de partition. Ce premier résultat permet d’étudier certains aspects de systèmes physiques dits intégrables comme la chaîne de Toda et plus particulièrement sa limite hydrodynamique. Ce deuxième résultat permet enfin d’étendre l’application de la méthode des équations de boucle dans le cas où les particules ne se concentrent pas sur un compact. Finalement, un dernier modèle est étudié, le modèle sinh. L’étude de ce modèle est motivée par la méthode de séparation des variables quantique où ce genre d’intégrales apparaît. Il constitue une généralisation des beta-ensembles classiques où l’effet confinant est plus faible que l’interaction et où cette dernière est plus compliqué. La mesure d’équilibre y est étudié et permet d’obtenir une certaine vérification d’une conjecture de Lukyanov sur le modèle sinh-Gordon quantique en 1+1 dimensions et volume fini
This thesis aims to extend mathematical techniques that extract the asymptotic behavior of certain multiple integrals as the number of integrals tends to infinity. A well-understood case is the partition function of classical beta-ensembles. Probabilistic techniques of large deviations and analysis of loop equations form the classical arsenal for its study and allow for a broad understanding of its asymptotic behavior. Non-trivial generalizations of this multiple integral are studied in this manuscript: the high-temperature regime of beta-ensembles and the sinh model. In the first model, the temperature proportional to the number of particles makes the entropy of the same order as the confining potential and the two-body interaction. This has multiple consequences: an unbounded support for the equilibrium measure contrary to the classical regime of beta-ensembles, and a much more delicate master operator to handle. A detailed study of its behavior allows for the demonstration of a central limit theorem and the asymptotic behavior of the logarithm of its partition function. This first result permits the study of certain aspects of so-called integrable physical systems like the Toda chain, and more specifically, its hydrodynamic limit. This second result finally extends the application of the method of loop equations to cases where particles do not concentrate on a compact set. Lastly, another model is studied, the sinh model. The study of this model is motivated by the quantum separation of variables method where such integrals appear. It constitutes a generalization of classical beta-ensembles where the confining effect is weaker than the interaction, and the latter is more complicated. The equilibrium measure is studied, leading to a certain verification of Lukyanov's conjecture on the quantum sinh-Gordon model in 1+1 dimensions and finite volume

We consider Bukhvostov-Lipatov model, an integrable quantum field theory in two dimensions that arises as an approximation to O(3) NLSM. We compute its vacuum energy on a cylinder with twisted boundary conditions in weak coupling limit using renormalized perturbation theory, and in the short distance limit using conformal perturbation theory. The exact solution of this model via coordinate Bethe ansatz is provided. Two different regularizations of Bethe ansatz equations are constructed. The vacuum state is constructed and the vacuum energy is computed within both regularizations using numerical methods. Bethe ansatz equations governing the vacuum state are shown to coincide with functional relations between connection coefficients of auxiliary linear problem for an integrable classical PDE known as modified sinh-Gordon equation. Based on this correspondence the system of nonlinear integral equations equivalent to the full system of Bethe ansatz equations is derived for massive and conformal cases. This system of NLIE was solved numerically. We have also used it to investigate analytically the properties of solution describing vacuum state. Finally, we have derived a formula that expresses the vacuum energy of Bukhvostov-Lipatov model in terms of the regularized area of constant mean curvature surface embedded into ADS3 space.