Pub Date : 2024-11-21DOI: 10.1016/j.physd.2024.134431
Wenxuan Guo , Qiang Zhang
It is well known that, influenced only by gravity, the fluid interface is unstable when a light fluid supports a heavy fluid and is stable when a heavy fluid supports a light fluid. The situation becomes much more complicated when a vertical electric field is externally applied to the dielectric fluids. We present a nonlinear perturbation solution for an unstable interface between two incompressible, inviscid, immiscible, and perfectly dielectric fluids in the presence of vertical electric fields and gravity in two dimensions. Our nonlinear stability analysis shows that even when the linear theory indicates that the interface is stable, this system is actually unstable. The destabilization effects of the vertical electric field always dominate when gravity provides stabilization effects. This is true even when the applied vertical electric field is very weak. Analytical expressions for the overall amplitude and velocity of the interface are derived up to an arbitrary order in terms of the initial perturbation amplitude and are displayed explicitly up to the fourth order. A comparison study between the predictions of the nonlinear perturbation solution and the numerical results shows that the derived solutions capture the primary nonlinear behavior of the unstable fluid interface. By analyzing the electrical force at the interface, we provide theoretical explanations for the nonlinear phenomena induced by the vertical electric field.
{"title":"Understanding the nonlinear behavior of Rayleigh–Taylor instability with a vertical electric field for perfect dielectric fluids","authors":"Wenxuan Guo , Qiang Zhang","doi":"10.1016/j.physd.2024.134431","DOIUrl":"10.1016/j.physd.2024.134431","url":null,"abstract":"<div><div>It is well known that, influenced only by gravity, the fluid interface is unstable when a light fluid supports a heavy fluid and is stable when a heavy fluid supports a light fluid. The situation becomes much more complicated when a vertical electric field is externally applied to the dielectric fluids. We present a nonlinear perturbation solution for an unstable interface between two incompressible, inviscid, immiscible, and perfectly dielectric fluids in the presence of vertical electric fields and gravity in two dimensions. Our nonlinear stability analysis shows that even when the linear theory indicates that the interface is stable, this system is actually unstable. The destabilization effects of the vertical electric field always dominate when gravity provides stabilization effects. This is true even when the applied vertical electric field is very weak. Analytical expressions for the overall amplitude and velocity of the interface are derived up to an arbitrary order in terms of the initial perturbation amplitude and are displayed explicitly up to the fourth order. A comparison study between the predictions of the nonlinear perturbation solution and the numerical results shows that the derived solutions capture the primary nonlinear behavior of the unstable fluid interface. By analyzing the electrical force at the interface, we provide theoretical explanations for the nonlinear phenomena induced by the vertical electric field.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134431"},"PeriodicalIF":2.7,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.physd.2024.134435
Peter J. Forrester , Nicholas S. Witte
The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the members of an eigen-sequence. When this eigen-sequence has rotational invariance, as for the eigen-angles of Dyson’s circular ensembles, recent work of Riser and Kanzieper has uncovered an exact identity expressing the power spectrum in terms of the generating function for the conditioned gap probability of having eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlevé VI system, imply a computational scheme for the corresponding power spectrum, and allow for the determination of its large limit. In the present work, these results are extended to the case of the circular orthogonal ensemble and circular symplectic ensemble, where the integrability is expressed through four particular Painlevé VI systems for finite , and two Painlevé III systems for the limit , and also via corresponding Fredholm determinants. The relation between the limiting power spectrum , where denotes the Fourier variable, and the limiting generating function for the conditioned gap probabilities is particularly direct, involving just a single integration over the gap endpoint in the latter. Interpreting this generating function as the characteristic function of a counting statistic allows for it to be shown that , where is the Dyson index.
{"title":"Power spectra of Dyson’s circular ensembles","authors":"Peter J. Forrester , Nicholas S. Witte","doi":"10.1016/j.physd.2024.134435","DOIUrl":"10.1016/j.physd.2024.134435","url":null,"abstract":"<div><div>The power spectrum is a Fourier series statistic associated with the covariances of the displacement from average positions of the members of an eigen-sequence. When this eigen-sequence has rotational invariance, as for the eigen-angles of Dyson’s circular ensembles, recent work of Riser and Kanzieper has uncovered an exact identity expressing the power spectrum in terms of the generating function for the conditioned gap probability of having <span><math><mrow><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></math></span> eigenvalues in an interval. These authors moreover showed how for the circular unitary ensemble integrability properties of the generating function, via a particular Painlevé VI system, imply a computational scheme for the corresponding power spectrum, and allow for the determination of its large <span><math><mi>N</mi></math></span> limit. In the present work, these results are extended to the case of the circular orthogonal ensemble and circular symplectic ensemble, where the integrability is expressed through four particular Painlevé VI systems for finite <span><math><mi>N</mi></math></span>, and two Painlevé III<span><math><msup><mrow></mrow><mrow><mo>′</mo></mrow></msup></math></span> systems for the limit <span><math><mrow><mi>N</mi><mo>→</mo><mi>∞</mi></mrow></math></span>, and also via corresponding Fredholm determinants. The relation between the limiting power spectrum <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>ω</mi></math></span> denotes the Fourier variable, and the limiting generating function for the conditioned gap probabilities is particularly direct, involving just a single integration over the gap endpoint in the latter. Interpreting this generating function as the characteristic function of a counting statistic allows for it to be shown that <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow><munder><mrow><mo>∼</mo></mrow><mrow><mi>ω</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mn>1</mn></mrow><mrow><mi>π</mi><mi>β</mi><mrow><mo>|</mo><mi>ω</mi><mo>|</mo></mrow></mrow></mfrac></mrow></math></span>, where <span><math><mi>β</mi></math></span> is the Dyson index.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134435"},"PeriodicalIF":2.7,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744735","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.physd.2024.134428
Wei-Wei Han , Rui Fang , William Layton
The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov–Prandtl 1-equation URANS model. In tests, the 1/2-equation model produced comparable velocity statistics to a full 1-equation model with lower computational complexity. There is little progress in the numerical analysis of URANS models due to the difficulties in treating the coupling between equations and the nonlinearities in highest-order terms. The numerical analysis herein on the 1/2-equation model has independent interest and is also a first numerical analysis step to address the couplings and nonlinearities in a full 1-equation model. This report develops a complete numerical analysis of the 1/2-equation model. Stability, convergence, and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate the predictions of the convergence theory.
{"title":"Numerical analysis of a 1/2-equation model of turbulence","authors":"Wei-Wei Han , Rui Fang , William Layton","doi":"10.1016/j.physd.2024.134428","DOIUrl":"10.1016/j.physd.2024.134428","url":null,"abstract":"<div><div>The recent 1/2-equation model of turbulence is a simplification of the standard Kolmogorov–Prandtl 1-equation URANS model. In tests, the 1/2-equation model produced comparable velocity statistics to a full 1-equation model with lower computational complexity. There is little progress in the numerical analysis of URANS models due to the difficulties in treating the coupling between equations and the nonlinearities in highest-order terms. The numerical analysis herein on the 1/2-equation model has independent interest and is also a first numerical analysis step to address the couplings and nonlinearities in a full 1-equation model. This report develops a complete numerical analysis of the 1/2-equation model. Stability, convergence, and error estimates are proven for a semi-discrete and fully discrete approximation. Finally, numerical tests are conducted to validate the predictions of the convergence theory.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134428"},"PeriodicalIF":2.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.physd.2024.134427
F.R. Waters , C.A. Yates , J.H.P. Dawes
The simplest particle-based mass-action models for Turing instability – i.e. those with only two component species undergoing instantaneous interactions of at most two particles, with the smallest number of distinct interactions – fall into a surprisingly small number of classes of reaction schemes. In previous work we have computed this classification, with different schemes distinguished by the structure of the interactions. Within a given class the reaction stoichiometry and rates remain as parameters that determine the linear and nonlinear evolution of the system.
Adopting the usual weakly nonlinear scalings and analysis reveals that, under suitable choices of reaction stoichiometry, and in nine of the 11 classes of minimal scheme exhibiting a spatially in-phase (“true activator-inhibitor”) Turing instability, stable patterns are indeed generated in open regions of parameter space via a generically supercritical bifurcation from the spatially uniform state. In three of these classes the instability is always supercritical while in six there is an open region in which it is subcritical. Intriguingly, however, in the remaining two classes of minimal scheme we require different weakly nonlinear scalings, since the coefficient in the usual cubic normal form unexpectedly vanishes identically. In these cases, a different set of asymptotic scalings is required.
We present a complete analysis through deriving the normal form for these two cases also, which involves quintic terms. This fifth-order normal form also captures the behaviour along the boundaries between the supercritical and subcritical cases of the cubic normal form. The details of these calculations reveal the distinct roles played by reaction rate parameters as compared to stoichiometric parameters.
We quantitatively validate our analysis via numerical simulations and confirm the two different scalings for the amplitude of predicted stable patterned states.
{"title":"Weakly nonlinear analysis of minimal models for Turing patterns","authors":"F.R. Waters , C.A. Yates , J.H.P. Dawes","doi":"10.1016/j.physd.2024.134427","DOIUrl":"10.1016/j.physd.2024.134427","url":null,"abstract":"<div><div>The simplest particle-based mass-action models for Turing instability – i.e. those with only two component species undergoing instantaneous interactions of at most two particles, with the smallest number of distinct interactions – fall into a surprisingly small number of classes of reaction schemes. In previous work we have computed this classification, with different schemes distinguished by the structure of the interactions. Within a given class the reaction stoichiometry and rates remain as parameters that determine the linear and nonlinear evolution of the system.</div><div>Adopting the usual weakly nonlinear scalings and analysis reveals that, under suitable choices of reaction stoichiometry, and in nine of the 11 classes of minimal scheme exhibiting a spatially in-phase (“true activator-inhibitor”) Turing instability, stable patterns are indeed generated in open regions of parameter space via a generically supercritical bifurcation from the spatially uniform state. In three of these classes the instability is always supercritical while in six there is an open region in which it is subcritical. Intriguingly, however, in the remaining two classes of minimal scheme we require different weakly nonlinear scalings, since the coefficient in the usual cubic normal form unexpectedly vanishes identically. In these cases, a different set of asymptotic scalings is required.</div><div>We present a complete analysis through deriving the normal form for these two cases also, which involves quintic terms. This fifth-order normal form also captures the behaviour along the boundaries between the supercritical and subcritical cases of the cubic normal form. The details of these calculations reveal the distinct roles played by reaction rate parameters as compared to stoichiometric parameters.</div><div>We quantitatively validate our analysis via numerical simulations and confirm the two different scalings for the amplitude of predicted stable patterned states.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134427"},"PeriodicalIF":2.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.physd.2024.134432
H. Christodoulidi , Ch. G. Antonopoulos
In this paper, we examine the dynamical and statistical properties of a mean-field Hamiltonian with on-site potentials, where particles interact via nonlinear global forces. The absence of linear dispersion triggers a variety of interesting dynamical features associated with very strong energy localisation, weak chaos and slow thermalisation processes. Particle excitations lead to energy packets that are mostly preserved over time. We study the route to thermalisation through the computation of the probability density distributions of the momenta of the system and their slow convergence into a Gaussian distribution in the context of non-extensive statistical mechanics and Tsallis entropy, a process that is further prolonged as the number of particles increases. In addition, we observe that the maximum Lyapunov exponent decays as a power–law with respect to the system size, indicating “integrable-like” behaviour in the thermodynamic limit. Finally, we give an analytic upper estimate for the growth of the maximum Lyapunov exponent in terms of the energy.
{"title":"Energy localisation and dynamics of a mean-field model with non-linear dispersion","authors":"H. Christodoulidi , Ch. G. Antonopoulos","doi":"10.1016/j.physd.2024.134432","DOIUrl":"10.1016/j.physd.2024.134432","url":null,"abstract":"<div><div>In this paper, we examine the dynamical and statistical properties of a mean-field Hamiltonian with on-site potentials, where particles interact via nonlinear global forces. The absence of linear dispersion triggers a variety of interesting dynamical features associated with very strong energy localisation, weak chaos and slow thermalisation processes. Particle excitations lead to energy packets that are mostly preserved over time. We study the route to thermalisation through the computation of the probability density distributions of the momenta of the system and their slow convergence into a Gaussian distribution in the context of non-extensive statistical mechanics and Tsallis entropy, a process that is further prolonged as the number of particles increases. In addition, we observe that the maximum Lyapunov exponent decays as a power–law with respect to the system size, indicating “integrable-like” behaviour in the thermodynamic limit. Finally, we give an analytic upper estimate for the growth of the maximum Lyapunov exponent in terms of the energy.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134432"},"PeriodicalIF":2.7,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-17DOI: 10.1016/j.physd.2024.134433
Guangrui Sun , Xingyi Wang , Yongliang Yang
In implicit large eddy simulations (ILES), it becomes increasingly clear that numerical errors are essential to simulation accuracy. Nevertheless, whether the numerical dissipation in a CFD solver can be regarded as a means of turbulence modeling cannot be known . In the present work, we propose a general method to quantify the numerical dissipation rate for arbitrary flow solvers. Unlike previous approaches in which the numerical dissipation is estimated from the perspective of kinetic energy transfer, our method focuses on direct comparisons with the SGS dissipation from explicit models. The new method is both self-contained and self-consistent, which can be applied to any numerical solver through a simple post-processing step in the physical space. We show that for two common techniques to introduce numerical dissipation (through numerical schemes and solution filtering), the quantification results help to determine if a simulation can be considered as a legitimate ILES run and provide direct guidance for designing better models. When the numerical dissipation is already significant, an improved ILES filtering approach is proposed, which reduces the native numerical dissipation and works better for low order codes. The methods are general and work well for different Reynolds numbers, grid resolutions, and numerical schemes.
{"title":"A direct quantification of numerical dissipation towards improved large eddy simulations","authors":"Guangrui Sun , Xingyi Wang , Yongliang Yang","doi":"10.1016/j.physd.2024.134433","DOIUrl":"10.1016/j.physd.2024.134433","url":null,"abstract":"<div><div>In implicit large eddy simulations (ILES), it becomes increasingly clear that numerical errors are essential to simulation accuracy. Nevertheless, whether the numerical dissipation in a CFD solver can be regarded as a means of turbulence modeling cannot be known <span><math><mi>a</mi></math></span> <span><math><mrow><mi>p</mi><mi>r</mi><mi>i</mi><mi>o</mi><mi>r</mi><mi>i</mi></mrow></math></span>. In the present work, we propose a general method to quantify the numerical dissipation rate for arbitrary flow solvers. Unlike previous approaches in which the numerical dissipation is estimated from the perspective of kinetic energy transfer, our method focuses on direct comparisons with the SGS dissipation from explicit models. The new method is both self-contained and self-consistent, which can be applied to any numerical solver through a simple post-processing step in the physical space. We show that for two common techniques to introduce numerical dissipation (through numerical schemes and solution filtering), the quantification results help to determine if a simulation can be considered as a legitimate ILES run and provide direct guidance for designing better models. When the numerical dissipation is already significant, an improved ILES filtering approach is proposed, which reduces the native numerical dissipation and works better for low order codes. The methods are general and work well for different Reynolds numbers, grid resolutions, and numerical schemes.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134433"},"PeriodicalIF":2.7,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-15DOI: 10.1016/j.physd.2024.134429
Zefu Feng , Kun Zhao , Shouming Zhou
This paper is devoted to the study of the existence and stability of non-trivial steady state solutions to the following coupled system of PDEs on the half-line : which is a model of chemotaxis of Keller–Segel type. When is subject to the no-flux boundary condition, equals a positive value at the origin, and assuming the functions vanish at the far field, a unique steady state is constructed under suitable restrictions on the system parameters, which is capable of describing fundamental phenomena in chemotaxis, such as spatial aggregation. Moreover, the steady state is shown to be nonlinearly asymptotically stable if carries zero mass, matches at the far field, and the initial perturbation is sufficiently small in weighted Sobolev spaces.
{"title":"Existence and stability of boundary spike layer solutions of an attractive chemotaxis model with singular sensitivity and nonlinear consumption rate of chemical stimuli","authors":"Zefu Feng , Kun Zhao , Shouming Zhou","doi":"10.1016/j.physd.2024.134429","DOIUrl":"10.1016/j.physd.2024.134429","url":null,"abstract":"<div><div>This paper is devoted to the study of the existence and stability of non-trivial steady state solutions to the following coupled system of PDEs on the half-line <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>: <span><span><span><span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>−</mo><mi>χ</mi><msub><mrow><mrow><mo>[</mo><mi>u</mi><msub><mrow><mrow><mo>(</mo><mo>ln</mo><mi>w</mi><mo>)</mo></mrow></mrow><mrow><mi>x</mi></mrow></msub><mo>]</mo></mrow></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo></mrow></math></span></span><span><span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>ɛ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>γ</mi></mrow></msup><msup><mrow><mi>w</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>,</mo></mrow></math></span></span></span></span> which is a model of chemotaxis of Keller–Segel type. When <span><math><mi>u</mi></math></span> is subject to the no-flux boundary condition, <span><math><mi>w</mi></math></span> equals a positive value at the origin, and assuming the functions vanish at the far field, a unique steady state <span><math><mrow><mo>(</mo><mi>U</mi><mo>,</mo><mi>W</mi><mo>)</mo></mrow></math></span> is constructed under suitable restrictions on the system parameters, which is capable of describing fundamental phenomena in chemotaxis, such as spatial aggregation. Moreover, the steady state is shown to be nonlinearly asymptotically stable if <span><math><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>U</mi><mo>)</mo></mrow></math></span> carries zero mass, <span><math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> matches <span><math><mrow><mi>W</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> at the far field, and the initial perturbation is sufficiently small in weighted Sobolev spaces.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"471 ","pages":"Article 134429"},"PeriodicalIF":2.7,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142744734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1016/j.physd.2024.134419
Jiaqing Shan, Maohua Li
In this paper, the Darboux transformation (DT) of the reverse space–time (RST) nonlocal short pulse equation is constructed by a hodograph transformation and the eigenfunctions of its Lax pair. The multi-soliton solutions of the RST nonlocal short pulse equation are produced through the DT, which can be expressed in terms of determinant representation. The correctness of DT and determinant representation of N-soliton solutions are proven. By taking different values of eigenvalues, bounded soliton solutions and unbounded soliton solutions can be obtained. In addition, based on the degenerate Darboux transformation, the -positon solutions of the RST nonlocal short pulse equation are computed from the determinant expression of the multi-soliton solution. The decomposition of positons, approximate trajectory and “phase shift” after collision are discussed explicitly. Furthermore, different kinds of mixed solutions are also presented, and the interaction properties between positons and solitons are investigated.
本文通过霍多图变换及其拉克斯对的特征函数，构建了反向时空（RST）非局域短脉冲方程的达布变换（Darboux transformation，DT）。通过 DT 生成 RST 非局部短脉冲方程的多孑子解，可以用行列式表示。证明了 DT 和行列式表示 N 玻利子解的正确性。通过取不同的特征值，可以得到有界孤子解和无界孤子解。此外，基于退化达尔布变换，从多孤子解的行列式表达计算出 RST 非局部短脉冲方程的 N 正子解。明确讨论了正子分解、近似轨迹和碰撞后的 "相移"。此外，还提出了不同种类的混合解，并研究了正子和孤子之间的相互作用特性。
{"title":"The dynamic of the positons for the reverse space–time nonlocal short pulse equation","authors":"Jiaqing Shan, Maohua Li","doi":"10.1016/j.physd.2024.134419","DOIUrl":"10.1016/j.physd.2024.134419","url":null,"abstract":"<div><div>In this paper, the Darboux transformation (DT) of the reverse space–time (RST) nonlocal short pulse equation is constructed by a hodograph transformation and the eigenfunctions of its Lax pair. The multi-soliton solutions of the RST nonlocal short pulse equation are produced through the DT, which can be expressed in terms of determinant representation. The correctness of DT and determinant representation of N-soliton solutions are proven. By taking different values of eigenvalues, bounded soliton solutions and unbounded soliton solutions can be obtained. In addition, based on the degenerate Darboux transformation, the <span><math><mi>N</mi></math></span>-positon solutions of the RST nonlocal short pulse equation are computed from the determinant expression of the multi-soliton solution. The decomposition of positons, approximate trajectory and “phase shift” after collision are discussed explicitly. Furthermore, different kinds of mixed solutions are also presented, and the interaction properties between positons and solitons are investigated.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"470 ","pages":"Article 134419"},"PeriodicalIF":2.7,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-12DOI: 10.1016/j.physd.2024.134426
Josep M. Cors , Miguel Garrido
For , we show the existence of symmetric periodic orbits of very large radii in the elliptic three-dimensional restricted -body problem when the primaries have equal masses and are arranged in a -gon central configuration. These periodic orbits are close to very large circular Keplerian orbits lying nearly a plane perpendicular to that of the primaries. They exist for a discrete sequence of values of the mean motion, no matter the value of the eccentricity of the primaries.
对于 N≥3，我们证明了在椭圆形三维受限 (N+1)- 体问题中，当 N 个基体质量相等并以 N 宫中心构型排列时，存在半径非常大的对称周期轨道。这些周期轨道接近于非常大的圆形开普勒轨道，几乎位于垂直于基体的平面上。无论主星的偏心率是多少，它们都存在于平均运动的离散值序列中。
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Pub Date : 2024-11-07DOI: 10.1016/j.physd.2024.134424
Javier E. Contreras-Reyes
The Jensen-variance (JV) information based on Jensen’s inequality and variance has been previously proposed to measure the distance between two random variables. Based on the relationship between JV distance and autocorrelation function of two weakly stationary process, the Jensen-autocovariance and Jensen-autocorrelation functions are proposed in this paper. Furthermore, the distance between two different weakly stationary processes is measured by the Jensen-cross-correlation function. Moreover, autocorrelation function is also considered for ARMA and ARFIMA processes, deriving explicit formulas for Jensen-autocorrelation function that only depends on model parametric space and lag, whose were also illustrated by numeric results. In order to study the usefulness of proposed functions, two real-life applications were considered: the Tree Ring and Southern Humboldt current ecosystem time series.
以前曾提出过基于詹森不等式和方差的詹森方差（JV）信息来测量两个随机变量之间的距离。根据 JV 距离与两个弱静止过程的自相关函数之间的关系，本文提出了 Jensen-自方差函数和 Jensen-自相关函数。此外，两个不同弱静止过程之间的距离用詹森-交叉相关函数来衡量。此外，还考虑了 ARMA 和 ARFIMA 过程的自相关函数，推导出了仅取决于模型参数空间和滞后期的詹森-自相关函数的明确公式，并通过数值结果对其进行了说明。为了研究拟议函数的实用性，考虑了两个实际应用：树环和南洪堡海流生态系统时间序列。
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