In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg--de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave--solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.
{"title":"On the Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations","authors":"Nikolay K. Vitanov","doi":"arxiv-2312.03621","DOIUrl":"https://doi.org/arxiv-2312.03621","url":null,"abstract":"In this article, we follow an idea that is opposite to the idea of Hopf and\u0000Cole: we use transformations in order to transform simpler linear or nonlinear\u0000differential equations (with known solutions) to more complicated nonlinear\u0000differential equations. In such a way, we can obtain numerous exact solutions\u0000of nonlinear differential equations. We apply this methodology to the classical\u0000parabolic differential equation (the wave equation), to the classical\u0000hyperbolic differential equation (the heat equation), and to the classical\u0000elliptic differential equation (Laplace equation). In addition, we use the\u0000methodology to obtain exact solutions of nonlinear ordinary differential\u0000equations by means of the solutions of linear differential equations and by\u0000means of the solutions of the nonlinear differential equations of Bernoulli and\u0000Riccati. Finally, we demonstrate the capacity of the methodology to lead to\u0000exact solutions of nonlinear partial differential equations on the basis of\u0000known solutions of other nonlinear partial differential equations. As an\u0000example of this, we use the Korteweg--de Vries equation and its solutions.\u0000Traveling wave solutions of nonlinear differential equations are of special\u0000interest in this article. We demonstrate the existence of the following\u0000phenomena described by some of the obtained solutions: (i) occurrence of the\u0000solitary wave--solitary antiwave from the solution, which is zero at the\u0000initial moment (analogy of an occurrence of particle and antiparticle from the\u0000vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves\u0000(analogy of splitting of a particle into two particles); (iii) soliton behavior\u0000of some of the obtained waves; (iv) existence of solitons which move with the\u0000same velocity despite the different shape and amplitude of the solitons.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138547624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Agnieszka Wierzchucka, Francesco Piazza, Pieter W. Claeys
The Tavis-Cummings model is a paradigmatic central-mode model where a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we here establish the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates and illustrates how these models support a collective spin description despite the presence of disorder.
{"title":"Integrability, multifractality, and two-photon dynamics in disordered Tavis-Cummings models","authors":"Agnieszka Wierzchucka, Francesco Piazza, Pieter W. Claeys","doi":"arxiv-2312.03833","DOIUrl":"https://doi.org/arxiv-2312.03833","url":null,"abstract":"The Tavis-Cummings model is a paradigmatic central-mode model where a set of\u0000two-level quantum emitters (spins) are coupled to a collective cavity mode.\u0000Here we study the eigenstate spectrum, its localization properties and the\u0000effect on dynamics, focusing on the two-excitation sector relevant for\u0000nonlinear photonics. These models admit two sources of disorder: in the\u0000coupling between the spins and the cavity and in the energy shifts of the\u0000individual spins. While this model was known to be exactly solvable in the\u0000limit of a homogeneous coupling and inhomogeneous energy shifts, we here\u0000establish the solvability in the opposite limit of a homogeneous energy shift\u0000and inhomogeneous coupling, presenting the exact solution and corresponding\u0000conserved quantities. We identify three different classes of eigenstates,\u0000exhibiting different degrees of multifractality and semilocalization closely\u0000tied to the integrable points, and study their stability to perturbations away\u0000from these solvable points. The dynamics of the cavity occupation number away\u0000from equilibrium, exhibiting boson bunching and a two-photon blockade, is\u0000explicitly related to the localization properties of the eigenstates and\u0000illustrates how these models support a collective spin description despite the\u0000presence of disorder.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138554879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The work deals with the qualification of semidiscrete hyperbolic type equations. We study a class of equations of the form $$frac{du_{n+1}}{dx}=fleft(frac{du_{n}}{dx},u_{n+1},u_{n}right),$$ here the unknown function $u_n(x)$ depends on one discrete $n$ and one continuous $x$ variables. Qualification is based on the requirement of the existence of higher symmetries. The case is considered when the symmetry is of order 5 in continuous directions. As a result, a list of four equations with the required conditions is obtained. For one of the found equations, a Lax representation is constructed.
{"title":"Classification of semidiscrete hyperbolic type equations. The case of fifth order symmetries","authors":"R. N. Garifullin","doi":"arxiv-2312.03745","DOIUrl":"https://doi.org/arxiv-2312.03745","url":null,"abstract":"The work deals with the qualification of semidiscrete hyperbolic type\u0000equations. We study a class of equations of the form\u0000$$frac{du_{n+1}}{dx}=fleft(frac{du_{n}}{dx},u_{n+1},u_{n}right),$$ here the\u0000unknown function $u_n(x)$ depends on one discrete $n$ and one continuous $x$\u0000variables. Qualification is based on the requirement of the existence of higher\u0000symmetries. The case is considered when the symmetry is of order 5 in\u0000continuous directions. As a result, a list of four equations with the required\u0000conditions is obtained. For one of the found equations, a Lax representation is\u0000constructed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138569360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct novel solutions to the set-theoretical entwining Yang-Baxter equation. These solutions are birational maps involving non-commutative dynamical variables which are elements of the Grassmann algebra of order $n$. The maps arise from refactorisation problems of Lax supermatrices associated to a nonlinear Schr"odinger equation. In this non-commutative setting, we construct a spectral curve associated to each of the obtained maps using the characteristic function of its monodromy supermatrix. We find generating functions of invariants (first integrals) for the entwining Yang-Baxter maps from the moduli of the spectral curves. Moreover, we show that a hierarchy of birational entwining Yang-Baxter maps with commutative variables can be obtained by fixing the order $n$ of the Grassmann algebra. We present the members of the hierarchy in the case $n=1$ (dual numbers) and $n=2$, and discuss their dynamical and integrability properties, such as Lax matrices, invariants, and measure preservation.
{"title":"Entwining Yang-Baxter maps over Grassmann algebras","authors":"P. Adamopoulou, G. Papamikos","doi":"arxiv-2311.18673","DOIUrl":"https://doi.org/arxiv-2311.18673","url":null,"abstract":"We construct novel solutions to the set-theoretical entwining Yang-Baxter\u0000equation. These solutions are birational maps involving non-commutative\u0000dynamical variables which are elements of the Grassmann algebra of order $n$.\u0000The maps arise from refactorisation problems of Lax supermatrices associated to\u0000a nonlinear Schr\"odinger equation. In this non-commutative setting, we\u0000construct a spectral curve associated to each of the obtained maps using the\u0000characteristic function of its monodromy supermatrix. We find generating\u0000functions of invariants (first integrals) for the entwining Yang-Baxter maps\u0000from the moduli of the spectral curves. Moreover, we show that a hierarchy of\u0000birational entwining Yang-Baxter maps with commutative variables can be\u0000obtained by fixing the order $n$ of the Grassmann algebra. We present the\u0000members of the hierarchy in the case $n=1$ (dual numbers) and $n=2$, and\u0000discuss their dynamical and integrability properties, such as Lax matrices,\u0000invariants, and measure preservation.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is based on the author's PhD thesis. The main result of the thesis is the use of the boost operator to develop a systematic method to construct new integrable spin chains with nearest-neighbour interaction and characterized by an R-matrix of non-difference form. This method has the advantage of being more feasible than directly solving the Yang-Baxter equation. We applied this approach to various contexts, in particular, in the realm of open quantum systems, we achieved the first classification of integrable Lindbladians. These operators describe the dynamics of physical systems in contact with a Markovian environment. Within this classification, we discovered a novel deformation of the Hubbard model spanning three sites of the spin chain. Additionally, we applied our method to classify models with $mathfrak{su}(2)oplus mathfrak{su}(2)$ symmetry and we recovered the matrix part of the S-matrix of $AdS_5 times S^5$ derived by requiring centrally extended $mathfrak{su}(2|2)$ symmetry. Furthermore, we focus on spin 1/2 chain on models of 8-Vertex type and we showed that the models of this class satisfy the free fermion condition. This enables us to express the transfer matrix associated to some of the models in a diagonal form, simplifying the computation of the eigenvalues and eigenvectors. The thesis is based on the works: 2003.04332, 2010.11231, 2011.08217, 2101.08279, 2207.14193, 2301.01612, 2305.01922.
{"title":"Yang-Baxter integrable open quantum systems","authors":"Chiara Paletta","doi":"arxiv-2312.00064","DOIUrl":"https://doi.org/arxiv-2312.00064","url":null,"abstract":"This work is based on the author's PhD thesis. The main result of the thesis\u0000is the use of the boost operator to develop a systematic method to construct\u0000new integrable spin chains with nearest-neighbour interaction and characterized\u0000by an R-matrix of non-difference form. This method has the advantage of being\u0000more feasible than directly solving the Yang-Baxter equation. We applied this\u0000approach to various contexts, in particular, in the realm of open quantum\u0000systems, we achieved the first classification of integrable Lindbladians. These\u0000operators describe the dynamics of physical systems in contact with a Markovian\u0000environment. Within this classification, we discovered a novel deformation of\u0000the Hubbard model spanning three sites of the spin chain. Additionally, we\u0000applied our method to classify models with $mathfrak{su}(2)oplus\u0000mathfrak{su}(2)$ symmetry and we recovered the matrix part of the S-matrix of\u0000$AdS_5 times S^5$ derived by requiring centrally extended $mathfrak{su}(2|2)$\u0000symmetry. Furthermore, we focus on spin 1/2 chain on models of 8-Vertex type\u0000and we showed that the models of this class satisfy the free fermion condition.\u0000This enables us to express the transfer matrix associated to some of the models\u0000in a diagonal form, simplifying the computation of the eigenvalues and\u0000eigenvectors. The thesis is based on the works: 2003.04332, 2010.11231,\u00002011.08217, 2101.08279, 2207.14193, 2301.01612, 2305.01922.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138544037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Robert ConteENS Paris-Saclay, France and U of Hong Kong, A. Michel GrundlandUQTR, Canada
We derive all the reductions of the system of two coupled sine-Gordon equations introduced by Konopelchenko and Rogers to ordinary differential equations. All these reductions are degeneracies of a master reduction to an equation found by Chazy "curious for its elegance", an algebraic transform of the most general sixth equation of Painlev'e. -- -- Nous 'etablissons toutes les r'eductions du syst`eme de deux 'equations coupl'ees de sine-Gordon introduit par Konopelchenko et Rogers `a des 'equations diff'erentielles ordinaires. Ces r'eductions sont toutes des d'eg'en'erescences d'une r'eduction ma{^i}tresse `a une 'equation jug'ee par Chazy "curieuse en raison de [son] 'el'egance", transform'ee alg'ebrique de la sixi`eme 'equation de Painlev'e la plus g'en'erale.
我们导出了由Konopelchenko和Rogers引入的两个耦合正弦- gordon方程组对常微分方程的所有化简。所有这些约简都是对Chazy发现的一个“对其优雅感到好奇”的方程的主约简的简并，这是painleve最一般的第六方程的代数变换。-- -- Nous 'etablissons吹捧les ' educations du system ' eme de deux 'equations ' couples ' es de sin - gordon介绍parkonopelchenko和Rogers ' a des 'equations diff ' entientielles ordinaires。Ces r '排出的书桌 '如 ' en '一个r erescences “马排出{ ^ 我}tresse '一个“equationjug ”ee par Chazy“curieuse en雷森(儿子) ' el ' egance”,变换 ' eealg ' ebrique de la泗溪“高速”方程de Painlev “e la + g ' ' erale。
{"title":"Réductions d'un système bidimensionnel de sine-Gordon à la sixième équation de Painlevé","authors":"Robert ConteENS Paris-Saclay, France and U of Hong Kong, A. Michel GrundlandUQTR, Canada","doi":"arxiv-2311.17469","DOIUrl":"https://doi.org/arxiv-2311.17469","url":null,"abstract":"We derive all the reductions of the system of two coupled sine-Gordon\u0000equations introduced by Konopelchenko and Rogers to ordinary differential\u0000equations. All these reductions are degeneracies of a master reduction to an\u0000equation found by Chazy \"curious for its elegance\", an algebraic transform of\u0000the most general sixth equation of Painlev'e. -- -- Nous 'etablissons toutes les r'eductions du syst`eme de deux 'equations\u0000coupl'ees de sine-Gordon introduit par Konopelchenko et Rogers `a des\u0000'equations diff'erentielles ordinaires. Ces r'eductions sont toutes des\u0000d'eg'en'erescences d'une r'eduction ma{^i}tresse `a une 'equation\u0000jug'ee par Chazy \"curieuse en raison de [son] 'el'egance\", transform'ee\u0000alg'ebrique de la sixi`eme 'equation de Painlev'e la plus g'en'erale.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543903","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We formulate the inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials via a Riemann-Hilbert problem approach. We also use the resulting formalism to solve the initial value problem for the nonlinear Schroedinger equation. We establish a uniqueness theorem for the solutions of the Riemann-Hilbert problem, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. Importantly, the formalism shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used.
{"title":"Spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schroedinger equation with periodic boundary conditions","authors":"Gino Biondini, Zechuan Zhang","doi":"arxiv-2311.18127","DOIUrl":"https://doi.org/arxiv-2311.18127","url":null,"abstract":"We formulate the inverse spectral theory for a self-adjoint one-dimensional\u0000Dirac operator associated periodic potentials via a Riemann-Hilbert problem\u0000approach. We also use the resulting formalism to solve the initial value\u0000problem for the nonlinear Schroedinger equation. We establish a uniqueness\u0000theorem for the solutions of the Riemann-Hilbert problem, which provides a new\u0000method for obtaining the potential from the spectral data. Two additional,\u0000scalar Riemann-Hilbert problems are also formulated that provide conditions for\u0000the periodicity in space and time of the solution generated by arbitrary sets\u0000of spectral data. The formalism applies for both finite-genus and\u0000infinite-genus potentials. Importantly, the formalism shows that only a single\u0000set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the\u0000potential of the Dirac operator and the corresponding solution of the\u0000defocusing NLS equation, in contrast with the representation of the solution of\u0000the NLS equation via the finite-genus formalism, in which two different sets of\u0000Dirichlet eigenvalues are used.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Discrete integrable systems are closely related to orthogonal polynomials and isospectral matrix transformations. In this paper, we use these relationships to propose a nonautonomous time-discretization of the Camassa-Holm (CH) peakon equation, which describes the motion of peakon waves, which are soliton waves with sharp peaks. We then validate our time-discretization, and clarify its asymptotic behavior as the discrete-time goes to infinity. We present numerical examples to demonstrate that the proposed discrete equation captures peakon wave motions.
{"title":"Discretization of Camassa-Holm peakon equation using orthogonal polynomials and matrix $LR$ transformations","authors":"R. Watanabe, M. Iwasaki, S. Tsujimoto","doi":"arxiv-2311.16582","DOIUrl":"https://doi.org/arxiv-2311.16582","url":null,"abstract":"Discrete integrable systems are closely related to orthogonal polynomials and\u0000isospectral matrix transformations. In this paper, we use these relationships\u0000to propose a nonautonomous time-discretization of the Camassa-Holm (CH) peakon\u0000equation, which describes the motion of peakon waves, which are soliton waves\u0000with sharp peaks. We then validate our time-discretization, and clarify its\u0000asymptotic behavior as the discrete-time goes to infinity. We present numerical\u0000examples to demonstrate that the proposed discrete equation captures peakon\u0000wave motions.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dmitry Shepelsky, Iryna Karpenvko, Stepan Bogdanov, Jaroslaw E. Prilepsky
We consider the Riemann--Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schr"odinger (NLS) equation, addressing the question of how the RH problem parameters can be retrieved from the solution. Within the RH approach, a finite-band solution to the NLS equation is given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.
{"title":"Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Riemann--Hilbert approach: inverse and direct problems","authors":"Dmitry Shepelsky, Iryna Karpenvko, Stepan Bogdanov, Jaroslaw E. Prilepsky","doi":"arxiv-2311.16902","DOIUrl":"https://doi.org/arxiv-2311.16902","url":null,"abstract":"We consider the Riemann--Hilbert (RH) approach to the construction of\u0000periodic finite-band solutions to the focusing nonlinear Schr\"odinger (NLS)\u0000equation, addressing the question of how the RH problem parameters can be\u0000retrieved from the solution. Within the RH approach, a finite-band solution to\u0000the NLS equation is given in terms of the solution of an associated RH problem,\u0000the jump conditions for which are characterized by specifying the endpoints of\u0000the arcs defining the contour of the RH problem and the constants (so-called\u0000phases) involved in the jump matrices. In our work, we solve the problem of\u0000retrieving the phases given the solution of the NLS equation evaluated at a\u0000fixed time. Our findings are corroborated by numerical examples of phases\u0000computation, demonstrating the viability of the method proposed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work proposes a closed formula for the leading term of the long-distance and large-time asymptotics in a cone of the space-like regime for the transverse dynamical two-point functions of the XXZ spin 1/2 chain at finite temperatures. The result follows from a simple analysis of the thermal form factor series for dynamical correlation functions. The leading asymptotics we obtain are driven by the Bethe Ansatz data associated with the first sub-leading Eigenvalue of the quantum transfer matrix.
{"title":"Space-like asymptotics of the thermal two-point functions of the XXZ spin-1/2 chain","authors":"F. Göhmann, K. K. Kozlowski","doi":"arxiv-2311.17196","DOIUrl":"https://doi.org/arxiv-2311.17196","url":null,"abstract":"This work proposes a closed formula for the leading term of the long-distance\u0000and large-time asymptotics in a cone of the space-like regime for the\u0000transverse dynamical two-point functions of the XXZ spin 1/2 chain at finite\u0000temperatures. The result follows from a simple analysis of the thermal form\u0000factor series for dynamical correlation functions. The leading asymptotics we\u0000obtain are driven by the Bethe Ansatz data associated with the first\u0000sub-leading Eigenvalue of the quantum transfer matrix.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138543897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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