Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations is reconsidered, where two `proper' discrete Ablowitz-Kaup-Newell-Segur equations and two `unproper' discrete Ablowitz-Kaup-Newell-Segur equations are derived. The `proper' equations admit local reduction, while the `unproper' equations admit nonlocal reduction. By imposing the local and nonlocal complex reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two local and nonlocal discrete complex modified Korteweg-de Vries equations are constructed. For the obtained local and nonlocal discrete complex modified Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are presented by solving the determining equation set. The dynamical behaviors of 1-soliton solution are analyzed and illustrated. Continuum limits of the resulting local and nonlocal discrete complex modified Korteweg-de Vries equations are discussed.
{"title":"Solutions of local and nonlocal discrete complex modified Korteweg-de Vries equations and continuum limits","authors":"Ya-Nan Hu, Shou-Feng Shen, Song-lin Zhao","doi":"arxiv-2404.14150","DOIUrl":"https://doi.org/arxiv-2404.14150","url":null,"abstract":"Cauchy matrix approach for the discrete Ablowitz-Kaup-Newell-Segur equations\u0000is reconsidered, where two `proper' discrete Ablowitz-Kaup-Newell-Segur\u0000equations and two `unproper' discrete Ablowitz-Kaup-Newell-Segur equations are\u0000derived. The `proper' equations admit local reduction, while the `unproper'\u0000equations admit nonlocal reduction. By imposing the local and nonlocal complex\u0000reductions on the obtained discrete Ablowitz-Kaup-Newell-Segur equations, two\u0000local and nonlocal discrete complex modified Korteweg-de Vries equations are\u0000constructed. For the obtained local and nonlocal discrete complex modified\u0000Korteweg-de Vries equations, soliton solutions and Jordan-block solutions are\u0000presented by solving the determining equation set. The dynamical behaviors of\u00001-soliton solution are analyzed and illustrated. Continuum limits of the\u0000resulting local and nonlocal discrete complex modified Korteweg-de Vries\u0000equations are discussed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798577","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 Navier-Stokes equations are paradigmatic equations describing hydrodynamics of an interacting system with microscopic interactions encoded in transport coefficients. In this work we show how the Navier-Stokes equations arise from the microscopic dynamics of nearly integrable $1d$ quantum many-body systems. We build upon the recently developed hydrodynamics of integrable models to study the effective Boltzmann equation with collision integral taking into account the non-integrable interactions. We compute the transport coefficients and find that the resulting Navier-Stokes equations have two regimes, which differ in the viscous properties of the resulting fluid. We illustrate the method by computing the transport coefficients for an experimentally relevant case of coupled 1d cold-atomic gases.
{"title":"Navier-Stokes equations for nearly integrable quantum gases","authors":"Maciej Łebek, Miłosz Panfil","doi":"arxiv-2404.14292","DOIUrl":"https://doi.org/arxiv-2404.14292","url":null,"abstract":"The Navier-Stokes equations are paradigmatic equations describing\u0000hydrodynamics of an interacting system with microscopic interactions encoded in\u0000transport coefficients. In this work we show how the Navier-Stokes equations\u0000arise from the microscopic dynamics of nearly integrable $1d$ quantum many-body\u0000systems. We build upon the recently developed hydrodynamics of integrable\u0000models to study the effective Boltzmann equation with collision integral taking\u0000into account the non-integrable interactions. We compute the transport\u0000coefficients and find that the resulting Navier-Stokes equations have two\u0000regimes, which differ in the viscous properties of the resulting fluid. We\u0000illustrate the method by computing the transport coefficients for an\u0000experimentally relevant case of coupled 1d cold-atomic gases.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798579","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 classical Landau--Lifshitz equation -- the simplest model of a ferromagnet -- provides an archetypal example for studying transport phenomena. In one-spatial dimension, integrability enables the classification of the spectrum of linear and nonlinear modes. An exact characterization of finite-temperature thermodynamics and transport has nonetheless remained elusive. We present an exact description of thermodynamic equilibrium states in terms of interacting modes. This is achieved by retrieving the classical Landau--Lifschitz model through the semiclassical limit of the integrable quantum spin-$S$ anisotropic Heisenberg chain at the level of the thermodynamic Bethe ansatz description. In the axial regime, the mode spectrum comprises solitons with unconventional statistics, whereas in the planar regime we additionally find two special types of modes of radiative and solitonic type. The obtained framework paves the way for analytical study of unconventional transport properties: as an example we study the finite-temperature spin Drude weight, finding excellent agreement with Monte Carlo simulations.
{"title":"Landau-Lifschitz magnets: exact thermodynamics and transport","authors":"Alvise Bastianello, Žiga Krajnik, Enej Ilievski","doi":"arxiv-2404.12106","DOIUrl":"https://doi.org/arxiv-2404.12106","url":null,"abstract":"The classical Landau--Lifshitz equation -- the simplest model of a\u0000ferromagnet -- provides an archetypal example for studying transport phenomena.\u0000In one-spatial dimension, integrability enables the classification of the\u0000spectrum of linear and nonlinear modes. An exact characterization of\u0000finite-temperature thermodynamics and transport has nonetheless remained\u0000elusive. We present an exact description of thermodynamic equilibrium states in\u0000terms of interacting modes. This is achieved by retrieving the classical\u0000Landau--Lifschitz model through the semiclassical limit of the integrable\u0000quantum spin-$S$ anisotropic Heisenberg chain at the level of the thermodynamic\u0000Bethe ansatz description. In the axial regime, the mode spectrum comprises\u0000solitons with unconventional statistics, whereas in the planar regime we\u0000additionally find two special types of modes of radiative and solitonic type.\u0000The obtained framework paves the way for analytical study of unconventional\u0000transport properties: as an example we study the finite-temperature spin Drude\u0000weight, finding excellent agreement with Monte Carlo simulations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"95 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623021","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}
Extended versions of the noncommutative(nc) KP equation and the nc mKP equation are constructed in a unified way, for which two types of quasideterminant solutions are also presented. In commutative setting, the quasideterminant solutions provide the known and unknown Wronskian and Grammian solutions for the bilinear KP equation with self-consistent sources and the bilinear mKP equation with self-consistent sources, respectively. Miura transformation is established for the extended nc KP and nc mKP equations.
{"title":"The extended versions of the noncommutative KP and mKP equations and Miura transformation","authors":"Muhammad Kashif, Li Chunxia, Cui Mengyuan","doi":"arxiv-2404.11391","DOIUrl":"https://doi.org/arxiv-2404.11391","url":null,"abstract":"Extended versions of the noncommutative(nc) KP equation and the nc mKP\u0000equation are constructed in a unified way, for which two types of\u0000quasideterminant solutions are also presented. In commutative setting, the\u0000quasideterminant solutions provide the known and unknown Wronskian and Grammian\u0000solutions for the bilinear KP equation with self-consistent sources and the\u0000bilinear mKP equation with self-consistent sources, respectively. Miura\u0000transformation is established for the extended nc KP and nc mKP equations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140615119","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 Ablowitz-Ladik chain is an integrable discretized version of the nonlinear Schr"{o}dinger equation. We report on a novel underlying Hamiltonian particle system with properties similar to the ones known for the classical Toda chain and Calogero fluid with $1/sinh^2$ pair interaction. Boundary conditions are imposed such that, both in the distant past and future, particles have a constant velocity. We establish the many-particle scattering for the Ablowitz-Ladik chain and obtain properties known for generic integrable many-body systems. For a specific choice of the chain, real initial data remain real in the course of time. Then, asymptotically, particles move in pairs with a velocity-dependent size and scattering shifts are governed by the fusion rule.
{"title":"Particle Scattering and Fusion for the Ablowitz-Ladik Chain","authors":"Alberto Brollo, Herbert Spohn","doi":"arxiv-2404.07095","DOIUrl":"https://doi.org/arxiv-2404.07095","url":null,"abstract":"The Ablowitz-Ladik chain is an integrable discretized version of the\u0000nonlinear Schr\"{o}dinger equation. We report on a novel underlying Hamiltonian\u0000particle system with properties similar to the ones known for the classical\u0000Toda chain and Calogero fluid with $1/sinh^2$ pair interaction. Boundary\u0000conditions are imposed such that, both in the distant past and future,\u0000particles have a constant velocity. We establish the many-particle scattering\u0000for the Ablowitz-Ladik chain and obtain properties known for generic integrable\u0000many-body systems. For a specific choice of the chain, real initial data remain\u0000real in the course of time. Then, asymptotically, particles move in pairs with\u0000a velocity-dependent size and scattering shifts are governed by the fusion\u0000rule.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580701","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 challenge of solving the initial value problem for the coupled Lakshmanan Porsezian Daniel equation, while considering nonzero boundary conditions at infinity, is addressed through the development of a suitable inverse scattering transform. Analytical properties of the Jost eigenfunctions are examined, along with the analysis of scattering coefficient characteristics. This analysis leads to the derivation of additional auxiliary eigenfunctions necessary for the comprehensive investigation of the fundamental eigenfunctions. Two symmetry conditions are discussed to study the eigenfunctions and scattering coefficients. These symmetry results are utilized to rigorously define the discrete spectrum and ascertain the corresponding symmetries of scattering datas. The inverse scattering problem is formulated by the Riemann-Hilbert problem. Then we can derive the exact solutions by coupled Lakshmanan Porsezian Daniel equation, the novel soliton solutions are derived and examined in detail.
{"title":"Inverse scattering transform for the coupled Lakshmanan-Porsezian-Daniel equation with nonzero boundary conditions","authors":"Peng-Fei Han, Ru-Suo Ye, Yi Zhang","doi":"arxiv-2404.03351","DOIUrl":"https://doi.org/arxiv-2404.03351","url":null,"abstract":"The challenge of solving the initial value problem for the coupled Lakshmanan\u0000Porsezian Daniel equation, while considering nonzero boundary conditions at\u0000infinity, is addressed through the development of a suitable inverse scattering\u0000transform. Analytical properties of the Jost eigenfunctions are examined, along\u0000with the analysis of scattering coefficient characteristics. This analysis\u0000leads to the derivation of additional auxiliary eigenfunctions necessary for\u0000the comprehensive investigation of the fundamental eigenfunctions. Two symmetry\u0000conditions are discussed to study the eigenfunctions and scattering\u0000coefficients. These symmetry results are utilized to rigorously define the\u0000discrete spectrum and ascertain the corresponding symmetries of scattering\u0000datas. The inverse scattering problem is formulated by the Riemann-Hilbert\u0000problem. Then we can derive the exact solutions by coupled Lakshmanan Porsezian\u0000Daniel equation, the novel soliton solutions are derived and examined in\u0000detail.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580703","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 propose a novel approach to tackle integrability problem for evolutionary differential-difference equations (D$Delta$Es) on free associative algebras, also referred to as nonabelian D$Delta$Es. This approach enables us to derive necessary integrability conditions, determine the integrability of a given equation, and make progress in the classification of integrable nonabelian D$Delta$Es. This work involves establishing symbolic representations for the nonabelian difference algebra, difference operators, and formal series, as well as introducing a novel quasi-local extension for the algebra of formal series within the context of symbolic representations. Applying this formalism, we solve the classification problem of integrable skew-symmetric quasi-linear nonabelian equations of orders $(-1,1)$, $(-2,2)$, and $(-3,3)$, consequently revealing some new equations in the process.
{"title":"Integrability of Nonabelian Differential-Difference Equations: the Symmetry Approach","authors":"Vladimir Novikov, Jing Ping Wang","doi":"arxiv-2404.02326","DOIUrl":"https://doi.org/arxiv-2404.02326","url":null,"abstract":"We propose a novel approach to tackle integrability problem for evolutionary\u0000differential-difference equations (D$Delta$Es) on free associative algebras,\u0000also referred to as nonabelian D$Delta$Es. This approach enables us to derive\u0000necessary integrability conditions, determine the integrability of a given\u0000equation, and make progress in the classification of integrable nonabelian\u0000D$Delta$Es. This work involves establishing symbolic representations for the\u0000nonabelian difference algebra, difference operators, and formal series, as well\u0000as introducing a novel quasi-local extension for the algebra of formal series\u0000within the context of symbolic representations. Applying this formalism, we\u0000solve the classification problem of integrable skew-symmetric quasi-linear\u0000nonabelian equations of orders $(-1,1)$, $(-2,2)$, and $(-3,3)$, consequently\u0000revealing some new equations in the process.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580790","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 article introduces contact germs that transform solutions of some partial differential equations into solutions of other equations. Parametric symmetries of differential equations generalizing point and contact symmetries are defined. New transformations and symmetries may depend on derivatives of arbitrary but finite order. The stationary Schr"odinger equations, acoustics and gas dynamics equations are considered as examples.
{"title":"Contact germs and partial differential equations","authors":"O. V. Kaptsov","doi":"arxiv-2404.01955","DOIUrl":"https://doi.org/arxiv-2404.01955","url":null,"abstract":"The article introduces contact germs that transform solutions of some partial\u0000differential equations into solutions of other equations. Parametric symmetries\u0000of differential equations generalizing point and contact symmetries are\u0000defined. New transformations and symmetries may depend on derivatives of\u0000arbitrary but finite order. The stationary Schr\"odinger equations, acoustics\u0000and gas dynamics equations are considered as examples.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"319 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140581034","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 good Boussinesq equation has several modified versions such as the modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma equation. This work builds the full relations among these equations by Miura transformation and invertible linear transformations and draws a pyramid diagram to demonstrate such relations. The direct and inverse spectral analysis shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation has simple pole at origin, the solution of Riemann-Hilbert problem for the good Boussinesq equation has double pole at origin, while the solution of Riemann-Hilbert problem for the modified Boussinesq equation and Mikhailov-Lenells equation doesn't have singularity at origin. Further, the large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz class initial value is studied by Deift-Zhou nonlinear steepest descent analysis. In such initial condition, the asymptotic expressions of the Hirota-Satsuma equation and good Boussinesq equation away from the origin are proposed and it is displayed that the leading term of asymptotic formulas match well with direct numerical simulations.
{"title":"Miura transformations and large-time behaviors of the Hirota-Satsuma equation","authors":"Deng-Shan Wang, Cheng Zhu, Xiaodong Zhu","doi":"arxiv-2404.01215","DOIUrl":"https://doi.org/arxiv-2404.01215","url":null,"abstract":"The good Boussinesq equation has several modified versions such as the\u0000modified Boussinesq equation, Mikhailov-Lenells equation and Hirota-Satsuma\u0000equation. This work builds the full relations among these equations by Miura\u0000transformation and invertible linear transformations and draws a pyramid\u0000diagram to demonstrate such relations. The direct and inverse spectral analysis\u0000shows that the solution of Riemann-Hilbert problem for Hirota-Satsuma equation\u0000has simple pole at origin, the solution of Riemann-Hilbert problem for the good\u0000Boussinesq equation has double pole at origin, while the solution of\u0000Riemann-Hilbert problem for the modified Boussinesq equation and\u0000Mikhailov-Lenells equation doesn't have singularity at origin. Further, the\u0000large-time asymptotic behaviors of the Hirota-Satsuma equation with Schwartz\u0000class initial value is studied by Deift-Zhou nonlinear steepest descent\u0000analysis. In such initial condition, the asymptotic expressions of the\u0000Hirota-Satsuma equation and good Boussinesq equation away from the origin are\u0000proposed and it is displayed that the leading term of asymptotic formulas match\u0000well with direct numerical simulations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"279 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580699","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}
To obtain new integrable nonlinear differential equations there are some well-known methods such as Lax equations with different Lax representations. There are also some other methods which are based on integrable scalar nonlinear partial differential equations. We show that some systems of integrable equations published recently are the ${cal M}_{2}$-extension of integrable scalar equations. For illustration we give Korteweg-de Vries, Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such an extension of integrable scalar equations we obtain some new integrable systems with recursion operators. We give also the soliton solutions of the system equations and integrable standard nonlocal and shifted nonlocal reductions of these systems.
{"title":"On SK and KK Integrable Systems","authors":"Metin Gürses, Aslı Pekcan","doi":"arxiv-2404.00671","DOIUrl":"https://doi.org/arxiv-2404.00671","url":null,"abstract":"To obtain new integrable nonlinear differential equations there are some\u0000well-known methods such as Lax equations with different Lax representations.\u0000There are also some other methods which are based on integrable scalar\u0000nonlinear partial differential equations. We show that some systems of\u0000integrable equations published recently are the ${cal M}_{2}$-extension of\u0000integrable scalar equations. For illustration we give Korteweg-de Vries,\u0000Kaup-Kupershmidt, and Sawada-Kotera equations as examples. By the use of such\u0000an extension of integrable scalar equations we obtain some new integrable\u0000systems with recursion operators. We give also the soliton solutions of the\u0000system equations and integrable standard nonlocal and shifted nonlocal\u0000reductions of these systems.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"138 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580795","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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