The non-Abelian ferromagnet recently introduced by the authors, consisting of atoms in the fundamental representation of $SU(N)$, is studied in the limit where $N$ becomes large and scales as the square root of the number of atoms $n$. This model exhibits additional phases, as well as two different temperature scales related by a factor $N!/!ln N$. The paramagnetic phase splits into a "dense" and a "dilute" phase, separated by a third-order transition and leading to a triple critical point in the scale parameter $n/N^2$ and the temperature, while the ferromagnetic phase exhibits additional structure, and a new paramagnetic-ferromagnetic metastable phase appears at the larger temperature scale. These phases can coexist, becoming stable or metastable as temperature varies. A generalized model in which the number of $SU(N)$-equivalent states enters the partition function with a nontrivial weight, relevant, e.g., when there is gauge invariance in the system, is also studied and shown to manifest similar phases, with the dense-dilute phase transition becoming second-order in the fully gauge invariant case.
{"title":"Triple critical point and emerging temperature scales in $SU(N)$ ferromagnetism at large $N$","authors":"Alexios P. Polychronakos, Konstantinos Sfetsos","doi":"arxiv-2408.08357","DOIUrl":"https://doi.org/arxiv-2408.08357","url":null,"abstract":"The non-Abelian ferromagnet recently introduced by the authors, consisting of\u0000atoms in the fundamental representation of $SU(N)$, is studied in the limit\u0000where $N$ becomes large and scales as the square root of the number of atoms\u0000$n$. This model exhibits additional phases, as well as two different\u0000temperature scales related by a factor $N!/!ln N$. The paramagnetic phase\u0000splits into a \"dense\" and a \"dilute\" phase, separated by a third-order\u0000transition and leading to a triple critical point in the scale parameter\u0000$n/N^2$ and the temperature, while the ferromagnetic phase exhibits additional\u0000structure, and a new paramagnetic-ferromagnetic metastable phase appears at the\u0000larger temperature scale. These phases can coexist, becoming stable or\u0000metastable as temperature varies. A generalized model in which the number of\u0000$SU(N)$-equivalent states enters the partition function with a nontrivial\u0000weight, relevant, e.g., when there is gauge invariance in the system, is also\u0000studied and shown to manifest similar phases, with the dense-dilute phase\u0000transition becoming second-order in the fully gauge invariant case.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194556","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}
Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra $mathcal{L}^{rm tor}_{r+1}(mathfrak{sl}_ell)$ via lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent this hierarchy in the form of Lax equations, and show that it is an extension of a certain reduction of the $ell$-component KP hierarchy.
{"title":"Integrable hierarchy for homogeneous realization of toroidal Lie algebra $mathcal{L}^{rm tor}_{r+1}(mathfrak{sl}_ell)$","authors":"Chao-Zhong Wu, Yi Yang","doi":"arxiv-2408.07376","DOIUrl":"https://doi.org/arxiv-2408.07376","url":null,"abstract":"Starting from a fairly explicit homogeneous realization of the toroidal Lie\u0000algebra $mathcal{L}^{rm tor}_{r+1}(mathfrak{sl}_ell)$ via lattice vertex\u0000algebra, we derive an integrable hierarchy of Hirota bilinear equations.\u0000Moreover, we represent this hierarchy in the form of Lax equations, and show\u0000that it is an extension of a certain reduction of the $ell$-component KP\u0000hierarchy.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194552","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 contradistinction to the case of massive excitations, the connection between integrability and the tree-level massless scattering matrix of integrable $sigma$-models is lost. Namely, in well-known 2-d integrable models the tree-level massless S-matrix exhibits particle production and fails to factorise. This is conjectured to happen due to IR ambiguities in the massless tree-level amplitudes. We present a definition of the massless S-matrix which has all the nice properties of integrable theories, there is no particle production and the S-matrix factorises. As an example, we present in detail the case of the $SU(2)$ principal chiral model (PCM).
与大质量激元的情况相反，可积分性与可积分的$sigma$模型的树级无质量散射矩阵之间失去了联系。也就是说，在著名的二维可积分模型中，树级无质量S矩阵表现出粒子产生，而无法因子化。据推测，这是由于质量树级振幅的红外模糊性造成的。我们提出了无质量 S 矩阵的定义，它具有可积分理论的所有优良特性，不存在粒子产 生，而且 S 矩阵能够因式分解。作为一个例子，我们详细介绍了$SU(2)$主手性模型（PCM）的情况。
{"title":"The massless S-matrix of integrable $σ$-models","authors":"George Georgiou","doi":"arxiv-2408.03673","DOIUrl":"https://doi.org/arxiv-2408.03673","url":null,"abstract":"In contradistinction to the case of massive excitations, the connection\u0000between integrability and the tree-level massless scattering matrix of\u0000integrable $sigma$-models is lost. Namely, in well-known 2-d integrable models\u0000the tree-level massless S-matrix exhibits particle production and fails to\u0000factorise. This is conjectured to happen due to IR ambiguities in the massless\u0000tree-level amplitudes. We present a definition of the massless S-matrix which\u0000has all the nice properties of integrable theories, there is no particle\u0000production and the S-matrix factorises. As an example, we present in detail the\u0000case of the $SU(2)$ principal chiral model (PCM).","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938398","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}
Leonardo Biagetti, Maciej Lebek, Milosz Panfil, Jacopo De Nardis
We consider quantum or classical many-body Hamiltonian systems, whose dynamics is given by an integrable, contact interactions, plus another, possibly long-range, generic two-body potential. We show how the dynamics of local observables is given in terms of a generalised version of Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, which we denote as gBBGKY, which is built for the densities, and their correlations, of the quasiparticles of the underlying integrable model. Unlike the usual cases of perturbation theory from free gases, the presence of local interactions in the integrable model "lifts" the so-called kinetic blocking, and the second layer of the hierarchy reproduces the dynamics at all time-scales. The latter consists of a fast pre-equilibration to a non-thermal steady state, and its subsequent thermalisation to a Gibbs ensemble. We show how the final relaxation is encoded into a Boltzmann scattering integral involving three or higher body-scatterings, and which, remarkably, is entirely determined by the diffusion constants of the underlying integrable model. We check our results with exact molecular dynamics simulations, finding perfect agreement. Our results show how gBBGKY can be successfully employed in quantum systems to compute scattering integrals and Fermi's golden rule transition rates.
{"title":"Generalised BBGKY hierarchy for near-integrable dynamics","authors":"Leonardo Biagetti, Maciej Lebek, Milosz Panfil, Jacopo De Nardis","doi":"arxiv-2408.00593","DOIUrl":"https://doi.org/arxiv-2408.00593","url":null,"abstract":"We consider quantum or classical many-body Hamiltonian systems, whose\u0000dynamics is given by an integrable, contact interactions, plus another,\u0000possibly long-range, generic two-body potential. We show how the dynamics of\u0000local observables is given in terms of a generalised version of\u0000Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, which we denote as gBBGKY, which\u0000is built for the densities, and their correlations, of the quasiparticles of\u0000the underlying integrable model. Unlike the usual cases of perturbation theory\u0000from free gases, the presence of local interactions in the integrable model\u0000\"lifts\" the so-called kinetic blocking, and the second layer of the hierarchy\u0000reproduces the dynamics at all time-scales. The latter consists of a fast\u0000pre-equilibration to a non-thermal steady state, and its subsequent\u0000thermalisation to a Gibbs ensemble. We show how the final relaxation is encoded\u0000into a Boltzmann scattering integral involving three or higher\u0000body-scatterings, and which, remarkably, is entirely determined by the\u0000diffusion constants of the underlying integrable model. We check our results\u0000with exact molecular dynamics simulations, finding perfect agreement. Our\u0000results show how gBBGKY can be successfully employed in quantum systems to\u0000compute scattering integrals and Fermi's golden rule transition rates.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141884009","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}
Elastic collisions of solitons generally have a finite phase shift. When the phase shift has a finitely large value, the two vertices of the (2+1)-dimensional 2-soliton are significantly separated due to the phase shift, accompanied by the formation of a local structure connecting the two V-shaped solitons. We define this local structure as the stem structure. This study systematically investigates the localized stem structures between two solitons in the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system. These stem structures, arising from quasi-resonant collisions between the solitons, exhibit distinct features of spatial locality and temporal invariance. We explore two scenarios: one characterized by weakly quasi-resonant collisions (i.e. $a_{12}approx 0$), and the other by strongly quasi-resonant collisions (i.e. $a_{12}approx +infty$). Through mathematical analysis, we extract comprehensive insights into the trajectories, amplitudes, and velocities of the soliton arms. Furthermore, we discuss the characteristics of the stem structures, including their length and extreme points. Our findings shed new light on the interaction between solitons in the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system.
{"title":"Localized stem structures in quasi-resonant two-soliton solutions for the asymmetric Nizhnik-Novikov-Veselov system","authors":"Feng Yuan, Jiguang Rao, Jingsong He, Yi Cheng","doi":"arxiv-2407.20875","DOIUrl":"https://doi.org/arxiv-2407.20875","url":null,"abstract":"Elastic collisions of solitons generally have a finite phase shift. When the\u0000phase shift has a finitely large value, the two vertices of the\u0000(2+1)-dimensional 2-soliton are significantly separated due to the phase shift,\u0000accompanied by the formation of a local structure connecting the two V-shaped\u0000solitons. We define this local structure as the stem structure. This study\u0000systematically investigates the localized stem structures between two solitons\u0000in the (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov system. These stem\u0000structures, arising from quasi-resonant collisions between the solitons,\u0000exhibit distinct features of spatial locality and temporal invariance. We\u0000explore two scenarios: one characterized by weakly quasi-resonant collisions\u0000(i.e. $a_{12}approx 0$), and the other by strongly quasi-resonant collisions\u0000(i.e. $a_{12}approx +infty$). Through mathematical analysis, we extract\u0000comprehensive insights into the trajectories, amplitudes, and velocities of the\u0000soliton arms. Furthermore, we discuss the characteristics of the stem\u0000structures, including their length and extreme points. Our findings shed new\u0000light on the interaction between solitons in the (2+1)-dimensional asymmetric\u0000Nizhnik-Novikov-Veselov system.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141873405","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 a class of simple, but nonlinear, systems of recursions involving $2$ dependent variables $x_{j}left( nright) $ is identified, such that the solutions of their initial-values problems -- with arbitrary initial data $x_{j}left( 0right) $ -- may be explicitly obtained.
{"title":"Solvable nonlinear systems of 2 recursions displaying interesting evolutions","authors":"Francesco Calogero","doi":"arxiv-2407.18270","DOIUrl":"https://doi.org/arxiv-2407.18270","url":null,"abstract":"In this paper a class of simple, but nonlinear, systems of recursions\u0000involving $2$ dependent variables $x_{j}left( nright) $ is identified, such\u0000that the solutions of their initial-values problems -- with arbitrary initial\u0000data $x_{j}left( 0right) $ -- may be explicitly obtained.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"857 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141872273","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}
José Luis Jaramillo, Michele Lenzi, Carlos F. Sopuerta
In this work we investigate the presence of integrable hidden structures in the dynamics of perturbed non-rotating black holes (BHs). This can also be considered as a first step in a wider program of an effective identification of ``slow'' and ``fast'' degrees of freedom (DoFs) in the (binary) BH dynamics, following a wave-mean flow perspective. The slow DoFs would be associated with a nonlinear integrable dynamics, on which the fast ones propagate following an effective linear dynamics. BH perturbation theory offers a natural ground to test these properties. Indeed, the decoupling of Einstein equations into wave master equations with a potential provides an instance of such splitting into (frozen) slow DoFs (background potential) over which the linear dynamics of the fast ones (perturbation master functions) evolve. It has been recently shown that these wave equations possess an infinite number of symmetries that correspond to the flow of the infinite hierarchy of Korteweg-de Vries (KdV) equations. Starting from these results, we systematically investigate the presence of integrable structures in BH perturbation theory. We first study them in Cauchy slices and then extend the analysis to hyperboloidal foliations. This second step introduces a splitting of the master equation into bulk and boundary contributions, unveiling an underlying structural relation with the slow and fast DoFs. This insight represents a first step to establish the integrable structures associated to the slow DoFs as bulk symmetries of the dynamics of perturbed BHs.
{"title":"Integrability in Perturbed Black Holes: Background Hidden Structures","authors":"José Luis Jaramillo, Michele Lenzi, Carlos F. Sopuerta","doi":"arxiv-2407.14196","DOIUrl":"https://doi.org/arxiv-2407.14196","url":null,"abstract":"In this work we investigate the presence of integrable hidden structures in\u0000the dynamics of perturbed non-rotating black holes (BHs). This can also be\u0000considered as a first step in a wider program of an effective identification of\u0000``slow'' and ``fast'' degrees of freedom (DoFs) in the (binary) BH dynamics,\u0000following a wave-mean flow perspective. The slow DoFs would be associated with\u0000a nonlinear integrable dynamics, on which the fast ones propagate following an\u0000effective linear dynamics. BH perturbation theory offers a natural ground to\u0000test these properties. Indeed, the decoupling of Einstein equations into wave\u0000master equations with a potential provides an instance of such splitting into\u0000(frozen) slow DoFs (background potential) over which the linear dynamics of the\u0000fast ones (perturbation master functions) evolve. It has been recently shown\u0000that these wave equations possess an infinite number of symmetries that\u0000correspond to the flow of the infinite hierarchy of Korteweg-de Vries (KdV)\u0000equations. Starting from these results, we systematically investigate the\u0000presence of integrable structures in BH perturbation theory. We first study\u0000them in Cauchy slices and then extend the analysis to hyperboloidal foliations.\u0000This second step introduces a splitting of the master equation into bulk and\u0000boundary contributions, unveiling an underlying structural relation with the\u0000slow and fast DoFs. This insight represents a first step to establish the\u0000integrable structures associated to the slow DoFs as bulk symmetries of the\u0000dynamics of perturbed BHs.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745634","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}
From a specific series of exchange conditions for a one-parameter Hamiltonian vector field, we establish an integrable hierarchy using Lax pairs derived from the dispersionless partial differential equation. An exterior differential form of the integrable hierarchy is introduced, further confirming the existence of the tau function. Subsequently, we present the twistor structure of the hierarchy. By constructing the nonlinear Riemann Hilbert problem for the equation, the structure of the solution to the equation is better understood.
从一参数哈密顿矢量场的一系列特定交换条件出发，我们利用从无分散偏微分方程导出的拉克斯对建立了可积分层次结构。我们引入了可积分层次结构的外微分形式，进一步证实了 tau 函数的存在。随后，我们介绍了该层次结构的扭曲结构。通过构建方程的非线性黎曼希尔伯特问题，我们更好地理解了方程解的结构。
{"title":"The integrable hierarchy and the nonlinear Riemann-Hilbert problem associated with one typical Einstein-Weyl physico-geometric dispersionless system","authors":"Ge Yi, Tangna Lv, Kelei Tian, Ying Xu","doi":"arxiv-2407.11515","DOIUrl":"https://doi.org/arxiv-2407.11515","url":null,"abstract":"From a specific series of exchange conditions for a one-parameter Hamiltonian\u0000vector field, we establish an integrable hierarchy using Lax pairs derived from\u0000the dispersionless partial differential equation. An exterior differential form\u0000of the integrable hierarchy is introduced, further confirming the existence of\u0000the tau function. Subsequently, we present the twistor structure of the\u0000hierarchy. By constructing the nonlinear Riemann Hilbert problem for the\u0000equation, the structure of the solution to the equation is better understood.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720608","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}
It is pointed, that the $16times16$ Hamiltonian density matrix, corresponding to XX spin chain in a staggered magnetic field, satisfies the Reshetikhin condition as well, as the two higher ones. All of them are necessary for the existence of the corresponding $R$-matrix. At the special imaginary value of the staggered field, the $R$-matrix is obtained explicitly. It is shown that the corresponding Hamiltonian has purely imaginary spectrum.
研究指出，与交错磁场中的 XX 自旋链相对应的 $16times16$ 哈密顿密度矩阵与两个更高的哈密顿密度矩阵一样，也满足雷谢提金条件。所有这些都是相应 $R$ 矩阵存在的必要条件。在交错磁场的特殊虚值下，R$R 矩阵被明确得到。
{"title":"R-matrix for the XX spin chain at the special imaginary value of staggered magnetic field","authors":"P. N. Bibikov","doi":"arxiv-2407.10395","DOIUrl":"https://doi.org/arxiv-2407.10395","url":null,"abstract":"It is pointed, that the $16times16$ Hamiltonian density matrix,\u0000corresponding to XX spin chain in a staggered magnetic field, satisfies the\u0000Reshetikhin condition as well, as the two higher ones. All of them are\u0000necessary for the existence of the corresponding $R$-matrix. At the special\u0000imaginary value of the staggered field, the $R$-matrix is obtained explicitly.\u0000It is shown that the corresponding Hamiltonian has purely imaginary spectrum.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720609","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 the present paper we reconsider the integrable case of the Hamiltonian $N$-species Volterra system, as it has been introduced by Vito Volterra in 1937, and significantly enrich the results already published in the ArXiv in 2019. In fact, we present a new approach to the construction of conserved quantities and comment about the solutions of the equations of motion; we display mostly new analytical and numerical results, starting from the classical predator-prey model till the general $N-$species model.
{"title":"The Volterra Integrable case. Novel analytical and numerical results","authors":"M. Scalia, O. Ragnisco, B. Tirozzi, F. Zullo","doi":"arxiv-2407.09155","DOIUrl":"https://doi.org/arxiv-2407.09155","url":null,"abstract":"In the present paper we reconsider the integrable case of the Hamiltonian\u0000$N$-species Volterra system, as it has been introduced by Vito Volterra in\u00001937, and significantly enrich the results already published in the ArXiv in\u00002019. In fact, we present a new approach to the construction of conserved\u0000quantities and comment about the solutions of the equations of motion; we\u0000display mostly new analytical and numerical results, starting from the\u0000classical predator-prey model till the general $N-$species model.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"66 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720673","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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