In this paper, we consider the fully parabolic nu’trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights.
{"title":"Global solvability and asymptotic behavior of solutions for a fully parabolic nutrient taxis system","authors":"Hanqi Huang, Guoqiang Ren, Xing Zhou","doi":"10.1063/5.0212819","DOIUrl":"https://doi.org/10.1063/5.0212819","url":null,"abstract":"In this paper, we consider the fully parabolic nu’trient taxis system: ut = d1Δu − ∇ · (ϕ(u, v)∇v), vt = d2Δv − ξug(v) − μv + r(x, t), x ∈ Ω, t > 0 under homogeneous Neumann boundary conditions in a convex bounded domain with smooth boundary. We show that the system possesses a global bounded classical solution in domains of arbitrary dimension and at least one global generalized solution in high-dimensional domain. In addition, the asymptotic behavior of generalized solutions is discussed. Our results not only generalize and partly improve upon previously known findings but also introduce new insights.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"19 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yongshuai Zhang, Deqin Qiu, Shoufeng Shen, Jingsong He
With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.
{"title":"The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles","authors":"Yongshuai Zhang, Deqin Qiu, Shoufeng Shen, Jingsong He","doi":"10.1063/5.0205072","DOIUrl":"https://doi.org/10.1063/5.0205072","url":null,"abstract":"With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"15 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the following nonlinear Schrödinger equations with critical growth: −Δu+V(|y|)u=uN+2N−2,u>0inRN, where V(|y|) is a bounded positive radial function in C1, N ≥ 5. By using a finite reduction argument, we show that if r2V(r) has either an isolated local maximum or an isolated local minimum at r0 > 0 with V(r0) > 0, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of O(3).
{"title":"New type of solutions for Schrödinger equations with critical growth","authors":"Yuan Gao, Yuxia Guo","doi":"10.1063/5.0206967","DOIUrl":"https://doi.org/10.1063/5.0206967","url":null,"abstract":"We consider the following nonlinear Schrödinger equations with critical growth: −Δu+V(|y|)u=uN+2N−2,u>0inRN, where V(|y|) is a bounded positive radial function in C1, N ≥ 5. By using a finite reduction argument, we show that if r2V(r) has either an isolated local maximum or an isolated local minimum at r0 > 0 with V(r0) > 0, there exists infinitely many non-radial large energy solutions which are invariant under some sub-groups of O(3).","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"127 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper the connection between stochastic optimal control and reinforcement learning is investigated. Our main motivation is to apply importance sampling to sampling rare events which can be reformulated as an optimal control problem. By using a parameterised approach the optimal control problem becomes a stochastic optimization problem which still raises some open questions regarding how to tackle the scalability to high-dimensional problems and how to deal with the intrinsic metastability of the system. To explore new methods we link the optimal control problem to reinforcement learning since both share the same underlying framework, namely a Markov Decision Process (MDP). For the optimal control problem we show how the MDP can be formulated. In addition we discuss how the stochastic optimal control problem can be interpreted in the framework of reinforcement learning. At the end of the article we present the application of two different reinforcement learning algorithms to the optimal control problem and a comparison of the advantages and disadvantages of the two algorithms.
{"title":"Connecting stochastic optimal control and reinforcement learning","authors":"J. Quer, Enric Ribera Borrell","doi":"10.1063/5.0140665","DOIUrl":"https://doi.org/10.1063/5.0140665","url":null,"abstract":"In this paper the connection between stochastic optimal control and reinforcement learning is investigated. Our main motivation is to apply importance sampling to sampling rare events which can be reformulated as an optimal control problem. By using a parameterised approach the optimal control problem becomes a stochastic optimization problem which still raises some open questions regarding how to tackle the scalability to high-dimensional problems and how to deal with the intrinsic metastability of the system. To explore new methods we link the optimal control problem to reinforcement learning since both share the same underlying framework, namely a Markov Decision Process (MDP). For the optimal control problem we show how the MDP can be formulated. In addition we discuss how the stochastic optimal control problem can be interpreted in the framework of reinforcement learning. At the end of the article we present the application of two different reinforcement learning algorithms to the optimal control problem and a comparison of the advantages and disadvantages of the two algorithms.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerned a fractional Schrödinger equation in whole space, for ɛ > 0 is a small parameter, ɛ2s(−Δ)su + V(x)u = |u|p−2u, where 12<s<1, N > 1 and 2<p<2NN−2s. We prove the non-degeneracy and uniqueness of bubble solutions by using local Pohozaev identity and finite dimensional reduction, which are the cornerstones to construct different type solutions.
{"title":"Uniqueness and non-degeneracy of solutions for nonlinear fractional Schrödinger equation with perturbation","authors":"Yuanda Wu, Yimin Zhang","doi":"10.1063/5.0208876","DOIUrl":"https://doi.org/10.1063/5.0208876","url":null,"abstract":"This paper concerned a fractional Schrödinger equation in whole space, for ɛ &gt; 0 is a small parameter, ɛ2s(−Δ)su + V(x)u = |u|p−2u, where 12&lt;s&lt;1, N &gt; 1 and 2&lt;p&lt;2NN−2s. We prove the non-degeneracy and uniqueness of bubble solutions by using local Pohozaev identity and finite dimensional reduction, which are the cornerstones to construct different type solutions.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"73 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the Cauchy problem for a highly nonlinear shallow water model arising from the full water waves with Coriolis effect. The existence of weak solutions to the equation in the lower order Sobolev space Hs(R) with 1<s≤32 is presented. Moreover, the local well-posedness of strong solutions in Sobolev space Hs(R) with s>32 is established by the pseudoparabolic regularization technique.
{"title":"The existence and uniqueness of weak solutions for a highly nonlinear shallow-water model with Coriolis effect","authors":"Shouming Zhou, Jie Xu","doi":"10.1063/5.0201600","DOIUrl":"https://doi.org/10.1063/5.0201600","url":null,"abstract":"In this paper, we consider the Cauchy problem for a highly nonlinear shallow water model arising from the full water waves with Coriolis effect. The existence of weak solutions to the equation in the lower order Sobolev space Hs(R) with 1&lt;s≤32 is presented. Moreover, the local well-posedness of strong solutions in Sobolev space Hs(R) with s&gt;32 is established by the pseudoparabolic regularization technique.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"25 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The method of nonlinearization of the Lax pair is developed for the Ablowitz-Kaup-Newell-Segur (AKNS) equation in the presence of space-inverse reductions. As a result, we obtain a new type of finite-dimensional Hamiltonian systems: they are nonlocal in the sense that the inverse of the space variable is involved. For such nonlocal Hamiltonian systems, we show that they preserve the Liouville integrability and they can be linearized on the Jacobi variety. We also show how to construct the algebro-geometric solutions to the AKNS equation with space-inverse reductions by virtue of our nonlocal finite-dimensional Hamiltonian systems. As an application, algebro-geometric solutions to the AKNS equation with the Dirichlet and with the Neumann boundary conditions, and algebro-geometric solutions to the nonlocal nonlinear Schrödinger (NLS) equation are obtained. nonlocal finite-dimensional integrable Hamiltonian system, algebro-geometric solution, Dirichlet (Neumann) boundary, nonlocal NLS equation.
{"title":"Integrable nonlocal finite-dimensional Hamiltonian systems related to the Ablowitz-Kaup-Newell-Segur system","authors":"Baoqiang Xia, Ruguang Zhou","doi":"10.1063/5.0200162","DOIUrl":"https://doi.org/10.1063/5.0200162","url":null,"abstract":"The method of nonlinearization of the Lax pair is developed for the Ablowitz-Kaup-Newell-Segur (AKNS) equation in the presence of space-inverse reductions. As a result, we obtain a new type of finite-dimensional Hamiltonian systems: they are nonlocal in the sense that the inverse of the space variable is involved. For such nonlocal Hamiltonian systems, we show that they preserve the Liouville integrability and they can be linearized on the Jacobi variety. We also show how to construct the algebro-geometric solutions to the AKNS equation with space-inverse reductions by virtue of our nonlocal finite-dimensional Hamiltonian systems. As an application, algebro-geometric solutions to the AKNS equation with the Dirichlet and with the Neumann boundary conditions, and algebro-geometric solutions to the nonlocal nonlinear Schrödinger (NLS) equation are obtained. nonlocal finite-dimensional integrable Hamiltonian system, algebro-geometric solution, Dirichlet (Neumann) boundary, nonlocal NLS equation.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"31 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compare two approaches which use K-theory for C*-algebras to classify symmetry protected topological phases of quantum systems described in the one particle approximation. In the approach by Kellendonk, which is more abstract and more general, the algebra remains unspecified and the symmetries are defined using gradings and real structures. In the approach by Alldridge et al., the algebra is physically motivated and the symmetries implemented by generators which commute with the Hamiltonian. Both approaches use van Daele’s version of K-theory. We show that the second approach is a special case of the first one. We highlight the role played by two of the symmetries: charge conservation and spin rotation symmetry.
我们比较了两种方法，这两种方法使用 C* 矩阵的 K 理论来对单粒子近似描述的量子系统的对称性保护拓扑相进行分类。Kellendonk 的方法更抽象、更通用，其代数仍未指定，对称性是用等级和实结构定义的。在 Alldridge 等人的方法中，代数是以物理为动机的，对称性是通过与哈密顿换算的生成器来实现的。这两种方法都使用了 van Daele 版本的 K 理论。我们证明第二种方法是第一种方法的特例。我们强调了其中两个对称性的作用：电荷守恒和自旋旋转对称。
{"title":"Comparison between two approaches to classify topological insulators using K-theory","authors":"Lorenzo Scaglione","doi":"10.1063/5.0197127","DOIUrl":"https://doi.org/10.1063/5.0197127","url":null,"abstract":"We compare two approaches which use K-theory for C*-algebras to classify symmetry protected topological phases of quantum systems described in the one particle approximation. In the approach by Kellendonk, which is more abstract and more general, the algebra remains unspecified and the symmetries are defined using gradings and real structures. In the approach by Alldridge et al., the algebra is physically motivated and the symmetries implemented by generators which commute with the Hamiltonian. Both approaches use van Daele’s version of K-theory. We show that the second approach is a special case of the first one. We highlight the role played by two of the symmetries: charge conservation and spin rotation symmetry.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"175 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ < 0 and μ > 0.
{"title":"Information theoretic measures in one-dimensional Dunkl oscillator","authors":"Debraj Nath, Niladri Ghosh, Amlan K. Roy","doi":"10.1063/5.0200405","DOIUrl":"https://doi.org/10.1063/5.0200405","url":null,"abstract":"We consider the solution of one dimensional Schrödinger Dunkl equation for energies and eigenfunctions. Then we provide analytical expressions for various information theoretic measures. For a given density function, quantities such as position expectation value, entropic moment, disequilibrium, Rényi entropy, Shannon entropy, Tsallis entropy, Fisher information are presented. Next, a few relative information measures corresponding to two density functions, like relative entropy, relative Fisher, relative Rényi, relative Tsallis, along with their associated Jensen divergences such as Jensen–Shannon divergence, Jensen–Fisher divergence, Jensen–Rényi divergence, Jensen–Tsallis divergence are treated. Sample results are provided in graphical form. Dependence of these quantities on the Dunkl parameter μ shows distinct features for μ &lt; 0 and μ &gt; 0.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"57 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study integral forms of the tensor products of Virasoro vertex operator algebras ⊗i∈IL(cpi,qi,0)⊗ni and their modules. On the one hand, with the tool of binary linear codes, we show the existence and give constructions of integral forms of the tensor product vertex operator algebras that contain the conformal vector. Several interesting and explicit examples of codes are presented. On the other hand, we investigate the integral form theory of modules over the vertex operator algebra ⊗i∈IL(cpi,qi,0)⊗ni. More precisely, we construct integral forms of modules and contragredient modules for these tensor product vertex operator algebras. The integrality of intertwining operators among integral forms of ⊗i∈IL(cpi,qi,0)⊗ni-modules are also obtained.
{"title":"Integral forms for tensor products of Virasoro vertex operator algebras and their modules","authors":"Hongyan Guo, Hongju Zhao","doi":"10.1063/5.0195338","DOIUrl":"https://doi.org/10.1063/5.0195338","url":null,"abstract":"In this paper, we study integral forms of the tensor products of Virasoro vertex operator algebras ⊗i∈IL(cpi,qi,0)⊗ni and their modules. On the one hand, with the tool of binary linear codes, we show the existence and give constructions of integral forms of the tensor product vertex operator algebras that contain the conformal vector. Several interesting and explicit examples of codes are presented. On the other hand, we investigate the integral form theory of modules over the vertex operator algebra ⊗i∈IL(cpi,qi,0)⊗ni. More precisely, we construct integral forms of modules and contragredient modules for these tensor product vertex operator algebras. The integrality of intertwining operators among integral forms of ⊗i∈IL(cpi,qi,0)⊗ni-modules are also obtained.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"7 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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