In this paper, we give an explicit nondegeneracy condition for the existence of Kolmogorov-Arnold-Moser (KAM) tori of an N-point vortex system on the plane by using the method of reduction via generalized Jacobi coordinates and matrix theory. Furthermore, by constructing a series of canonical transformations to reduce the degree of freedom of the Hamiltonian, we obtain a new simplified Hamiltonian system. Finally, we give the equivalent relationship between the relative equilibrium point of the original system and the equilibrium point of the new system.
{"title":"Explicit nondegeneracy conditions of KAM tori for the planar N-point vortex systems","authors":"Xuanqing Xiong, Qihuai Liu","doi":"10.1063/5.0138452","DOIUrl":"https://doi.org/10.1063/5.0138452","url":null,"abstract":"In this paper, we give an explicit nondegeneracy condition for the existence of Kolmogorov-Arnold-Moser (KAM) tori of an N-point vortex system on the plane by using the method of reduction via generalized Jacobi coordinates and matrix theory. Furthermore, by constructing a series of canonical transformations to reduce the degree of freedom of the Hamiltonian, we obtain a new simplified Hamiltonian system. Finally, we give the equivalent relationship between the relative equilibrium point of the original system and the equilibrium point of the new system.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83395313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The solutions to degenerate dynamical systems of the form A(x)ẋ=f(x) are studied by considering the equation as a differential inclusion. The set Z={det(A(x))=0}, called the singular set, is assumed to have an empty interior. The reasons leading us to the definition of the sets used for differential inclusion are exposed in detail. This definition is then applied on the one hand to generic cases and on the other hand to the particular cases resulting from physics, which can be found in Saavedra, Troncoso, and Zanelli [J. Math. Phys. 42, 4383 (2001)]. It is shown that generalized solutions may enter, leave, or remain in the singular locus.
{"title":"Generalized solutions to degenerate dynamical systems","authors":"P. Jouan, U. Serres","doi":"10.1063/5.0144432","DOIUrl":"https://doi.org/10.1063/5.0144432","url":null,"abstract":"The solutions to degenerate dynamical systems of the form A(x)ẋ=f(x) are studied by considering the equation as a differential inclusion. The set Z={det(A(x))=0}, called the singular set, is assumed to have an empty interior. The reasons leading us to the definition of the sets used for differential inclusion are exposed in detail. This definition is then applied on the one hand to generic cases and on the other hand to the particular cases resulting from physics, which can be found in Saavedra, Troncoso, and Zanelli [J. Math. Phys. 42, 4383 (2001)]. It is shown that generalized solutions may enter, leave, or remain in the singular locus.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79611467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space E3 in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [Peng et al., Phys. Rev. Lett. 114, 010601 (2015)], they were so far not addressed in the growing literature on superintegrability. Here, we venture into this field by searching for additional first-order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.
本文对量子力学中三维欧几里得空间E3上带磁场的超可积哈密顿系统的研究作出了贡献。在实验证实了复杂电磁场的物理相关性之后，数学界对复杂电磁场的兴趣日益浓厚[Peng et al.， Phys.]。Rev. Lett. 114, 010601(2015)]，到目前为止，它们还没有在越来越多的关于超可积性的文献中得到解决。在这里，我们通过寻找圆柱型可积系统的附加一阶运动积分来探索这一领域。我们发现已知的系统可以通过允许复杂耦合常数扩展到这个领域。除此之外，我们还发现了一个新的系统，它的运动积分也具有复常数。所有这些系统都是多重可分的。由于非厄米设置和失去规范不变性，对这些系统进行严格的数学分析是具有挑战性的。我们正式着手，并将这些问题的解决作为一项公开挑战。
{"title":"Cylindrical first-order superintegrability with complex magnetic fields","authors":"O. Kubů, L. Šnobl","doi":"10.1063/5.0138095","DOIUrl":"https://doi.org/10.1063/5.0138095","url":null,"abstract":"This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space E3 in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [Peng et al., Phys. Rev. Lett. 114, 010601 (2015)], they were so far not addressed in the growing literature on superintegrability. Here, we venture into this field by searching for additional first-order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88479403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the existence of normalized ground states for nonlinear fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities in R3. To overcome the special difficulties created by the nonlocal term and fractional Sobolev critical term, we develop a perturbed Pohožaev method based on the Brézis–Lieb lemma and monotonicity trick. Using the Pohožaev manifold decomposition and fibering map, we prove the existence of a positive normalized ground state. Moreover, the asymptotic behavior of the obtained normalized solutions is also explored. These conclusions extend some known ones in previous papers.
{"title":"Normalized ground states for fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities","authors":"L. Kong, Haibo Chen","doi":"10.1063/5.0098126","DOIUrl":"https://doi.org/10.1063/5.0098126","url":null,"abstract":"In this paper, we study the existence of normalized ground states for nonlinear fractional Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities in R3. To overcome the special difficulties created by the nonlocal term and fractional Sobolev critical term, we develop a perturbed Pohožaev method based on the Brézis–Lieb lemma and monotonicity trick. Using the Pohožaev manifold decomposition and fibering map, we prove the existence of a positive normalized ground state. Moreover, the asymptotic behavior of the obtained normalized solutions is also explored. These conclusions extend some known ones in previous papers.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"120 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90267594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies the existence and decay estimates of weak solutions to the surface growth equation. First, the global existence of weak solutions is obtained by the approximation method introduced by Majda and Bertozzi [Vorticity and Incompressible Flow (Cambridge University Press, 2001)]. Then, we derive the L2-decay rates of weak solutions via the Fourier splitting method under the assumption that u0∈L1(R)∩L2(R). For more general cases, i.e., u0∈L2(R), the behavior of weak solutions in L2 is obtained by the spectral theory of self-adjoint operators.
研究了表面生长方程弱解的存在性和衰减估计。首先，利用Majda和Bertozzi [Vorticity and Incompressible Flow (Cambridge University Press, 2001)]引入的近似方法，得到弱解的全局存在性。然后，在假设u0∈L1(R)∩L2(R)的前提下，通过傅里叶分裂方法导出弱解的L2衰减率。对于更一般的情况，即u0∈L2(R)，利用自伴随算子的谱理论得到了L2中弱解的性质。
{"title":"Large time behavior of weak solutions to the surface growth equation","authors":"Xuewen Wang, Chenggang Liu, Yanqing Wang, P. Han","doi":"10.1063/5.0136559","DOIUrl":"https://doi.org/10.1063/5.0136559","url":null,"abstract":"This paper studies the existence and decay estimates of weak solutions to the surface growth equation. First, the global existence of weak solutions is obtained by the approximation method introduced by Majda and Bertozzi [Vorticity and Incompressible Flow (Cambridge University Press, 2001)]. Then, we derive the L2-decay rates of weak solutions via the Fourier splitting method under the assumption that u0∈L1(R)∩L2(R). For more general cases, i.e., u0∈L2(R), the behavior of weak solutions in L2 is obtained by the spectral theory of self-adjoint operators.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"49 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74990965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we define a new type restricted quantum group Ūq(sl2*) and determine its Hopf Poincaré-Birkhoff-Witt-deformations Ūq(sl2*,κ) in which Ūq(sl2*,0)=Ūq(sl2*) and the classical restricted Drinfeld–Jimbo quantum group Ūq(sl2) is included. We show that Ūq(sl2*) is a basic Hopf algebra, then uniformly realize Ūq(sl2*) and Ūq(sl2) via some quotients of (deformed) preprojective algebras corresponding to the Gabriel quiver of Ūq(sl2*). Moreover, we obtain a uniform tensor-categorical realization of Ūq(sl2*) and Ūq(sl2), which is consistent with the above-mentioned Hopf-algebraic realization.
{"title":"A new type restricted quantum group","authors":"Yongjun Xu, Jialei Chen","doi":"10.1063/5.0142193","DOIUrl":"https://doi.org/10.1063/5.0142193","url":null,"abstract":"In this paper, we define a new type restricted quantum group Ūq(sl2*) and determine its Hopf Poincaré-Birkhoff-Witt-deformations Ūq(sl2*,κ) in which Ūq(sl2*,0)=Ūq(sl2*) and the classical restricted Drinfeld–Jimbo quantum group Ūq(sl2) is included. We show that Ūq(sl2*) is a basic Hopf algebra, then uniformly realize Ūq(sl2*) and Ūq(sl2) via some quotients of (deformed) preprojective algebras corresponding to the Gabriel quiver of Ūq(sl2*). Moreover, we obtain a uniform tensor-categorical realization of Ūq(sl2*) and Ūq(sl2), which is consistent with the above-mentioned Hopf-algebraic realization.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78501020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the possible emergence of traveling waves within an evolving interstellar gas cloud. To model this evolution, we use Euler–Poisson equations with the additional assumptions that the gas is incompressible, stratified, and self-gravitating. Within this framework, we establish that when the cloud has low density, the speed of these traveling waves is low. We suggest that the self-gravitational coalescence of embedded solid matter in the gas to form larger aggregates, such as cometary nuclei, may occur in the vicinity of wave crests where the mass density is highest. This idea is consistent with the widely agreed mechanism for planetary formation in proto-planetary disks, namely, that the accumulation of solids to form larger planetoids is initiated at the location of pressure maxima in the gas disk.
{"title":"Traveling waves in an evolving interstellar gas cloud","authors":"M. Humi","doi":"10.1063/5.0127453","DOIUrl":"https://doi.org/10.1063/5.0127453","url":null,"abstract":"This paper considers the possible emergence of traveling waves within an evolving interstellar gas cloud. To model this evolution, we use Euler–Poisson equations with the additional assumptions that the gas is incompressible, stratified, and self-gravitating. Within this framework, we establish that when the cloud has low density, the speed of these traveling waves is low. We suggest that the self-gravitational coalescence of embedded solid matter in the gas to form larger aggregates, such as cometary nuclei, may occur in the vicinity of wave crests where the mass density is highest. This idea is consistent with the widely agreed mechanism for planetary formation in proto-planetary disks, namely, that the accumulation of solids to form larger planetoids is initiated at the location of pressure maxima in the gas disk.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"08 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85978888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the two-fluid incompressible Navier–Stokes–Maxwell system in three dimensional space. The analysis shows that the effect of the Lorentz force induced by the electromagnetic field leads to some different structures of the spectrum. Moreover, the detailed analysis of the Green’s function to the linearized system is made with applications to derive the optimal time decay rate of the solution converging to the steady state.
{"title":"The two-fluid incompressible Navier–Stokes–Maxwell system: Green’s function and optimal decay rate","authors":"G. Wang, Mingying Zhong","doi":"10.1063/5.0132274","DOIUrl":"https://doi.org/10.1063/5.0132274","url":null,"abstract":"In this paper, we consider the two-fluid incompressible Navier–Stokes–Maxwell system in three dimensional space. The analysis shows that the effect of the Lorentz force induced by the electromagnetic field leads to some different structures of the spectrum. Moreover, the detailed analysis of the Green’s function to the linearized system is made with applications to derive the optimal time decay rate of the solution converging to the steady state.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80770320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the nonlinear stability of contact waves for the Cauchy problem to the compressible Navier–Stokes equations for a reacting mixture in one dimension. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the convergence rate by using anti-derivative methods and elaborated energy estimates.
{"title":"Decay rate to contact discontinuities for the one-dimensional compressible Navier–Stokes equations with a reacting mixture","authors":"Lishuang Peng, Yong Li","doi":"10.1063/5.0104769","DOIUrl":"https://doi.org/10.1063/5.0104769","url":null,"abstract":"In this paper, we investigate the nonlinear stability of contact waves for the Cauchy problem to the compressible Navier–Stokes equations for a reacting mixture in one dimension. If the corresponding Riemann problem for the compressible Euler system admits a contact discontinuity solution, it is shown that the contact wave is nonlinearly stable, while the strength of the contact discontinuity and the initial perturbation are suitably small. Especially, we obtain the convergence rate by using anti-derivative methods and elaborated energy estimates.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"30 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88409265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A full compressible Navier–Stokes–Poisson system models the motion of viscous ions under the effect of variable temperature and plays important roles in the study of self-gravitational viscous gaseous stars and in simulations of charged particles in semiconductor devices and plasmas physics. We establish the time-asymptotic nonlinear stability of a planar rarefaction wave to the initial value problem of a three-dimensional full compressible Navier–Stokes–Poisson equation when the initial data are a small perturbation of the planar rarefaction wave. The proof is given by a delicate energy method, which involves highly non-trivial a priori bounds due to the effect of the self-consistent electric field. This appears as the first result on the nonlinear stability of wave patterns to the full compressible Navier–Stokes–Poisson system in multi-dimensions.
{"title":"Stability of the planar rarefaction wave to three-dimensional full\u0000 compressible Navier–Stokes–Poisson system","authors":"Yeping Li, Yujuan Chen, Zhengzheng Chen","doi":"10.1063/5.0137502","DOIUrl":"https://doi.org/10.1063/5.0137502","url":null,"abstract":"A full compressible Navier–Stokes–Poisson system models the motion of viscous ions under the effect of variable temperature and plays important roles in the study of self-gravitational viscous gaseous stars and in simulations of charged particles in semiconductor devices and plasmas physics. We establish the time-asymptotic nonlinear stability of a planar rarefaction wave to the initial value problem of a three-dimensional full compressible Navier–Stokes–Poisson equation when the initial data are a small perturbation of the planar rarefaction wave. The proof is given by a delicate energy method, which involves highly non-trivial a priori bounds due to the effect of the self-consistent electric field. This appears as the first result on the nonlinear stability of wave patterns to the full compressible Navier–Stokes–Poisson system in multi-dimensions.","PeriodicalId":50141,"journal":{"name":"Journal of Mathematical Physics Analysis Geometry","volume":"67 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76971956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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