A seventh order ordinary differential equation (ODE) arising by reduction of the Drinfeld-Sokolov hierarchyis shown to be identical to a similarity reduction of an equationin the hierarchy of Sawada-Kotera.We also exhibit its link with a particular F-VI,a fourth order ODE isolated by Cosgrove which is likely to define a higher order Painlev'e function.
{"title":"On an equation arising by reduction of the Drinfeld-Sokolov hierarchy","authors":"R. ConteENS Paris Saclay","doi":"arxiv-2405.12606","DOIUrl":"https://doi.org/arxiv-2405.12606","url":null,"abstract":"A seventh order ordinary differential equation (ODE) arising by reduction of\u0000the Drinfeld-Sokolov hierarchyis shown to be identical to a similarity\u0000reduction of an equationin the hierarchy of Sawada-Kotera.We also exhibit its\u0000link with a particular F-VI,a fourth order ODE isolated by Cosgrove which is\u0000likely to define a higher order Painlev'e function.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151530","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 considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schr"odinger equations. The well-known example of this kind of generalization is the Davey-Stewartson equation. It turns out that the finite-field reductions of these lattices, obtained by imposing cutoff boundary conditions of an appropriate type, are Darboux integrable, i.e., they have complete sets of characteristic integrals. An algorithm for constructing complete sets of characteristic integrals of finite field systems using Lax pairs and Miura-type transformations is discussed.
{"title":"On integrable reductions of two-dimensional Toda-type lattices","authors":"I. T. Habibullin, A. U. Sakieva","doi":"arxiv-2405.10666","DOIUrl":"https://doi.org/arxiv-2405.10666","url":null,"abstract":"The article considers lattices of the two-dimensional Toda type, which can be\u0000interpreted as dressing chains for spatially two-dimensional generalizations of\u0000equations of the class of nonlinear Schr\"odinger equations. The well-known\u0000example of this kind of generalization is the Davey-Stewartson equation. It\u0000turns out that the finite-field reductions of these lattices, obtained by\u0000imposing cutoff boundary conditions of an appropriate type, are Darboux\u0000integrable, i.e., they have complete sets of characteristic integrals. An\u0000algorithm for constructing complete sets of characteristic integrals of finite\u0000field systems using Lax pairs and Miura-type transformations is discussed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151555","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}
Magnetic hopfions are localized magnetic solitons with non-zero 3D topological charge (Hopf index). Here I present an analytical calculation of the toroidal magnetic hopfion vector potential, emergent magnetic field, the Hopf index, and the magnetization configuration. The calculation method is based on the concept of the spinor representation of the Hopf mapping. The hopfions with arbitrary values of the azimuthal and poloidal vorticities are considered. The special role of the toroidal coordinates and their connection with the emergent vector po tential gauge are demonstrated. The hopfion magnetization field is found explicitly for the arbitrary Hopf indices. It is shown that the Hopf charge density can be represented as a Jacobian of the transformation from the toroidal to the cylindrical coordinates.
{"title":"Emergent magnetic field and vector potential of the toroidal magnetic hopfions","authors":"Konstantin Y. Guslienko","doi":"arxiv-2405.10811","DOIUrl":"https://doi.org/arxiv-2405.10811","url":null,"abstract":"Magnetic hopfions are localized magnetic solitons with non-zero 3D\u0000topological charge (Hopf index). Here I present an analytical calculation of\u0000the toroidal magnetic hopfion vector potential, emergent magnetic field, the\u0000Hopf index, and the magnetization configuration. The calculation method is\u0000based on the concept of the spinor representation of the Hopf mapping. The\u0000hopfions with arbitrary values of the azimuthal and poloidal vorticities are\u0000considered. The special role of the toroidal coordinates and their connection\u0000with the emergent vector po tential gauge are demonstrated. The hopfion\u0000magnetization field is found explicitly for the arbitrary Hopf indices. It is\u0000shown that the Hopf charge density can be represented as a Jacobian of the\u0000transformation from the toroidal to the cylindrical coordinates.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141151529","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 study matrix Lax representations (MLRs) for differential-difference (lattice) equations. For a given equation, two MLRs are said to be gauge equivalent if one of them can be obtained from the other by means of a matrix gauge transformation. We present results on the following questions: 1. When is a given MLR gauge equivalent to an MLR suitable for constructing differential-difference Miura-type transformations by the method of [G. Berkeley, S. Igonin, J. Phys. A (2016), arXiv:1512.09123]? 2. When is a given MLR gauge equivalent to a trivial MLR? Furthermore, we present new examples of integrable differential-difference equations with Miura-type transformations.
我们研究微分-差分（网格）方程的矩阵拉克斯表示（MLR）。对于一个给定方程，如果两个 MLR 中的一个可以通过矩阵量规变换从另一个得到，那么这两个 MLR 可以说是量规等价的。我们将介绍有关以下问题的结果：1.给定的 MLR 何时与适合通过[G.Berkeley, S. Igonin, J. Phys. A (2016), arXiv:1512.09123] 方法构造微分差分米乌拉型变换的 MLR 轨距等价？2.给定的 MLR 量规何时等价于微不足道的 MLR？此外，我们还提出了具有米乌拉型变换的可积分微分-差分方程的新例子。
{"title":"On Lax representations under the gauge equivalence relation and Miura-type transformations for lattice equations","authors":"Sergei Igonin","doi":"arxiv-2405.08579","DOIUrl":"https://doi.org/arxiv-2405.08579","url":null,"abstract":"We study matrix Lax representations (MLRs) for differential-difference\u0000(lattice) equations. For a given equation, two MLRs are said to be gauge\u0000equivalent if one of them can be obtained from the other by means of a matrix\u0000gauge transformation. We present results on the following questions: 1. When is a given MLR gauge equivalent to an MLR suitable for constructing\u0000differential-difference Miura-type transformations by the method of [G.\u0000Berkeley, S. Igonin, J. Phys. A (2016), arXiv:1512.09123]? 2. When is a given MLR gauge equivalent to a trivial MLR? Furthermore, we present new examples of integrable differential-difference\u0000equations with Miura-type transformations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141059138","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 a recent work Sharma and Bhosale [Phys. Rev. B, 109, 014412 (2024)], $N$-spin Floquet model having infinite range Ising interaction was introduced. In this paper, we generalized the strength of interaction to $J$, such that $J=1$ case reduces to the aforementioned work. We show that for $J=1/2$ the model still exhibits integrability for an even number of qubits only. We analytically solve the cases of $6$, $8$, $10$, and $12$ qubits, finding its eigensystem, dynamics of entanglement for various initial states, and the unitary evolution operator. These quantities exhibit the signature of quantum integrability (QI). For the general case of even-$N > 12$ qubits, we conjuncture the presence of QI using the numerical evidences such as spectrum degeneracy, and the exact periodic nature of both the entanglement dynamics and the time-evolved unitary operator. We numerically show the absence of QI for odd $N$ by observing a violation of the signatures of QI. We analytically and numerically find that the maximum value of time-evolved concurrence ($C_{mbox{max}}$) decreases with $N$, indicating the multipartite nature of entanglement. Possible experiments to verify our results are discussed.
{"title":"Signatures of Integrability and Exactly Solvable Dynamics in an Infinite-Range Many-Body Floquet Spin System","authors":"Harshit Sharma, Udaysinh T. Bhosale","doi":"arxiv-2405.15797","DOIUrl":"https://doi.org/arxiv-2405.15797","url":null,"abstract":"In a recent work Sharma and Bhosale [Phys. Rev. B, 109, 014412 (2024)],\u0000$N$-spin Floquet model having infinite range Ising interaction was introduced.\u0000In this paper, we generalized the strength of interaction to $J$, such that\u0000$J=1$ case reduces to the aforementioned work. We show that for $J=1/2$ the\u0000model still exhibits integrability for an even number of qubits only. We\u0000analytically solve the cases of $6$, $8$, $10$, and $12$ qubits, finding its\u0000eigensystem, dynamics of entanglement for various initial states, and the\u0000unitary evolution operator. These quantities exhibit the signature of quantum\u0000integrability (QI). For the general case of even-$N > 12$ qubits, we\u0000conjuncture the presence of QI using the numerical evidences such as spectrum\u0000degeneracy, and the exact periodic nature of both the entanglement dynamics and\u0000the time-evolved unitary operator. We numerically show the absence of QI for\u0000odd $N$ by observing a violation of the signatures of QI. We analytically and\u0000numerically find that the maximum value of time-evolved concurrence\u0000($C_{mbox{max}}$) decreases with $N$, indicating the multipartite nature of\u0000entanglement. Possible experiments to verify our results are discussed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141170132","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 classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps $R:(x,y)mapsto (u(x,y),v(x,y)),$ where $u, v$ are two rational functions on two arguments, that serve as solutions of the pentagon equation. Furthermore, provided a pentagon map that admits a partial inverse, we obtain genuine entwining pentagon set theoretical solutions.
{"title":"On quadrirational pentagon maps","authors":"Charalampos Evripidou, Pavlos Kassotakis, Anastasios Tongas","doi":"arxiv-2405.04945","DOIUrl":"https://doi.org/arxiv-2405.04945","url":null,"abstract":"We classify rational solutions of a specific type of the set theoretical\u0000version of the pentagon equation. That is, we find all quadrirational maps\u0000$R:(x,y)mapsto (u(x,y),v(x,y)),$ where $u, v$ are two rational functions on\u0000two arguments, that serve as solutions of the pentagon equation. Furthermore,\u0000provided a pentagon map that admits a partial inverse, we obtain genuine\u0000entwining pentagon set theoretical solutions.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930360","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}
Some simple nonlinear recursions which can be completely managed are identified and the behaviour of all their solutions is ascertained.
确定了一些可以完全管理的简单非线性递归，并确定了其所有解的行为。
{"title":"Simple recursions displaying interesting evolutions","authors":"Francesco Calogero","doi":"arxiv-2405.00370","DOIUrl":"https://doi.org/arxiv-2405.00370","url":null,"abstract":"Some simple nonlinear recursions which can be completely managed are\u0000identified and the behaviour of all their solutions is ascertained.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140833988","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}
Shallow water waves phenomena in nature attract the attention of scholars and play an important role in fields such as tsunamis, tidal waves, solitary waves, and hydraulic engineering. Hereby, for the shallow water waves phenomena in various natural environments, we study the KdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation. Based on the Bell polynomial theory, the B{"a}cklund transformation, Lax pair and infinite conservation laws of the KdV-CBS equation are derived, and it is proved that it is completely integrable in Lax pair sense. Various types of mixed solutions are constructed by using a combination of Homoclinic test method and Mathematica symbolic computations. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of the KdV-CBS equation. We hope that our research results could help the researchers understand the nonlinear complex phenomena of the shallow water waves in oceans, rivers and coastal areas.
{"title":"Investigation of shallow water waves near the coast or in lake environments via the KdV-Calogero-Bogoyavlenskii-Schiff equation","authors":"Peng-Fei Han, Yi Zhang","doi":"arxiv-2404.18697","DOIUrl":"https://doi.org/arxiv-2404.18697","url":null,"abstract":"Shallow water waves phenomena in nature attract the attention of scholars and\u0000play an important role in fields such as tsunamis, tidal waves, solitary waves,\u0000and hydraulic engineering. Hereby, for the shallow water waves phenomena in\u0000various natural environments, we study the KdV-Calogero-Bogoyavlenskii-Schiff\u0000(KdV-CBS) equation. Based on the Bell polynomial theory, the B{\"a}cklund\u0000transformation, Lax pair and infinite conservation laws of the KdV-CBS equation\u0000are derived, and it is proved that it is completely integrable in Lax pair\u0000sense. Various types of mixed solutions are constructed by using a combination\u0000of Homoclinic test method and Mathematica symbolic computations. These findings\u0000have important significance for the discipline, offering vital insights into\u0000the intricate dynamics of the KdV-CBS equation. We hope that our research\u0000results could help the researchers understand the nonlinear complex phenomena\u0000of the shallow water waves in oceans, rivers and coastal areas.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140834245","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 symmetric (2+1)-dimensional Lotka-Volterra equation with self-consistent sources is constructed using the source generation producure, whose solutions are expressed in terms of pfaffians. As special cases of the pfaffian solutions, different types of explicit solutions are presented, including dromions, soliton solutions and breather solutions.
{"title":"The symmetric (2+1)-dimensional Lotka-Volterra equation with self-consistent sources","authors":"Mengyuan Cui, Chunxia Li, Yuqin Yao","doi":"arxiv-2404.14969","DOIUrl":"https://doi.org/arxiv-2404.14969","url":null,"abstract":"The symmetric (2+1)-dimensional Lotka-Volterra equation with self-consistent\u0000sources is constructed using the source generation producure, whose solutions\u0000are expressed in terms of pfaffians. As special cases of the pfaffian\u0000solutions, different types of explicit solutions are presented, including\u0000dromions, soliton solutions and breather solutions.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798586","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}
As local and nonlocal reductions of a discrete second-order Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr"odinger type equations are considered. Through the bilinearization reduction method, we construct double Casoratian solutions of the reduced discrete nonlinear Schr"odinger type equations, including soliton solutions and Jordan-block solutions.Dynamics of the obtained one-soliton and two-soliton solutions are analyzed and illustrated. Moreover,both semi-continuous limit and full continuous limit, are applied to obtain solutions of the local and nonlocal semi-discrete nonlinear Schr"odinger type equations, as well as the local and nonlocal continuous nonlinear Schr"odinger type equations.
{"title":"Discrete nonlinear Schrödinger type equations: Solutions and continuum limits","authors":"Song-lin Zhao, Xiao-hui Feng, Wei Feng","doi":"arxiv-2404.14060","DOIUrl":"https://doi.org/arxiv-2404.14060","url":null,"abstract":"As local and nonlocal reductions of a discrete second-order\u0000Ablowitz-Kaup-Newell-Segur equation, two discrete nonlinear Schr\"odinger type\u0000equations are considered. Through the bilinearization reduction method, we\u0000construct double Casoratian solutions of the reduced discrete nonlinear\u0000Schr\"odinger type equations, including soliton solutions and Jordan-block\u0000solutions.Dynamics of the obtained one-soliton and two-soliton solutions are\u0000analyzed and illustrated. Moreover,both semi-continuous limit and full\u0000continuous limit, are applied to obtain solutions of the local and nonlocal\u0000semi-discrete nonlinear Schr\"odinger type equations, as well as the local and\u0000nonlocal continuous nonlinear Schr\"odinger type equations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140798813","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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