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GBDT version of the Darboux transformation for the matrix coupled dispersionless equations (local and non-local cases) 矩阵耦合无色散方程(局部和非局部情况)的GBDT Darboux变换
Journal of Integrable Systems Pub Date : 2020-01-01 DOI: 10.1093/integr/xyaa004
R. Popovych, A. Sakhnovich
{"title":"GBDT version of the Darboux transformation for the matrix coupled dispersionless equations (local and non-local cases)","authors":"R. Popovych, A. Sakhnovich","doi":"10.1093/integr/xyaa004","DOIUrl":"https://doi.org/10.1093/integr/xyaa004","url":null,"abstract":"\u0000 We introduce matrix coupled (local and non-local) dispersionless equations, construct GBDT (generalized Bäcklund-Darboux transformation) for these equations, derive wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and study their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and non-local dispersionless equations as well.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115290340","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
Geometric curve flows in low dimensional Cayley–Klein geometries 低维Cayley-Klein几何中的几何曲线流动
Journal of Integrable Systems Pub Date : 2020-01-01 DOI: 10.1093/integr/xyaa003
Joe Benson, F. Valiquette
{"title":"Geometric curve flows in low dimensional Cayley–Klein geometries","authors":"Joe Benson, F. Valiquette","doi":"10.1093/integr/xyaa003","DOIUrl":"https://doi.org/10.1093/integr/xyaa003","url":null,"abstract":"\u0000 Using the method of equivariant moving frames, we derive the evolution equations for the curvature invariants of arc-length parametrized curves under arc-length preserving geometric flows in two-, three- and four-dimensional Cayley–Klein geometries. In two and three dimensions, we obtain recursion operators, which show that the curvature evolution equations obtained are completely integrable.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129542983","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 solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation 非平稳椭圆Calogero-Sutherland方程的精确积分解
Journal of Integrable Systems Pub Date : 2019-08-01 DOI: 10.1093/integr/xyaa001
F. Atai, E. Langmann
{"title":"Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation","authors":"F. Atai, E. Langmann","doi":"10.1093/integr/xyaa001","DOIUrl":"https://doi.org/10.1093/integr/xyaa001","url":null,"abstract":"\u0000 We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero–Sutherland model (also known as elliptic Knizhnik–Zamolodchikov–Bernard equation). Our solutions provide integral representations of elliptic generalizations of the Jack polynomials.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115672689","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: 4
Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions 非平稳rujsenaars函数的仿射筛选算子、仿射Laumon空间和猜想
Journal of Integrable Systems Pub Date : 2019-01-01 DOI: 10.1093/integr/xyz010
J. Shiraishi
{"title":"Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions","authors":"J. Shiraishi","doi":"10.1093/integr/xyz010","DOIUrl":"https://doi.org/10.1093/integr/xyz010","url":null,"abstract":"\u0000 Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series $f^{widehat{mathfrak gl}_N}(x,p|s,kappa|q,t)$ which we call the non-stationary Ruijsenaars function. We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters $s$ and $kappa$ are suitably chosen, the limit $trightarrow q$ of $f^{widehat{mathfrak gl}_N}(x,p|s,kappa|q,q/t)$ gives us the dominant integrable characters of $widehat{mathfrak sl}_N$ multiplied by $1/(p^N;p^N)_infty$ (i.e. the $widehat{mathfrak gl}_1$ character). Several conjectures are presented for $f^{widehat{mathfrak gl}_N}(x,p|s,kappa|q,t)$, including the bispectral and the Poincaré dualities, and the evaluation formula. The main conjecture asserts that (i) one can normalize $f^{widehat{mathfrak gl}_N}(x,p|s,kappa|q,t)$ in such a way that the limit $kapparightarrow 1$ exists, and (ii) the limit $f^{{rm st.},widehat{mathfrak gl}_N}(x,p|s|q,t)$ gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine $q$-difference Toda operator ${mathcal T}^{widehat{mathfrak gl}_N}(kappa)$ is introduced, which comes as an outcome of the study of the Poincaré duality conjecture in the affine Toda limit $trightarrow 0$. The main conjecture is examined also in the limiting cases of the affine $q$-difference Toda ($trightarrow 0$), and the elliptic Calogero–Sutherland ($q,trightarrow 1$) equations.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132889394","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: 21
Geometry of Bäcklund transformations II: Monge–Ampère invariants Bäcklund变换的几何II: monge - ampantere不变量
Journal of Integrable Systems Pub Date : 2019-01-01 DOI: 10.1093/integr/xyz011
Yuhao Hu
{"title":"Geometry of Bäcklund transformations II: Monge–Ampère invariants","authors":"Yuhao Hu","doi":"10.1093/integr/xyz011","DOIUrl":"https://doi.org/10.1093/integr/xyz011","url":null,"abstract":"\u0000 This article is concerned with the question: For which pairs of hyperbolic Euler–Lagrange systems in the plane does there exist a rank-$1$ Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler–Lagrange systems. In addition, we discover a class of Bäcklund transformations relating two hyperbolic Euler–Lagrange systems of distinct types.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128951351","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: 2
Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type $boldsymbol{A^{(1)}_{n-1}}$ 类型为$boldsymbol{A^{(1)}_{n-1}}$的量子仿射代数反射方程的集理论解
Journal of Integrable Systems Pub Date : 2019-01-01 DOI: 10.1093/integr/xyz013
A. Kuniba, M. Okado
{"title":"Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type $boldsymbol{A^{(1)}_{n-1}}$","authors":"A. Kuniba, M. Okado","doi":"10.1093/integr/xyz013","DOIUrl":"https://doi.org/10.1093/integr/xyz013","url":null,"abstract":"\u0000 A trick to obtain a solution to the set-theoretical reflection equation from a known one to the Yang–Baxter equation is applied to crystals and geometric crystals associated to the quantum affine algebra of type $A^{(1)}_{n-1}$.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125683787","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: 2
On symmetric primitive potentials 关于对称原势
Journal of Integrable Systems Pub Date : 2018-12-26 DOI: 10.1093/integr/xyz006
P. Nabelek, D. Zakharov, V. Zakharov
{"title":"On symmetric primitive potentials","authors":"P. Nabelek, D. Zakharov, V. Zakharov","doi":"10.1093/integr/xyz006","DOIUrl":"https://doi.org/10.1093/integr/xyz006","url":null,"abstract":"\u0000 The concept of a primitive potential for the Schrödinger operator on the line was introduced in Dyachenko et al. (2016, Phys. D, 333, 148–156), Zakharov, Dyachenko et al. (2016, Lett. Math. Phys., 106, 731–740) and Zakharov, Zakharov et al. (2016, Phys. Lett. A, 380, 3881–3885). Such a potential is determined by a pair of positive functions on a finite interval, called the dressing functions, which are not uniquely determined by the potential. The potential is constructed by solving a contour problem on the complex plane. In this article, we consider a reduction where the dressing functions are equal. We show that in this case, the resulting potential is symmetric, and describe how to analytically compute the potential as a power series. In addition, we establish that if the dressing functions are both equal to one, then the resulting primitive potential is the elliptic one-gap potential.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"105 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116068842","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: 7
Non-interlacing peakon solutions of the Geng–Xue equation 庚雪方程的非交错峰解
Journal of Integrable Systems Pub Date : 2018-12-20 DOI: 10.1093/INTEGR/XYZ007
Budor Shuaib, Hans Lundmark
{"title":"Non-interlacing peakon solutions of the Geng–Xue equation","authors":"Budor Shuaib, Hans Lundmark","doi":"10.1093/INTEGR/XYZ007","DOIUrl":"https://doi.org/10.1093/INTEGR/XYZ007","url":null,"abstract":"The aim of the present article is to derive explicit formulas for arbitrary non-overlapping pure peakon solutions of the Geng–Xue (GX) equation, a two-component generalization of Novikov’s cubically non-linear Camassa–Holm type equation. By performing limiting procedures on the previously known formulas for so-called interlacing peakon solutions, where the peakons in the two component occur alternatingly, we turn some of the peakons into zero-amplitude ‘ghostpeakons’, in such a way that the remaining ordinary peakons occur in any desired configuration. A novel feature compared to the interlacing case is that the Lax pairs for the GX equation do not provide all the constants of motion necessary for the integration of the system. We also study the large-time asymptotics of the non-interlacing solutions. As in the interlacing case, the peakon amplitudes grow or decay exponentially, and their logarithms display phase shifts similar to those for the positions. Moreover, within a group of adjacent peakons in one component, all peakons but one have the same asymptotic velocity. A curious phenomenon occurs when the number of such peakon groups is odd, namely that the sets of incoming and outgoing velocities are unequal.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114920676","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: 5
KZ equations and Bethe subalgebras in generalized Yangians related to compatible $R$-matrices 相容R -矩阵下广义yangian中的KZ方程和Bethe子代数
Journal of Integrable Systems Pub Date : 2018-12-12 DOI: 10.1093/INTEGR/XYZ005
D. Gurevich, P. Saponov, D. Talalaev
{"title":"KZ equations and Bethe subalgebras in generalized Yangians related to compatible $R$-matrices","authors":"D. Gurevich, P. Saponov, D. Talalaev","doi":"10.1093/INTEGR/XYZ005","DOIUrl":"https://doi.org/10.1093/INTEGR/XYZ005","url":null,"abstract":"\u0000 The notion of compatible braidings was introduced in Isaev et al. (1999, J. Phys. A, 32, L115–L121). On the base of this notion, the authors of Isaev et al. (1999, J. Phys. A, 32, L115–L121) defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogues of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach, we introduce certain new algebras called generalized Yangians and define analogues of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutative Bethe subalgebra in each generalized Yangian. Besides, we define some analogues (also arising from couples of compatible braidings) of the Knizhnik–Zamolodchikov equation—classical and quantum.\u0000 Communicated by: Alexander Veselov","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124684359","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: 4
Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers 循环颤振的复三角rujsenaars - schneider模型的自旋版本
Journal of Integrable Systems Pub Date : 2018-11-21 DOI: 10.1093/integr/xyz008
M. Fairon
{"title":"Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers","authors":"M. Fairon","doi":"10.1093/integr/xyz008","DOIUrl":"https://doi.org/10.1093/integr/xyz008","url":null,"abstract":"We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with $mgeq 2$ vertices. Their global Poisson structure is characterized by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case $m=1$ corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars–Schneider (RS) system. This generalizes to new spin cases recent works on classical integrable systems in the RS family.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125909105","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: 11
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