Pub Date : 2024-08-13DOI: 10.1088/1361-6544/ad6b6e
José F Alves, Krerley Oliveira, Eduardo Santana
In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critical sets. The strategy here consists in using a finite number of inducing schemes with a Markov structure in infinitely many symbols to code the dynamics, to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. We apply our results to the important class of multidimensional Viana maps.
{"title":"Equilibrium states for hyperbolic potentials via inducing schemes *","authors":"José F Alves, Krerley Oliveira, Eduardo Santana","doi":"10.1088/1361-6544/ad6b6e","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6b6e","url":null,"abstract":"In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critical sets. The strategy here consists in using a finite number of inducing schemes with a Markov structure in infinitely many symbols to code the dynamics, to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. We apply our results to the important class of multidimensional Viana maps.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-11DOI: 10.1088/1361-6544/ad6949
Janne Nurminen
In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold where the metric is of the form . Here is a simple Riemannian metric on , e is the Euclidean metric on and c a smooth positive function. We show that if the associated Dirichlet-to-Neumann maps corresponding to metrics g and agree, then the Taylor series of the conformal factor at is equal to a positive constant. We also show a partial data result when n = 3.
在这项工作中，我们研究了黎曼流形上最小曲面方程的逆问题，其中的度量形式为 。这里是一个简单的黎曼流形，e 是欧几里得流形，c 是一个光滑的正函数。我们证明，如果对应于度量 g 的相关迪里希勒到诺伊曼映射一致，那么保角因子 at 的泰勒级数等于一个正常数。我们还展示了 n = 3 时的部分数据结果。
{"title":"An inverse problem for the minimal surface equation in the presence of a riemannian metric","authors":"Janne Nurminen","doi":"10.1088/1361-6544/ad6949","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6949","url":null,"abstract":"In this work we study an inverse problem for the minimal surface equation on a Riemannian manifold where the metric is of the form . Here is a simple Riemannian metric on , e is the Euclidean metric on and c a smooth positive function. We show that if the associated Dirichlet-to-Neumann maps corresponding to metrics g and agree, then the Taylor series of the conformal factor at is equal to a positive constant. We also show a partial data result when n = 3.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"27 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.1088/1361-6544/ad67a0
Yu L Sachkov
Left-invariant Lorentzian structures on the 2D solvable non-Abelian Lie group are studied. Sectional curvature, attainable sets, Lorentzian length maximizers, distance, spheres, and infinitesimal isometries are described.
{"title":"Lorentzian distance on the Lobachevsky plane *","authors":"Yu L Sachkov","doi":"10.1088/1361-6544/ad67a0","DOIUrl":"https://doi.org/10.1088/1361-6544/ad67a0","url":null,"abstract":"Left-invariant Lorentzian structures on the 2D solvable non-Abelian Lie group are studied. Sectional curvature, attainable sets, Lorentzian length maximizers, distance, spheres, and infinitesimal isometries are described.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"43 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.1088/1361-6544/ad68ba
A N W Hone, J A G Roberts and P Vanhaecke
Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the g = 2 case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus . The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case (g = 1), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices.
{"title":"A family of integrable maps associated with the Volterra lattice","authors":"A N W Hone, J A G Roberts and P Vanhaecke","doi":"10.1088/1361-6544/ad68ba","DOIUrl":"https://doi.org/10.1088/1361-6544/ad68ba","url":null,"abstract":"Recently Gubbiotti, Joshi, Tran and Viallet classified birational maps in four dimensions admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. The last three of the maps so obtained were shown to be Liouville integrable, in the sense of admitting a non-degenerate Poisson bracket with two first integrals in involution. Here we show how the first of these three Liouville integrable maps corresponds to genus 2 solutions of the infinite Volterra lattice, being the g = 2 case of a family of maps associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus . The continued fraction method provides explicit Hankel determinant formulae for tau functions of the solutions, together with an algebro-geometric description via a Lax representation for each member of the family, associating it with an algebraic completely integrable system. In particular, in the elliptic case (g = 1), as a byproduct we obtain Hankel determinant expressions for the solutions of the Somos-5 recurrence, but different to those previously derived by Chang, Hu and Xin. By applying contraction to the Stieltjes fraction, we recover integrable maps associated with Jacobi continued fractions on hyperelliptic curves, that one of us considered previously, as well as the Miura-type transformation between the Volterra and Toda lattices.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-07DOI: 10.1088/1361-6544/ad694c
Boris Khesin
The non-transitivity without extra constraints in the Euler equation in any dimension is almost evident and can be derived, e.g. from Morse theory.
欧拉方程在任何维度上没有额外约束的非传递性几乎是显而易见的，而且可以从莫尔斯理论等方面推导出来。
{"title":"The Euler non-mixing made easy","authors":"Boris Khesin","doi":"10.1088/1361-6544/ad694c","DOIUrl":"https://doi.org/10.1088/1361-6544/ad694c","url":null,"abstract":"The non-transitivity without extra constraints in the Euler equation in any dimension is almost evident and can be derived, e.g. from Morse theory.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"198 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-06DOI: 10.1088/1361-6544/ad68b9
Zachary Bradshaw
Recently, strong evidence has accumulated that some solutions to the Navier–Stokes equations in physically meaningful classes are not unique. The primary purpose of this paper is to establish necessary properties for the error of hypothetical non-unique Navier–Stokes flows under conditions motivated by the scaling of the equations. Our first set of results show that some scales are necessarily active—comparable in norm to the full error—as solutions separate. ‘Scale’ is interpreted in several ways, namely via algebraic bounds, the Fourier transform and discrete volume elements. These results include a new type of uniqueness criteria which is stated in terms of the error. The second result is a conditional predictability criteria for the separation of small perturbations. An implication is that the error necessarily activates at larger scales as flows de-correlate. The last result says that the error of the hypothetical non-unique Leray–Hopf solutions of Jia and Šverák locally grows in a self-similar fashion. Consequently, within the Leray–Hopf class, energy can hypothetically de-correlate at a rate which is faster than linear. This contrasts numerical work on predictability which identifies a linear rate. Our results suggest that this discrepancy may be explained by the fact that non-uniqueness might arise from perturbation around a singular flow.
{"title":"Remarks on the separation of Navier–Stokes flows","authors":"Zachary Bradshaw","doi":"10.1088/1361-6544/ad68b9","DOIUrl":"https://doi.org/10.1088/1361-6544/ad68b9","url":null,"abstract":"Recently, strong evidence has accumulated that some solutions to the Navier–Stokes equations in physically meaningful classes are not unique. The primary purpose of this paper is to establish necessary properties for the error of hypothetical non-unique Navier–Stokes flows under conditions motivated by the scaling of the equations. Our first set of results show that some scales are necessarily active—comparable in norm to the full error—as solutions separate. ‘Scale’ is interpreted in several ways, namely via algebraic bounds, the Fourier transform and discrete volume elements. These results include a new type of uniqueness criteria which is stated in terms of the error. The second result is a conditional predictability criteria for the separation of small perturbations. An implication is that the error necessarily activates at larger scales as flows de-correlate. The last result says that the error of the hypothetical non-unique Leray–Hopf solutions of Jia and Šverák locally grows in a self-similar fashion. Consequently, within the Leray–Hopf class, energy can hypothetically de-correlate at a rate which is faster than linear. This contrasts numerical work on predictability which identifies a linear rate. Our results suggest that this discrepancy may be explained by the fact that non-uniqueness might arise from perturbation around a singular flow.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"12 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-06DOI: 10.1088/1361-6544/ad68bb
Jeremy P Parker
We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for when σ = 10 and , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.
{"title":"The Lorenz system as a gradient-like system","authors":"Jeremy P Parker","doi":"10.1088/1361-6544/ad68bb","DOIUrl":"https://doi.org/10.1088/1361-6544/ad68bb","url":null,"abstract":"We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc do not exist. This condition is based upon the existence of an auxiliary function defined over the state space of the system, in a way analogous to a Lyapunov function for the stability of an equilibrium. For polynomial systems, Lyapunov functions can be found computationally by using sum-of-squares optimisation. We demonstrate this method by finding such an auxiliary function for the Lorenz system. We are able to show that the system is gradient-like for when σ = 10 and , significantly extending previous results. The results are rigorously validated by a novel procedure: First, an approximate numerical solution is found using finite-precision floating-point sum-of-squares optimisation. We then prove that there exists an exact solution close to this using interval arithmetic.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"38 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-06DOI: 10.1088/1361-6544/ad694a
Davide Sclosa
Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this case linearizing the system at the equilibrium leads to a Jacobian matrix which is zero in every entry. We call these equilibria completely degenerate. We prove that they exist for certain intrinsic frequencies if and only if the underlying graph is bipartite, and that they do not exist for generic intrinsic frequencies. In the case of zero intrinsic frequencies, we prove that they exist if and only if the graph has an Euler circuit such that the number of steps between any two visits at the same vertex is a multiple of 4. The simplest example is the cycle graph with 4 vertices. We prove that graphs with this property exist for every number of vertices and that they become asymptotically rare for N large. Regarding stability, we prove that for any choice of intrinsic frequencies, any coupling strength and any graph with at least one edge, completely degenerate equilibria are not Lyapunov stable. As a corollary, we obtain that stable equilibria in Kuramoto Networks must have at least one strictly negative eigenvalue.
仓本网络包含非双曲平衡点，其稳定性有时难以确定。我们考虑的是所有雅各布特征值均为零的极端情况。在这种情况下，平衡点处的系统线性化会导致雅各布矩阵的每个条目都为零。我们称这种平衡为完全退化平衡。我们将证明，当且仅当底层图是双方形时，对于某些固有频率，它们是存在的，而对于一般固有频率，它们是不存在的。在固有频率为零的情况下，我们证明只有当且仅当图形具有欧拉回路，使得在同一顶点的任意两次访问之间的步数是 4 的倍数时，它们才存在。最简单的例子是具有 4 个顶点的循环图。我们证明，具有这种性质的图对于任何顶点数都是存在的，而且当 N 较大时，这种图会逐渐变得稀少。关于稳定性，我们证明，对于任何固有频率、任何耦合强度和任何至少有一条边的图，完全退化的均衡都不是李雅普诺夫稳定的。作为推论，我们得出仓本网络中的稳定均衡必须至少有一个严格的负特征值。
{"title":"Completely degenerate equilibria of the Kuramoto model on networks","authors":"Davide Sclosa","doi":"10.1088/1361-6544/ad694a","DOIUrl":"https://doi.org/10.1088/1361-6544/ad694a","url":null,"abstract":"Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this case linearizing the system at the equilibrium leads to a Jacobian matrix which is zero in every entry. We call these equilibria completely degenerate. We prove that they exist for certain intrinsic frequencies if and only if the underlying graph is bipartite, and that they do not exist for generic intrinsic frequencies. In the case of zero intrinsic frequencies, we prove that they exist if and only if the graph has an Euler circuit such that the number of steps between any two visits at the same vertex is a multiple of 4. The simplest example is the cycle graph with 4 vertices. We prove that graphs with this property exist for every number of vertices and that they become asymptotically rare for N large. Regarding stability, we prove that for any choice of intrinsic frequencies, any coupling strength and any graph with at least one edge, completely degenerate equilibria are not Lyapunov stable. As a corollary, we obtain that stable equilibria in Kuramoto Networks must have at least one strictly negative eigenvalue.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"41 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1361-6544/ad673f
Rafael A Bilbao, Marlon Oliveira, Eduardo Santana
The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the w-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.
对于确定性系统而言，扩展性概念及其广义（度量扩展性、度量正扩展性、连续广义扩展性、可数度量扩展性）是众所周知的，对于研究重要的行为类型（如混沌行为）来说是一个有用的属性。本研究旨在将这些概念扩展到随机环境中，证明相对正熵与随机膨胀度量之间的关系，并应用它来证明如果一个随机动力学系统具有正的相对拓扑熵，那么 w 稳定类的条件度量为零。我们还证明了存在一种既不变又扩张的概率度量。此外，我们还得到了随机扩张度量概念与随机可数扩张系统概念之间的关系。
{"title":"Random expansive measures","authors":"Rafael A Bilbao, Marlon Oliveira, Eduardo Santana","doi":"10.1088/1361-6544/ad673f","DOIUrl":"https://doi.org/10.1088/1361-6544/ad673f","url":null,"abstract":"The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the <italic toggle=\"yes\">w</italic>-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"49 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1088/1361-6544/ad673e
Yingdu Dong, Xiong Li
In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.
{"title":"Response solutions for elliptic-hyperbolic equations with nonlinearities and periodic external forces","authors":"Yingdu Dong, Xiong Li","doi":"10.1088/1361-6544/ad673e","DOIUrl":"https://doi.org/10.1088/1361-6544/ad673e","url":null,"abstract":"In this paper, we focus on the existence of response solutions, i.e. periodic solutions with the same frequencies as the external forces, for elliptic-hyperbolic partial differential equations with nonlinearities and periodic forces. The main tools are Lyapunov–Schmidt reduction and Nash–Moser iteration scheme, both of which have demonstrated success in hyperbolic scenarios. At each step of the iteration, the Galerkin approximation of the equation is solved. The new issue is that the spectral theory of the generalized Sturm–Liouville problem is employed, which also introduces new difficulties for estimations at each step. Under appropriate non-resonance conditions on the frequency, the existence of response solutions for the model will be established.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":"22 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141870565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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