The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra lattices are proposed and studied by performing nonisopectral deformations on the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials without specific weight functions, respectively. Under stationary reductions, matrix discrete Painlev'{e} I and matrix asymmetric discrete Painlev'{e} I equations are derived separately not only from the noncommutative nonisospectral lattices themselves, but also from their Lax pairs. The rationality of the stationary reduction has been justified in the sense that quasideterminant solutions are provided for the corresponding matrix discrete Painlev'{e} equations.
通过对没有特定权函数的矩阵正交多项式和矩阵对称正交多项式分别进行非等谱变形，提出并研究了非等谱托达和洛特卡-伏特线方程的非交换类似物。在静态还原条件下，矩阵离散 Painlev'{e} I 和矩阵非对称离散 Painlev'{e} I 方程不仅可以从非交换正谱网格本身，而且可以从它们的 Lax 对分别得到。从为相应的矩阵离散 Painlev'{e} 方程提供等差数列解的意义上，证明了静态还原的合理性。
{"title":"Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlevé equations","authors":"Anhui Yan, Chunxia Li","doi":"arxiv-2407.08486","DOIUrl":"https://doi.org/arxiv-2407.08486","url":null,"abstract":"The noncommutative analogues of the nonisospectral Toda and Lotka-Volterra\u0000lattices are proposed and studied by performing nonisopectral deformations on\u0000the matrix orthogonal polynomials and matrix symmetric orthogonal polynomials\u0000without specific weight functions, respectively. Under stationary reductions,\u0000matrix discrete Painlev'{e} I and matrix asymmetric discrete Painlev'{e} I\u0000equations are derived separately not only from the noncommutative\u0000nonisospectral lattices themselves, but also from their Lax pairs. The\u0000rationality of the stationary reduction has been justified in the sense that\u0000quasideterminant solutions are provided for the corresponding matrix discrete\u0000Painlev'{e} equations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614472","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 consider the nonlinear Schr"{o}dinger (NLS) equation on the half-line subjecting to a class of boundary conditions preserve the integrability of the model. For such a half-line problem, the Poisson brackets of the corresponding scattering data are computed, and the variables of action-angle type are constructed. These action-angle variables completely trivialize the dynamics of the NLS equation on the half-line.
{"title":"Action-angle variables for the nonlinear Schrödinger equation on the half-line","authors":"Baoqiang Xia","doi":"arxiv-2407.06916","DOIUrl":"https://doi.org/arxiv-2407.06916","url":null,"abstract":"We consider the nonlinear Schr\"{o}dinger (NLS) equation on the half-line\u0000subjecting to a class of boundary conditions preserve the integrability of the\u0000model. For such a half-line problem, the Poisson brackets of the corresponding\u0000scattering data are computed, and the variables of action-angle type are\u0000constructed. These action-angle variables completely trivialize the dynamics of\u0000the NLS equation on the half-line.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573778","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}
In this paper, we consider a soliton gas of the focusing modified Korteweg-de Vries generated from the $N$-soliton solutions under the nonzero background. The spectral soliton density is chosen on the pure imaginary axis, excluding the branch cut $Sigma_{c}=left[-i, iright]$. In the limit $Ntoinfty$, we establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou nonlinear steepest-descent method, this soliton gas under the nonzero background will decay to a constant background as $xto+infty$, while its asymptotics as $xto-infty$ can be expressed with a Riemann-Theta function, attached to a Riemann surface with genus-two. We also analyze the large $t$ asymptotics over the entire spatial domain, which is divided into three distinct asymptotic regions depending on the ratio $xi=frac{x}{t}$. Using the similar method, we provide the leading-order asymptotic behaviors for these three regions and exhibit the dynamics of large $t$ asymptotics.
{"title":"A modified Korteweg-de Vries equation soliton gas under the nonzero background","authors":"Xiaoen Zhang, Liming Ling","doi":"arxiv-2407.05384","DOIUrl":"https://doi.org/arxiv-2407.05384","url":null,"abstract":"In this paper, we consider a soliton gas of the focusing modified Korteweg-de\u0000Vries generated from the $N$-soliton solutions under the nonzero background.\u0000The spectral soliton density is chosen on the pure imaginary axis, excluding\u0000the branch cut $Sigma_{c}=left[-i, iright]$. In the limit $Ntoinfty$, we\u0000establish the Riemann-Hilbert problem of the soliton gas. Using the Deift-Zhou\u0000nonlinear steepest-descent method, this soliton gas under the nonzero\u0000background will decay to a constant background as $xto+infty$, while its\u0000asymptotics as $xto-infty$ can be expressed with a Riemann-Theta function,\u0000attached to a Riemann surface with genus-two. We also analyze the large $t$\u0000asymptotics over the entire spatial domain, which is divided into three\u0000distinct asymptotic regions depending on the ratio $xi=frac{x}{t}$. Using the\u0000similar method, we provide the leading-order asymptotic behaviors for these\u0000three regions and exhibit the dynamics of large $t$ asymptotics.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141573779","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 symmetry approach to the classification of evolution integrable partial differential equations (see, for example~cite{MikShaSok91}) produces an infinite series of functions, defined in terms of the right hand side, that are conserved densities of any equation having infinitely many infinitesimal symmetries. For instance, the function $frac{partial f}{partial u_{x}}$ has to be a conserved density of any integrable equation of the~KdV type~$u_t=u_{xxx}+f(u,u_x)$. This fact imposes very strong conditions on the form of the function~$f$. In this paper we construct similar canonical densities for equations of the Boussinesq type. In order to do that, we write the equations as evolution systems and generalise the formal diagonalisation procedure proposed in cite{MSY} to these systems.
{"title":"Integrability conditions for Boussinesq type systems","authors":"Rafael Hernandez Heredero, Vladimir Sokolov","doi":"arxiv-2406.19919","DOIUrl":"https://doi.org/arxiv-2406.19919","url":null,"abstract":"The symmetry approach to the classification of evolution integrable partial\u0000differential equations (see, for example~cite{MikShaSok91}) produces an\u0000infinite series of functions, defined in terms of the right hand side, that are\u0000conserved densities of any equation having infinitely many infinitesimal\u0000symmetries. For instance, the function $frac{partial f}{partial u_{x}}$ has\u0000to be a conserved density of any integrable equation of the~KdV\u0000type~$u_t=u_{xxx}+f(u,u_x)$. This fact imposes very strong conditions on the\u0000form of the function~$f$. In this paper we construct similar canonical\u0000densities for equations of the Boussinesq type. In order to do that, we write\u0000the equations as evolution systems and generalise the formal diagonalisation\u0000procedure proposed in cite{MSY} to these systems.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502687","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}
In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously proposed in the work of Habibullin et al. (2016 {it J. Phys. A: Math. Theor.} {bf 57} 015203). This approach is based on the use of the well-known theory of Laplace transforms. The article completes the proof that for any known integrable equation of sine-Gordon type, the sequence of Laplace transforms associated with its linearization admits a third-order finite-field reduction. It is shown that the found reductions are closely related to the Lax pair and recursion operators for both characteristic directions of the given hyperbolic equation. Previously unknown Lax pairs and recursion operators were constructed.
{"title":"Reduction of the Laplace sequence and sine-Gordon type equations","authors":"K I Faizulina, A R Khakimova","doi":"arxiv-2406.19837","DOIUrl":"https://doi.org/arxiv-2406.19837","url":null,"abstract":"In this work, we continue the development of methods for constructing Lax\u0000pairs and recursion operators for nonlinear integrable hyperbolic equations of\u0000soliton type, previously proposed in the work of Habibullin et al. (2016 {it\u0000J. Phys. A: Math. Theor.} {bf 57} 015203). This approach is based on the use\u0000of the well-known theory of Laplace transforms. The article completes the proof\u0000that for any known integrable equation of sine-Gordon type, the sequence of\u0000Laplace transforms associated with its linearization admits a third-order\u0000finite-field reduction. It is shown that the found reductions are closely\u0000related to the Lax pair and recursion operators for both characteristic\u0000directions of the given hyperbolic equation. Previously unknown Lax pairs and\u0000recursion operators were constructed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528740","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}
An exact analytical solution of the nonlinear boson diffusion equation (NBDE) is presented. It accounts for the time evolution towards the Bose-Einstein equilibrium distribution through inelastic and elastic collisions in case of constant transport coefficients. As a currently interesting application, gluon scattering in relativistic heavy-ion collisions is investigated. An estimate of time-dependent gluon-condensate formation in overoccupied systems through number-conserving elastic scatterings in Pb-Pb collisions at relativistic energies is given.
{"title":"Exact solution of the nonlinear boson diffusion equation for gluon scattering","authors":"L. Möhringer, G. Wolschin","doi":"arxiv-2406.11017","DOIUrl":"https://doi.org/arxiv-2406.11017","url":null,"abstract":"An exact analytical solution of the nonlinear boson diffusion equation (NBDE)\u0000is presented. It accounts for the time evolution towards the Bose-Einstein\u0000equilibrium distribution through inelastic and elastic collisions in case of\u0000constant transport coefficients. As a currently interesting application, gluon\u0000scattering in relativistic heavy-ion collisions is investigated. An estimate of\u0000time-dependent gluon-condensate formation in overoccupied systems through\u0000number-conserving elastic scatterings in Pb-Pb collisions at relativistic\u0000energies is given.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872277","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 inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions at infinity is thoroughly discussed. We delve into the analytical properties of the Jost eigenfunctions and scrutinize the characteristics of the scattering coefficients. To enhance our investigation of the fundamental eigenfunctions, we have derived additional auxiliary eigenfunctions with the help of the adjoint problem. Two symmetry conditions are studied to constrain the behavior of the eigenfunctions and scattering coefficients. Utilizing these symmetries, we precisely delineate the discrete spectrum and establish the associated symmetries of the scattering data. By framing the inverse problem within the context of the Riemann-Hilbert problem, we develop suitable jump conditions to express the eigenfunctions. Consequently, we deduce the pure soliton solutions from the defocusing-defocusing coupled Hirota equations, and the double-poles solutions are provided explicitly for the first time in this work.
{"title":"Inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions: double-pole solutions","authors":"Peng-Fei Han, Wen-Xiu Ma, Ru-Suo Ye, Yi Zhang","doi":"arxiv-2406.08189","DOIUrl":"https://doi.org/arxiv-2406.08189","url":null,"abstract":"The inverse scattering transform for the defocusing-defocusing coupled Hirota\u0000equations with non-zero boundary conditions at infinity is thoroughly\u0000discussed. We delve into the analytical properties of the Jost eigenfunctions\u0000and scrutinize the characteristics of the scattering coefficients. To enhance\u0000our investigation of the fundamental eigenfunctions, we have derived additional\u0000auxiliary eigenfunctions with the help of the adjoint problem. Two symmetry\u0000conditions are studied to constrain the behavior of the eigenfunctions and\u0000scattering coefficients. Utilizing these symmetries, we precisely delineate the\u0000discrete spectrum and establish the associated symmetries of the scattering\u0000data. By framing the inverse problem within the context of the Riemann-Hilbert\u0000problem, we develop suitable jump conditions to express the eigenfunctions.\u0000Consequently, we deduce the pure soliton solutions from the\u0000defocusing-defocusing coupled Hirota equations, and the double-poles solutions\u0000are provided explicitly for the first time in this work.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502688","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 non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations with self-consistent sources are established and solved. Two families of quasideterminant solutions are presented for the non-Abelian two-dimensional Toda lattice with self-consistent sources. By employing periodic and quasi-periodic reductions, a matrix sine-Gordon equation with self-consistent sources is constructed for the first time, for which exact solutions in terms of quasideterminants are derived.
{"title":"The non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations with self-consistent sources","authors":"Mengyuan Cui, Chunxia Li","doi":"arxiv-2406.05634","DOIUrl":"https://doi.org/arxiv-2406.05634","url":null,"abstract":"The non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations\u0000with self-consistent sources are established and solved. Two families of\u0000quasideterminant solutions are presented for the non-Abelian two-dimensional\u0000Toda lattice with self-consistent sources. By employing periodic and\u0000quasi-periodic reductions, a matrix sine-Gordon equation with self-consistent\u0000sources is constructed for the first time, for which exact solutions in terms\u0000of quasideterminants are derived.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141502689","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}
One of the most important tasks in mathematics and physics is to connect differential geometry and nonlinear differential equations. In the study of nonlinear optics, integrable nonlinear differential equations such as the nonlinear Schr"odinger equation (NLSE) and higher-order NLSE (HNLSE) play crucial roles. Because of the medium's balance between dispersion and nonlinearity, all of these systems display soliton solutions. The soliton surfaces, or manifolds, connected to these integrable systems hold significance in numerous areas of mathematics and physics. We examine the use of soliton theory in differential geometry in this paper. We build the two-dimensional soliton surface in the three-dimensional Euclidean space by taking into account the Fokas-Lenells Derivative nonlinear Schr"odinger equation (also known as the gauged Fokas-Lenells equation). The same is constructed by us using the Sym-Tafel formula. The first and second fundamental forms, surface area, and Gaussian curvature are obtained using a Lax representation of the gauged FLE.
{"title":"Fokas-Lenells Derivative nonlinear Schrödinger equation its associated soliton surfaces and Gaussian curvature","authors":"Sagardeep Talukdar, Riki Dutta, Gautam Kumar Saharia, Sudipta Nandy","doi":"arxiv-2406.03203","DOIUrl":"https://doi.org/arxiv-2406.03203","url":null,"abstract":"One of the most important tasks in mathematics and physics is to connect\u0000differential geometry and nonlinear differential equations. In the study of\u0000nonlinear optics, integrable nonlinear differential equations such as the\u0000nonlinear Schr\"odinger equation (NLSE) and higher-order NLSE (HNLSE) play\u0000crucial roles. Because of the medium's balance between dispersion and\u0000nonlinearity, all of these systems display soliton solutions. The soliton\u0000surfaces, or manifolds, connected to these integrable systems hold significance\u0000in numerous areas of mathematics and physics. We examine the use of soliton\u0000theory in differential geometry in this paper. We build the two-dimensional\u0000soliton surface in the three-dimensional Euclidean space by taking into account\u0000the Fokas-Lenells Derivative nonlinear Schr\"odinger equation (also known as\u0000the gauged Fokas-Lenells equation). The same is constructed by us using the\u0000Sym-Tafel formula. The first and second fundamental forms, surface area, and\u0000Gaussian curvature are obtained using a Lax representation of the gauged FLE.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548620","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}
In this work, we analyze the asymptotic behaviors of high-order rogue wave solutions with multiple large parameters and discover novel rogue wave patterns, including claw-like, OTR-type, TTR-type, semi-modified TTR-type, and their modified patterns. A correlation is established between these rogue wave patterns and the root structures of the Adler--Moser polynomials with multiple roots. At the positions in the $(x,t)$-plane corresponding to single roots of the Adler--Moser polynomials, these high-order rogue wave patterns asymptotically approach first-order rogue waves. At the positions in the $(x,t)$-plane corresponding to multiple roots of the Adler--Moser polynomials, these rogue wave patterns asymptotically tend toward lower-order fundamental rogue waves, dispersed first-order rogue waves, or mixed structures of these rogue waves. These structures are related to the root structures of special Adler--Moser polynomials with new free parameters, such as the Yablonskii--Vorob'ev polynomial hierarchy, among others. Notably, the positions of the fundamental lower-order rogue waves or mixed structures in these rogue wave patterns can be controlled freely under specific conditions.
{"title":"Rogue wave patterns associated with Adler--Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation","authors":"Huian Lin, Liming Ling","doi":"arxiv-2405.19602","DOIUrl":"https://doi.org/arxiv-2405.19602","url":null,"abstract":"In this work, we analyze the asymptotic behaviors of high-order rogue wave\u0000solutions with multiple large parameters and discover novel rogue wave\u0000patterns, including claw-like, OTR-type, TTR-type, semi-modified TTR-type, and\u0000their modified patterns. A correlation is established between these rogue wave\u0000patterns and the root structures of the Adler--Moser polynomials with multiple\u0000roots. At the positions in the $(x,t)$-plane corresponding to single roots of\u0000the Adler--Moser polynomials, these high-order rogue wave patterns\u0000asymptotically approach first-order rogue waves. At the positions in the\u0000$(x,t)$-plane corresponding to multiple roots of the Adler--Moser polynomials,\u0000these rogue wave patterns asymptotically tend toward lower-order fundamental\u0000rogue waves, dispersed first-order rogue waves, or mixed structures of these\u0000rogue waves. These structures are related to the root structures of special\u0000Adler--Moser polynomials with new free parameters, such as the\u0000Yablonskii--Vorob'ev polynomial hierarchy, among others. Notably, the positions\u0000of the fundamental lower-order rogue waves or mixed structures in these rogue\u0000wave patterns can be controlled freely under specific conditions.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190966","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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