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Noncommutative nonisospectral Toda and Lotka-Volterra lattices, and matrix discrete Painlevé equations 非交换非等谱托达和洛特卡-沃尔特拉晶格，以及矩阵离散潘列维方程
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-07-11 DOI: arxiv-2407.08486
Anhui Yan, Chunxia Li
{"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}
引用次数: 0
Action-angle variables for the nonlinear Schrödinger equation on the half-line 半线上非线性薛定谔方程的作用角变量
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-07-09 DOI: arxiv-2407.06916
Baoqiang Xia
{"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}
引用次数: 0
A modified Korteweg-de Vries equation soliton gas under the nonzero background 非零背景下修正的科特韦格-德弗里斯方程孤子气体
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-07-07 DOI: arxiv-2407.05384
Xiaoen Zhang, Liming Ling
{"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}
引用次数: 0
Integrability conditions for Boussinesq type systems Boussinesq 型系统的可积分性条件
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-06-28 DOI: arxiv-2406.19919
Rafael Hernandez Heredero, Vladimir Sokolov
{"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}
引用次数: 0
Reduction of the Laplace sequence and sine-Gordon type equations 拉普拉斯序列和正弦-戈登方程的还原
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-06-28 DOI: arxiv-2406.19837
K I Faizulina, A R Khakimova
{"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}
引用次数: 0
Exact solution of the nonlinear boson diffusion equation for gluon scattering 胶子散射的非线性玻色子扩散方程的精确解
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-06-16 DOI: arxiv-2406.11017
L. Möhringer, G. Wolschin
{"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}
引用次数: 0
Inverse scattering transform for the defocusing-defocusing coupled Hirota equations with non-zero boundary conditions: double-pole solutions 具有非零边界条件的散焦-聚焦耦合广田方程的反散射变换：双极解
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-06-12 DOI: arxiv-2406.08189
Peng-Fei Han, Wen-Xiu Ma, Ru-Suo Ye, Yi Zhang
{"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}
引用次数: 0
The non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations with self-consistent sources 非阿贝尔二维户田晶格和具有自洽源的矩阵正弦-戈登方程
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-06-09 DOI: arxiv-2406.05634
Mengyuan Cui, Chunxia Li
{"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}
引用次数: 0
Fokas-Lenells Derivative nonlinear Schrödinger equation its associated soliton surfaces and Gaussian curvature Fokas-Lenells 衍生非线性薛定谔方程及其相关孤子面和高斯曲率
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-06-05 DOI: arxiv-2406.03203
Sagardeep Talukdar, Riki Dutta, Gautam Kumar Saharia, Sudipta Nandy
{"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}
引用次数: 0
Rogue wave patterns associated with Adler--Moser polynomials featuring multiple roots in the nonlinear Schrödinger equation 非线性薛定谔方程中与具有多根特征的阿德勒--莫泽多项式相关的流波模式
arXiv - PHYS - Exactly Solvable and Integrable Systems Pub Date : 2024-05-30 DOI: arxiv-2405.19602
Huian Lin, Liming Ling
{"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}
引用次数: 0
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