Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant difference. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role for yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane.
{"title":"On complex dynamics in a Suris's integrable map","authors":"Yasutaka Hanada, Akira Shudo","doi":"arxiv-2403.20023","DOIUrl":"https://doi.org/arxiv-2403.20023","url":null,"abstract":"Quantum tunneling in a two-dimensional integrable map is studied. The orbits\u0000of the map are all confined to the curves specified by the one-dimensional\u0000Hamiltonian. It is found that the behavior of tunneling splitting for the\u0000integrable map and the associated Hamiltonian system is qualitatively the same,\u0000with only a slight difference in magnitude. However, the tunneling tails of the\u0000wave functions, obtained by superposing the eigenfunctions that form the\u0000doublet, exhibit significant difference. To explore the origin of the\u0000difference, we observe the classical dynamics in the complex plane and find\u0000that the existence of branch points appearing in the potential function of the\u0000integrable map could play the role for yielding non-trivial behavior in the\u0000tunneling tail. The result highlights the subtlety of quantum tunneling, which\u0000cannot be captured in nature only by the dynamics in the real plane.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580793","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 report new rogue wave patterns whose wave crests form closed or open curves in the spatial plane, which we call rogue curves, in the Davey-Stewartson I equation. These rogue curves come in various striking shapes, such as rings, double rings, and many others. They emerge from a uniform background (possibly with a few lumps on it), reach high amplitude in such striking shapes, and then disappear into the same background again. We reveal that these rogue curves would arise when an internal parameter in bilinear expressions of the rogue waves is real and large. Analytically, we show that these rogue curves are predicted by root curves of certain types of double-real-variable polynomials. We compare analytical predictions of rogue curves to true solutions and demonstrate good agreement between them.
我们报告了新的流氓波模式，其波峰在空间平面上形成封闭或开放的曲线，我们称之为戴维-斯图尔特森 I 方程中的流氓曲线。这些无赖曲线有各种条纹形状，如环形、双环形等。它们从一个均匀的背景（可能上面有一些肿块）中出现，在这些引人注目的形状中达到高振幅，然后又消失在相同的背景中。我们发现，当流氓波线性表达式中的一个内部参数是真实且较大时，就会出现这些流氓曲线。分析表明，这些流氓曲线是由某些类型的双实变多项式的根曲线预测的。我们将流氓曲线的分析预测与真实解进行了比较，结果表明两者之间具有良好的一致性。
{"title":"Rogue curves in the Davey-Stewartson I equation","authors":"Bo Yang, Jianke Yang","doi":"arxiv-2403.18770","DOIUrl":"https://doi.org/arxiv-2403.18770","url":null,"abstract":"We report new rogue wave patterns whose wave crests form closed or open\u0000curves in the spatial plane, which we call rogue curves, in the\u0000Davey-Stewartson I equation. These rogue curves come in various striking\u0000shapes, such as rings, double rings, and many others. They emerge from a\u0000uniform background (possibly with a few lumps on it), reach high amplitude in\u0000such striking shapes, and then disappear into the same background again. We\u0000reveal that these rogue curves would arise when an internal parameter in\u0000bilinear expressions of the rogue waves is real and large. Analytically, we\u0000show that these rogue curves are predicted by root curves of certain types of\u0000double-real-variable polynomials. We compare analytical predictions of rogue\u0000curves to true solutions and demonstrate good agreement between them.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140324676","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}
A non-abelian generalisation of a birational representation of affine Weyl groups and their application to the discrete dynamical systems is presented. By using this generalisation, non-commutative analogs for the discrete systems of $A_n^{(1)}$, $n geq 2$ type and of $d$-Painlev'e equations with an additive dynamic were derived. A coalescence cascade of the later is also discussed.
{"title":"Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlevé equations","authors":"Irina Bobrova","doi":"arxiv-2403.18463","DOIUrl":"https://doi.org/arxiv-2403.18463","url":null,"abstract":"A non-abelian generalisation of a birational representation of affine Weyl\u0000groups and their application to the discrete dynamical systems is presented. By\u0000using this generalisation, non-commutative analogs for the discrete systems of\u0000$A_n^{(1)}$, $n geq 2$ type and of $d$-Painlev'e equations with an additive\u0000dynamic were derived. A coalescence cascade of the later is also discussed.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"558 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140324432","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 consider a matrix nonlinear partial differential equation that generalizes Heisenberg ferromagnet equation. This generalized Heisenberg ferromagnet equation is completely integrable with a linear bundle Lax pair related to the pseudo-unitary algebra. This allows us to explicitly derive particular solutions by using dressing technique. We shall discuss two classes of solutions over constant background: soliton-like solutions and quasi-rational solutions. Both classes have their analogues in the case of the Heisenberg ferromagnet equation related to the same Lie algebra.
{"title":"A Generic Nonlinear Evolution Equation of Magnetic Type II. Particular Solutions","authors":"T. Valchev","doi":"arxiv-2403.18165","DOIUrl":"https://doi.org/arxiv-2403.18165","url":null,"abstract":"We consider a matrix nonlinear partial differential equation that generalizes\u0000Heisenberg ferromagnet equation. This generalized Heisenberg ferromagnet\u0000equation is completely integrable with a linear bundle Lax pair related to the\u0000pseudo-unitary algebra. This allows us to explicitly derive particular\u0000solutions by using dressing technique. We shall discuss two classes of\u0000solutions over constant background: soliton-like solutions and quasi-rational\u0000solutions. Both classes have their analogues in the case of the Heisenberg\u0000ferromagnet equation related to the same Lie algebra.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325838","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}
Y. F. Adans, A. R. Aguirre, J. F. Gomes, G. V. Lobo, A. H. Zimerman
The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra $hat lie $ into graded subspaces defined by a grading operator $Q$. In this paper we shall discuss explicitly the simplest case of the affine $hat {sl}(2)$ Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$. Each of these elements in turn, defines a time evolution equation according to time $t=t_{2n+1}$. An interesting observation is that for negative grades, the zero curvature representation allows both, even or odd sub-hierarchies. In both cases, the flows are non-local leading to integro-differential equations. Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions, the negative even admits strictly non-zero vacuum solutions. Soliton solutions can be constructed by gauge transforming the zero curvature from the vacuum into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine $hat {sl}(2,1)$ super-algebra.
{"title":"SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations","authors":"Y. F. Adans, A. R. Aguirre, J. F. Gomes, G. V. Lobo, A. H. Zimerman","doi":"arxiv-2403.16285","DOIUrl":"https://doi.org/arxiv-2403.16285","url":null,"abstract":"The construction of Integrable Hierarchies in terms of zero curvature\u0000representation provides a systematic construction for a series of integrable\u0000non-linear evolution equations (flows) which shares a common affine Lie\u0000algebraic structure. The integrable hierarchies are then classified in terms of\u0000a decomposition of the underlying affine Lie algebra $hat lie $ into graded\u0000subspaces defined by a grading operator $Q$. In this paper we shall discuss\u0000explicitly the simplest case of the affine $hat {sl}(2)$ Kac-Moody algebra\u0000within the principal gradation given rise to the KdV and mKdV hierarchies and\u0000extend to supersymmetric models. It is known that the positive mKdV sub-hierachy is associated to some\u0000positive odd graded abelian subalgebra with elements denoted by $E^{(2n+1)}$.\u0000Each of these elements in turn, defines a time evolution equation according to\u0000time $t=t_{2n+1}$. An interesting observation is that for negative grades, the\u0000zero curvature representation allows both, even or odd sub-hierarchies. In both\u0000cases, the flows are non-local leading to integro-differential equations.\u0000Whilst positive and negative odd sub-hierarchies admit zero vacuum solutions,\u0000the negative even admits strictly non-zero vacuum solutions. Soliton solutions\u0000can be constructed by gauge transforming the zero curvature from the vacuum\u0000into a non trivial configuration (dressing method). Inspired by the dressing transformation method, we have constructed a\u0000gauge-Miura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative\u0000grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd\u0000and its consecutive even) of mKdV are mapped into a single odd KdV flow. These\u0000results are extended to supersymmetric hierarchies based upon the affine $hat\u0000{sl}(2,1)$ super-algebra.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"273 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140298947","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}
Matrix differential-difference Lax pairs play an essential role in the theory of integrable nonlinear differential-difference equations. We present sufficient conditions for the possibility to simplify such a Lax pair by matrix gauge transformations. Furthermore, we describe a procedure for such a simplification and present applications of it to constructing new integrable equations connected by (non-invertible) discrete substitutions to known equations with Lax pairs. Suppose that one has three (possibly multicomponent) equations $E$, $E_1$, $E_2$, a discrete substitution from $E_1$ to $E$, and a discrete substitution from $E_2$ to $E_1$. Then $E_1$ and $E_2$ can be called a modified version of $E$ and a doubly modified version of $E$, respectively. We demonstrate how the above-mentioned procedure helps (in the considered examples) to construct modified and doubly modified versions of a given equation possessing a Lax pair satisfying certain conditions. The considered examples include scalar equations of Itoh-Narita-Bogoyavlensky type and $2$-component equations related to the Toda lattice. Several new integrable equations and discrete substitutions are presented.
{"title":"Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations","authors":"Sergei Igonin","doi":"arxiv-2403.12022","DOIUrl":"https://doi.org/arxiv-2403.12022","url":null,"abstract":"Matrix differential-difference Lax pairs play an essential role in the theory\u0000of integrable nonlinear differential-difference equations. We present\u0000sufficient conditions for the possibility to simplify such a Lax pair by matrix\u0000gauge transformations. Furthermore, we describe a procedure for such a\u0000simplification and present applications of it to constructing new integrable\u0000equations connected by (non-invertible) discrete substitutions to known\u0000equations with Lax pairs. Suppose that one has three (possibly multicomponent) equations $E$, $E_1$,\u0000$E_2$, a discrete substitution from $E_1$ to $E$, and a discrete substitution\u0000from $E_2$ to $E_1$. Then $E_1$ and $E_2$ can be called a modified version of\u0000$E$ and a doubly modified version of $E$, respectively. We demonstrate how the\u0000above-mentioned procedure helps (in the considered examples) to construct\u0000modified and doubly modified versions of a given equation possessing a Lax pair\u0000satisfying certain conditions. The considered examples include scalar equations of Itoh-Narita-Bogoyavlensky\u0000type and $2$-component equations related to the Toda lattice. Several new\u0000integrable equations and discrete substitutions are presented.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168637","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}
Integrability of N=1 supersymmetric Ruijsenaars-Schneider three-body models based upon the potentials W(x)=2/x, W(x)=2/sin(x), and W(x)=2/sinh(x) is proven. The problem of constructing an algebraically resolvable set of Grassmann-odd constants of motion is reduced to finding a triplet of vectors such that all their scalar products can be expressed in terms of the original bosonic first integrals. The supersymmetric generalizations are used to build novel integrable (iso)spin extensions of the respective Ruijsenaars-Schneider three-body systems.
{"title":"Remarks on integrability of N=1 supersymmetric Ruijsenaars-Schneider three-body models","authors":"Anton Galajinsky","doi":"arxiv-2403.09204","DOIUrl":"https://doi.org/arxiv-2403.09204","url":null,"abstract":"Integrability of N=1 supersymmetric Ruijsenaars-Schneider three-body models\u0000based upon the potentials W(x)=2/x, W(x)=2/sin(x), and W(x)=2/sinh(x) is\u0000proven. The problem of constructing an algebraically resolvable set of\u0000Grassmann-odd constants of motion is reduced to finding a triplet of vectors\u0000such that all their scalar products can be expressed in terms of the original\u0000bosonic first integrals. The supersymmetric generalizations are used to build\u0000novel integrable (iso)spin extensions of the respective Ruijsenaars-Schneider\u0000three-body systems.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140148384","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 paper establishes a direct linearization scheme for the SU(2) anti-self-dual Yang-Mills (ASDYM) equation.The scheme starts from a set of linear integral equations with general measures and plane wave factors. After introducing infinite-dimensional matrices as master functions, we are able to investigate evolution relations and recurrence relations of these functions, which lead us to the unreduced ASDYM equation. It is then reduced to the ASDYM equation in the Euclidean space and two ultrahyperbolic spaces by reductions to meet the reality conditions and gauge conditions, respectively. Special solutions can be obtained by choosing suitable measures.
{"title":"Direct linearization of the SU(2) anti-self-dual Yang-Mills equation in various spaces","authors":"Shangshuai Li, Da-jun Zhang","doi":"arxiv-2403.06055","DOIUrl":"https://doi.org/arxiv-2403.06055","url":null,"abstract":"The paper establishes a direct linearization scheme for the SU(2)\u0000anti-self-dual Yang-Mills (ASDYM) equation.The scheme starts from a set of\u0000linear integral equations with general measures and plane wave factors. After\u0000introducing infinite-dimensional matrices as master functions, we are able to\u0000investigate evolution relations and recurrence relations of these functions,\u0000which lead us to the unreduced ASDYM equation. It is then reduced to the ASDYM\u0000equation in the Euclidean space and two ultrahyperbolic spaces by reductions to\u0000meet the reality conditions and gauge conditions, respectively. Special\u0000solutions can be obtained by choosing suitable measures.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105585","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 consider here the class of fully-nonlinear symmetry-integrable third-order evolution equations in 1+1 dimensions that were proposed recently in {it Open Communications in Nonlinear Mathematical Physics}, vol. 2, 216--228 (2022). In particular, we report all zero-order and higher-order potentialisations for this class of equations using their integrating factors (or multipliers) up to order four. Chains of connecting evolution equations are also obtained by multi-potentialisations.
{"title":"Potentialisations of a class of fully-nonlinear symmetry-integrable evolution equations","authors":"Marianna Euler, Norbert Euler","doi":"arxiv-2403.05722","DOIUrl":"https://doi.org/arxiv-2403.05722","url":null,"abstract":"We consider here the class of fully-nonlinear symmetry-integrable third-order\u0000evolution equations in 1+1 dimensions that were proposed recently in {it Open\u0000Communications in Nonlinear Mathematical Physics}, vol. 2, 216--228 (2022). In\u0000particular, we report all zero-order and higher-order potentialisations for\u0000this class of equations using their integrating factors (or multipliers) up to\u0000order four. Chains of connecting evolution equations are also obtained by\u0000multi-potentialisations.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105614","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}
Suvendu Barik, Alexander. S. Garkun, Vladimir Gritsev
We explore the algebraic structure of a particular ansatz of Yang Baxter Equation which is inspired from the Bethe Ansatz treatment of the ASEP spin-model. Various classes of Hamiltonian density arriving from two types of R-Matrices are found which also appear as solutions of constant YBE. We identify the idempotent and nilpotent categories of such constant R-Matrices and perform a rank-1 numerical search for the lowest dimension. A summary of finalised results reveals general non-hermitian spin-1/2 chain models.
我们探讨了杨-巴克斯特方程（Yang BaxterEquation）的一种特殊解析的代数结构，这种解析的灵感来自于对 ASEPspin 模型的 Bethe Ansatz 处理。我们发现了从两类 R 矩阵中得到的各类哈密顿密度，这些哈密顿密度也作为恒定杨百翰方程的解出现。我们确定了这类恒定 R 矩的等价和零等价类别，并对最低维度进行了秩-1 数值搜索。对最终结果的总结揭示了一般的非全息自旋-1/2 链模型。
{"title":"Novel approach of exploring ASEP-like models through the Yang Baxter Equation","authors":"Suvendu Barik, Alexander. S. Garkun, Vladimir Gritsev","doi":"arxiv-2403.03159","DOIUrl":"https://doi.org/arxiv-2403.03159","url":null,"abstract":"We explore the algebraic structure of a particular ansatz of Yang Baxter\u0000Equation which is inspired from the Bethe Ansatz treatment of the ASEP\u0000spin-model. Various classes of Hamiltonian density arriving from two types of\u0000R-Matrices are found which also appear as solutions of constant YBE. We\u0000identify the idempotent and nilpotent categories of such constant R-Matrices\u0000and perform a rank-1 numerical search for the lowest dimension. A summary of\u0000finalised results reveals general non-hermitian spin-1/2 chain models.","PeriodicalId":501592,"journal":{"name":"arXiv - PHYS - Exactly Solvable and Integrable Systems","volume":"97 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140046019","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：