Elastic collisions of solitons generally have a finite phase shift. When the phase shift has a finitely large value, the two vertices of the (2 + 1)-dimensional two-soliton are significantly separated due to the phase shift, accompanied by the formation of a local structure connecting the two V-shaped solitons. We define this local structure as the stem structure. This study systematically investigates the localized stem structures between two solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. These stem structures, arising from quasi-resonant collisions between the solitons, exhibit distinct features of spatial locality and temporal invariance. We explore two scenarios: one characterized by weakly quasi-resonant collisions (i.e. a12 ≈ 0), and the other by strongly quasi-resonant collisions (i.e. a12 ≈ +∞). Through mathematical analysis, we extract comprehensive insights into the trajectories, amplitudes, and velocities of the soliton arms. Furthermore, we discuss the characteristics of the stem structures, including their length and extreme points. Our findings shed new light on the interaction between solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system.
{"title":"Localized stem structures in quasi-resonant two-soliton solutions for the asymmetric Nizhnik–Novikov–Veselov system","authors":"Feng Yuan, Jiguang Rao, Jingsong He, Yi Cheng","doi":"10.1063/5.0218541","DOIUrl":"https://doi.org/10.1063/5.0218541","url":null,"abstract":"Elastic collisions of solitons generally have a finite phase shift. When the phase shift has a finitely large value, the two vertices of the (2 + 1)-dimensional two-soliton are significantly separated due to the phase shift, accompanied by the formation of a local structure connecting the two V-shaped solitons. We define this local structure as the stem structure. This study systematically investigates the localized stem structures between two solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system. These stem structures, arising from quasi-resonant collisions between the solitons, exhibit distinct features of spatial locality and temporal invariance. We explore two scenarios: one characterized by weakly quasi-resonant collisions (i.e. a12 ≈ 0), and the other by strongly quasi-resonant collisions (i.e. a12 ≈ +∞). Through mathematical analysis, we extract comprehensive insights into the trajectories, amplitudes, and velocities of the soliton arms. Furthermore, we discuss the characteristics of the stem structures, including their length and extreme points. Our findings shed new light on the interaction between solitons in the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov system.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"19 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207539","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, a compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The purpose of this study is twofold. First, it is shown that the system is locally well-posed. For that purpose, the existence of classical solutions which are perturbation of constant states is established. Second, it is proved that in the particular case of subsonic equilibrium states, sufficiently small perturbations decay globally in time. In order to prove this stability property, the linearized system around the subsonic state is examined. Using an appropriately constructed compensating matrix symbol in the Fourier space, it is proved that solutions to the linear system decay globally in time, underlying a dissipative mechanism of regularity gain type. These linear decay estimates, together with the local existence result, imply the global existence and the decay of perturbations to constant subsonic equilibrium states as solutions to the full nonlinear system.
{"title":"Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity","authors":"Ramón G. Plaza, Delyan Zhelyazov","doi":"10.1063/5.0172774","DOIUrl":"https://doi.org/10.1063/5.0172774","url":null,"abstract":"In this paper, a compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The purpose of this study is twofold. First, it is shown that the system is locally well-posed. For that purpose, the existence of classical solutions which are perturbation of constant states is established. Second, it is proved that in the particular case of subsonic equilibrium states, sufficiently small perturbations decay globally in time. In order to prove this stability property, the linearized system around the subsonic state is examined. Using an appropriately constructed compensating matrix symbol in the Fourier space, it is proved that solutions to the linear system decay globally in time, underlying a dissipative mechanism of regularity gain type. These linear decay estimates, together with the local existence result, imply the global existence and the decay of perturbations to constant subsonic equilibrium states as solutions to the full nonlinear system.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"62 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207545","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}
As an extension to the paper by Breuer et al., Ann. Henri Poincare 22, 3763 (2021), we study the linear statistics for the eigenvalues of the Schrödinger operator with random decaying potential with order O(x−α) (α > 0) at infinity. We first prove similar statements as in Breuer et al., Ann. Henri Poincare 22, 3763 (2021) for the trace of f(H), where f belongs to a class of analytic functions: there exists a critical exponent αc such that the fluctuation of the trace of f(H) converges in probability for α > αc, and satisfies a central limit theorem statement for α ≤ αc, where αc differs depending on f. Furthermore we study the asymptotic behavior of its expectation value.
{"title":"Eigenvalue fluctuations of 1-dimensional random Schrödinger operators","authors":"Takuto Mashiko, Yuma Marui, Naoki Maruyama, Fumihiko Nakano","doi":"10.1063/5.0125197","DOIUrl":"https://doi.org/10.1063/5.0125197","url":null,"abstract":"As an extension to the paper by Breuer et al., Ann. Henri Poincare 22, 3763 (2021), we study the linear statistics for the eigenvalues of the Schrödinger operator with random decaying potential with order O(x−α) (α > 0) at infinity. We first prove similar statements as in Breuer et al., Ann. Henri Poincare 22, 3763 (2021) for the trace of f(H), where f belongs to a class of analytic functions: there exists a critical exponent αc such that the fluctuation of the trace of f(H) converges in probability for α > αc, and satisfies a central limit theorem statement for α ≤ αc, where αc differs depending on f. Furthermore we study the asymptotic behavior of its expectation value.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"22 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207549","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 develop a method for the random sampling of (multimode) Gaussian states in terms of their covariance matrix, which we refer to as a random quantum covariance matrix (RQCM). We analyze the distribution of marginals and demonstrate that the eigenvalues of an RQCM converge to a shifted semicircular distribution in the limit of a large number of modes. We provide insights into the entanglement of such states based on the positive partial transpose criteria. Additionally, we show that the symplectic eigenvalues of an RQCM converge to a probability distribution that can be characterized using free probability. We present numerical estimates for the probability of a RQCM being separable and, if not, its extendibility degree, for various parameter values and mode bipartitions.
{"title":"Generating random Gaussian states","authors":"Leevi Leppäjärvi, Ion Nechita, Ritabrata Sengupta","doi":"10.1063/5.0202147","DOIUrl":"https://doi.org/10.1063/5.0202147","url":null,"abstract":"We develop a method for the random sampling of (multimode) Gaussian states in terms of their covariance matrix, which we refer to as a random quantum covariance matrix (RQCM). We analyze the distribution of marginals and demonstrate that the eigenvalues of an RQCM converge to a shifted semicircular distribution in the limit of a large number of modes. We provide insights into the entanglement of such states based on the positive partial transpose criteria. Additionally, we show that the symplectic eigenvalues of an RQCM converge to a probability distribution that can be characterized using free probability. We present numerical estimates for the probability of a RQCM being separable and, if not, its extendibility degree, for various parameter values and mode bipartitions.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"72 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207334","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 near superposition of squeezed vacuum states (NSVS) is investigated in this article. The state appears to be a superposition of a squeezed vacuum state (SVS) and a derivative-squeezed vacuum state. We have shown that NSVS is significantly different from any regular superposition of two SVSs. NSVS, like SVS, displays only even photons, but with different distributions. In some cases, NSVS has no vacuum state. NSVS displays sub-Poissonian statistics for small values of the squeezing parameter. NSVS reveals linear and amplitude-squared squeezing, with amplitude-squared squeezing surpassing SVS in most cases. The minimum uncertainty is explored, and a possible method for generating NSVS is explained. We have discovered that NSVS exhibits a similar behavior for all phase differences except when it equals precisely zero. This phenomenon has been identified and could potentially enable more sensitive measurements.
{"title":"Properties of near superposition of two squeezed vacuum states","authors":"Anas Othman","doi":"10.1063/5.0186279","DOIUrl":"https://doi.org/10.1063/5.0186279","url":null,"abstract":"The near superposition of squeezed vacuum states (NSVS) is investigated in this article. The state appears to be a superposition of a squeezed vacuum state (SVS) and a derivative-squeezed vacuum state. We have shown that NSVS is significantly different from any regular superposition of two SVSs. NSVS, like SVS, displays only even photons, but with different distributions. In some cases, NSVS has no vacuum state. NSVS displays sub-Poissonian statistics for small values of the squeezing parameter. NSVS reveals linear and amplitude-squared squeezing, with amplitude-squared squeezing surpassing SVS in most cases. The minimum uncertainty is explored, and a possible method for generating NSVS is explained. We have discovered that NSVS exhibits a similar behavior for all phase differences except when it equals precisely zero. This phenomenon has been identified and could potentially enable more sensitive measurements.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"16 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207548","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 a generalized Dullin–Gottwald–Holm equation. The equation admits single peakons and multi-peakons. Using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, we prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space.
在本文中，我们考虑了一个广义的 Dullin-Gottwald-Holm 方程。该方程包含单峰子和多峰子。利用能量论证并结合单峰子轨道稳定性与局部能量规范单调性的方法，我们证明了 N 个充分解耦的峰子之和在能量空间中是轨道稳定的。
{"title":"Orbital stability of multi-peakons for a generalized Dullin–Gottwald–Holm equation","authors":"Jiajing Wang, Tongjie Deng, Kelei Zhang","doi":"10.1063/5.0164490","DOIUrl":"https://doi.org/10.1063/5.0164490","url":null,"abstract":"In this paper, we consider a generalized Dullin–Gottwald–Holm equation. The equation admits single peakons and multi-peakons. Using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, we prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207551","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}
A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.
{"title":"Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom","authors":"E. M. Novikova","doi":"10.1063/5.0172309","DOIUrl":"https://doi.org/10.1063/5.0172309","url":null,"abstract":"A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"201 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207333","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 prove the existence of global in time small data solutions of semilinear Klein–Gordon equations in space-time with a static Schwarzschild radius in the expanding universe.
{"title":"Waves in cosmological background with static Schwarzschild radius in the expanding universe","authors":"Karen Yagdjian","doi":"10.1063/5.0166195","DOIUrl":"https://doi.org/10.1063/5.0166195","url":null,"abstract":"In this paper, we prove the existence of global in time small data solutions of semilinear Klein–Gordon equations in space-time with a static Schwarzschild radius in the expanding universe.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"4 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207550","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 a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder [J. Math. Phys. 36(8), 3902–3921 (1995)], the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines the interactions of a multi-dimensional system.
{"title":"Inverse scattering for repulsive potential and strong singular interactions","authors":"Atsuhide Ishida","doi":"10.1063/5.0215713","DOIUrl":"https://doi.org/10.1063/5.0215713","url":null,"abstract":"In a previous work of 2014 on a quantum system governed by the repulsive Hamiltonian, the author proved uniqueness for short-range interactions described by a scattering operator consisting of regular and singular parts. In this paper, the singular part is assumed to have much stronger singularities and the same uniqueness theorem is proved. By applying the time-dependent method invented by Enss and Weder [J. Math. Phys. 36(8), 3902–3921 (1995)], the high-velocity limit for a wider class of the scattering operator with stronger singularities also uniquely determines the interactions of a multi-dimensional system.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142207335","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}
Effect of cold plasma on the form of rays propagating in the equatorial plane of a rotating black hole is investigated. Two kinds of regions in the radius–impact parameter plane allowed for the rays are constructed: for radiation with a given frequency at infinity and for radiation with a given “telescope frequency” seen by a local observer. The form of allowed regions for locally nonrotating observers as well as observers falling freely from infinity is established. The allowed regions contain rays which directly reach the horizon, or there exists a “neck” connecting the forbidden regions such that the rays coming from infinity cannot reach the horizon. In case we considered a set of observers at various radii instead of the neck we find two different regions – from one the rays reach the horizon and not infinity and from the other one they reach infinity, but not the horizon. The results are analyzed by analytical methods and illustrated by figures constructed numerically.
{"title":"Radiation in the black hole–plasma system: Propagation in equatorial plane","authors":"Vladimír Balek, Barbora Bezděková, Jiří Bičák","doi":"10.1063/5.0200901","DOIUrl":"https://doi.org/10.1063/5.0200901","url":null,"abstract":"Effect of cold plasma on the form of rays propagating in the equatorial plane of a rotating black hole is investigated. Two kinds of regions in the radius–impact parameter plane allowed for the rays are constructed: for radiation with a given frequency at infinity and for radiation with a given “telescope frequency” seen by a local observer. The form of allowed regions for locally nonrotating observers as well as observers falling freely from infinity is established. The allowed regions contain rays which directly reach the horizon, or there exists a “neck” connecting the forbidden regions such that the rays coming from infinity cannot reach the horizon. In case we considered a set of observers at various radii instead of the neck we find two different regions – from one the rays reach the horizon and not infinity and from the other one they reach infinity, but not the horizon. The results are analyzed by analytical methods and illustrated by figures constructed numerically.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"33 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141948659","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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