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Keywords = regularized trace formulae

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7 pages, 252 KB  
Article
Trace Formulae for Second-Order Differential Pencils with a Frozen Argument
by Yi-Teng Hu and Murat Şat
Mathematics 2023, 11(18), 3996; https://doi.org/10.3390/math11183996 - 20 Sep 2023
Cited by 3 | Viewed by 1267
Abstract
This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, [...] Read more.
This paper deals with second-order differential pencils with a fixed frozen argument on a finite interval. We obtain the trace formulae under four boundary conditions: Dirichlet–Dirichlet, Neumann–Neumann, Dirichlet–Neumann, Neumann–Dirichlet. Although the boundary conditions and the corresponding asymptotic behaviour of the eigenvalues are different, the trace formulae have the same form which reveals the impact of the frozen argument. Full article
(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)
22 pages, 759 KB  
Article
The Multicomponent Higher-Order Chen–Lee–Liu System: The Riemann–Hilbert Problem and Its N-Soliton Solution
by Yong Zhang, Huanhe Dong and Yong Fang
Fractal Fract. 2022, 6(6), 327; https://doi.org/10.3390/fractalfract6060327 - 13 Jun 2022
Cited by 2 | Viewed by 2582
Abstract
It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. [...] Read more.
It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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17 pages, 296 KB  
Article
A Second Regularized Trace Formula for a Fourth Order Differential Operator
by Erdal Gül and Aylan Ceyhan
Symmetry 2021, 13(4), 629; https://doi.org/10.3390/sym13040629 - 9 Apr 2021
Cited by 9 | Viewed by 1982
Abstract
In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for [...] Read more.
In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator. Full article
(This article belongs to the Section Mathematics)
24 pages, 32575 KB  
Article
Variation Characteristics and Transportation of Aerosol, NO2, SO2, and HCHO in Coastal Cities of Eastern China: Dalian, Qingdao, and Shanghai
by Xiaomei Li, Pinhua Xie, Ang Li, Jin Xu, Zhaokun Hu, Hongmei Ren, Hongyan Zhong, Bo Ren, Xin Tian, Yeyuan Huang, Wenxuan Chai, Shuai Wang and Qingbo Li
Remote Sens. 2021, 13(5), 892; https://doi.org/10.3390/rs13050892 - 27 Feb 2021
Cited by 8 | Viewed by 2983
Abstract
This paper studied the method for converting the aerosol extinction to the mass concentration of particulate matter (PM) and obtained the spatio-temporal distribution and transportation of aerosol, nitrogen dioxide (NO2), sulfur dioxide (SO2), and formaldehyde (HCHO) based on multi-axis [...] Read more.
This paper studied the method for converting the aerosol extinction to the mass concentration of particulate matter (PM) and obtained the spatio-temporal distribution and transportation of aerosol, nitrogen dioxide (NO2), sulfur dioxide (SO2), and formaldehyde (HCHO) based on multi-axis differential optical absorption spectroscopy (MAX-DOAS) observations in Dalian (38.85°N, 121.36°E), Qingdao (36.35°N, 120.69°E), and Shanghai (31.60°N, 121.80°E) from 2019 to 2020. The PM2.5 measured by the in situ instrument and the PM2.5 simulated by the conversion formula showed a good correlation. The correlation coefficients R were 0.93 (Dalian), 0.90 (Qingdao), and 0.88 (Shanghai). A regular seasonality of the three trace gases is found, but not for aerosols. Considerable amplitudes in the weekly cycles were determined for NO2 and aerosols, but not for SO2 and HCHO. The aerosol profiles were nearly Gaussian, and the shapes of the trace gas profiles were nearly exponential, except for SO2 in Shanghai and HCHO in Qingdao. PM2.5 presented the largest transport flux, followed by NO2 and SO2. The main transport flux was the output flux from inland to sea in spring and winter. The MAX-DOAS and the Copernicus Atmosphere Monitoring Service (CAMS) models’ results were compared. The overestimation of NO2 and SO2 by CAMS is due to its overestimation of near-surface gas volume mixing ratios. Full article
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