Pub Date : 2024-09-17DOI: 10.1088/1361-6544/ad76f3
Yu Ilyashenko
Consider a germ of a holomorphic vector field at the origin on the coordinate complex plane. This germ is called a saddle-node if the origin is its singular point, one of its eigenvalues at zero is zero, and the other is not. A saddle-node germ is real if its restriction to the real plane is real. The monodromy transformation for this germ has a multiplier at zero equal to 1. The germ of this map is parabolic and admits a ‘normalizing cochain’. In this note we express the Dulac map of any real saddle-node up to a left composition with a real germ through one component of the cochain normalizing the monodromy transformation.
{"title":"Dulac maps of real saddle-nodes","authors":"Yu Ilyashenko","doi":"10.1088/1361-6544/ad76f3","DOIUrl":"https://doi.org/10.1088/1361-6544/ad76f3","url":null,"abstract":"Consider a germ of a holomorphic vector field at the origin on the coordinate complex plane. This germ is called a saddle-node if the origin is its singular point, one of its eigenvalues at zero is zero, and the other is not. A saddle-node germ is real if its restriction to the real plane is real. The monodromy transformation for this germ has a multiplier at zero equal to 1. The germ of this map is parabolic and admits a ‘normalizing cochain’. In this note we express the Dulac map of any real saddle-node up to a left composition with a real germ through one component of the cochain normalizing the monodromy transformation.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1981, Frisch and Morf (1981 Phys. Rev. A 23 2673–705) postulated the existence of complex singularities in solutions of Navier–Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier–Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of fluid dynamics on log-lattices, where the computational burden is logarithmic concerning ordinary fluid simulations. We analyze properties of potential complex singularities in both 1D and 3D models for lattices of different spacings. Dominant complex singularities are tracked using the singularity strip method to obtain new scalings regarding the approach to the real axis and the influence of normal, hypo and hyper dissipation.
{"title":"Tracking complex singularities of fluids on log-lattices","authors":"Quentin Pikeroen, Amaury Barral, Guillaume Costa, Ciro Campolina, Alexei Mailybaev and Berengere Dubrulle","doi":"10.1088/1361-6544/ad7661","DOIUrl":"https://doi.org/10.1088/1361-6544/ad7661","url":null,"abstract":"In 1981, Frisch and Morf (1981 Phys. Rev. A 23 2673–705) postulated the existence of complex singularities in solutions of Navier–Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier–Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of fluid dynamics on log-lattices, where the computational burden is logarithmic concerning ordinary fluid simulations. We analyze properties of potential complex singularities in both 1D and 3D models for lattices of different spacings. Dominant complex singularities are tracked using the singularity strip method to obtain new scalings regarding the approach to the real axis and the influence of normal, hypo and hyper dissipation.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-16DOI: 10.1088/1361-6544/ad7808
Jinjun Li, Wanxiang Zeng and Min Wu
We show that the lower discrete Hausdorff dimension of any spectrum for Moran measure is bounded by the Hausdorff dimension of its support.
我们证明，任何莫兰量度谱的离散豪斯多夫维度下限都受其支撑的豪斯多夫维度约束。
{"title":"Lower discrete Hausdorff dimension of spectra for Moran measure","authors":"Jinjun Li, Wanxiang Zeng and Min Wu","doi":"10.1088/1361-6544/ad7808","DOIUrl":"https://doi.org/10.1088/1361-6544/ad7808","url":null,"abstract":"We show that the lower discrete Hausdorff dimension of any spectrum for Moran measure is bounded by the Hausdorff dimension of its support.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-16DOI: 10.1088/1361-6544/ad7807
Zhengyu Yin and Zubiao Xiao
Let G be an infinite countable amenable group and P a polyhedron with the topological dimension . We construct a minimal subshift (X, G) of such that its mean topological dimension is equal to . This result answers the question of Dou (2017 Discrete Contin. Dyn. Syst.37 1411–24). Moreover, it extends the work of Jin and Qiao (2023 arXiv:2102.10339) for -action.
设 G 是一个无限可数的可配位群，P 是一个拓扑维度为 的多面体。我们构造一个最小子移位（X，G），使得它的平均拓扑维度等于 。 这个结果回答了 Dou（2017 Discrete Contin. Dyn. Syst.37 1411-24）的问题。此外，它还扩展了 Jin 和 Qiao（2023 arXiv:2102.10339）关于 - 作用的工作。
{"title":"Minimal amenable subshift with full mean topological dimension","authors":"Zhengyu Yin and Zubiao Xiao","doi":"10.1088/1361-6544/ad7807","DOIUrl":"https://doi.org/10.1088/1361-6544/ad7807","url":null,"abstract":"Let G be an infinite countable amenable group and P a polyhedron with the topological dimension . We construct a minimal subshift (X, G) of such that its mean topological dimension is equal to . This result answers the question of Dou (2017 Discrete Contin. Dyn. Syst.37 1411–24). Moreover, it extends the work of Jin and Qiao (2023 arXiv:2102.10339) for -action.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-15DOI: 10.1088/1361-6544/ad6b70
Claude Baesens, Marc Homs-Dones and Robert S MacKay
We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of ‘simplest’ diagrams proposed by Baesens and MacKay (2018 Nonlinearity31 2928–81). This shows that the proposed class is realisable.
{"title":"Example of simplest bifurcation diagram for a monotone family of vector fields on a torus *","authors":"Claude Baesens, Marc Homs-Dones and Robert S MacKay","doi":"10.1088/1361-6544/ad6b70","DOIUrl":"https://doi.org/10.1088/1361-6544/ad6b70","url":null,"abstract":"We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of ‘simplest’ diagrams proposed by Baesens and MacKay (2018 Nonlinearity31 2928–81). This shows that the proposed class is realisable.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-12DOI: 10.1088/1361-6544/ad75dd
Nadia Loy and Benoît Perthame
Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, kinetic model with BGK type operators implementing a velocity-jump process for cell migration, directed by the nonlocal sensing of either an external signal or the cell population density itself. We describe, in an asymptotic regime, the precise profile of these concentrations which, at the macroscale, are Dirac masses. Because Dirac concentrations look like Gaussian potentials, we use the Hopf–Cole transform to calculate the potential adapted to the problem. This potential, as in other similar situations, is obtained through the viscosity solutions of a Hamilton–Jacobi equation. We begin with the linear case, when the heterogeneous external signal is given, and we show that the concentration profile obtained after the diffusion approximation is not correct and is a simple eikonal approximation of the true H–J equation. Its heterogeneous nature leads us to develop a new analysis of the implicit equation defining the Hamiltonian and a new condition to circumvent the ‘dimensionality problem’. In the nonlinear case, when the signal occurs from the cell density itself, it is shown that the already observed linear instability (pattern formation) occurs when the Hamiltonian is convex-concave, a striking new feature of our approach.
{"title":"A Hamilton–Jacobi approach to nonlocal kinetic equations","authors":"Nadia Loy and Benoît Perthame","doi":"10.1088/1361-6544/ad75dd","DOIUrl":"https://doi.org/10.1088/1361-6544/ad75dd","url":null,"abstract":"Highly concentrated patterns have been observed in a spatially heterogeneous, nonlocal, kinetic model with BGK type operators implementing a velocity-jump process for cell migration, directed by the nonlocal sensing of either an external signal or the cell population density itself. We describe, in an asymptotic regime, the precise profile of these concentrations which, at the macroscale, are Dirac masses. Because Dirac concentrations look like Gaussian potentials, we use the Hopf–Cole transform to calculate the potential adapted to the problem. This potential, as in other similar situations, is obtained through the viscosity solutions of a Hamilton–Jacobi equation. We begin with the linear case, when the heterogeneous external signal is given, and we show that the concentration profile obtained after the diffusion approximation is not correct and is a simple eikonal approximation of the true H–J equation. Its heterogeneous nature leads us to develop a new analysis of the implicit equation defining the Hamiltonian and a new condition to circumvent the ‘dimensionality problem’. In the nonlinear case, when the signal occurs from the cell density itself, it is shown that the already observed linear instability (pattern formation) occurs when the Hamiltonian is convex-concave, a striking new feature of our approach.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1088/1361-6544/ad76f5
Shikun Cui and Zhen Wang
In this paper, we develop the numerical inverse scattering transform (NIST) for solving the derivative nonlinear Schrödinger (DNLS) equation. The key technique involves formulating a Riemann–Hilbert problem that is associated with the initial value problem and solving it numerically. Before solving the Riemann–Hilbert problem (RHP), two essential operations need to be carried out. Firstly, high-precision numerical calculations are performed on the scattering data. Secondly, the RHP is deformed using the Deift–Zhou nonlinear steepest descent method. The DNLS equation has a continuous spectrum consisting of the real and imaginary axes and features three saddle points, which introduces complexity not encountered in previous NIST approaches. In our numerical inverse scattering method, we divide the (x, t)-plane into three regions and propose specific deformations for each region. These strategies not only help reduce computational costs but also minimise errors in the calculations. Unlike traditional numerical methods, the NIST does not rely on time-stepping to compute the solution. Instead, it directly solves the associated Riemann–Hilbert problem. This unique characteristic of the NIST eliminates convergence issues typically encountered in other numerical approaches and proves to be more effective, especially for long-time simulations.
{"title":"Numerical inverse scattering transform for the derivative nonlinear Schrödinger equation","authors":"Shikun Cui and Zhen Wang","doi":"10.1088/1361-6544/ad76f5","DOIUrl":"https://doi.org/10.1088/1361-6544/ad76f5","url":null,"abstract":"In this paper, we develop the numerical inverse scattering transform (NIST) for solving the derivative nonlinear Schrödinger (DNLS) equation. The key technique involves formulating a Riemann–Hilbert problem that is associated with the initial value problem and solving it numerically. Before solving the Riemann–Hilbert problem (RHP), two essential operations need to be carried out. Firstly, high-precision numerical calculations are performed on the scattering data. Secondly, the RHP is deformed using the Deift–Zhou nonlinear steepest descent method. The DNLS equation has a continuous spectrum consisting of the real and imaginary axes and features three saddle points, which introduces complexity not encountered in previous NIST approaches. In our numerical inverse scattering method, we divide the (x, t)-plane into three regions and propose specific deformations for each region. These strategies not only help reduce computational costs but also minimise errors in the calculations. Unlike traditional numerical methods, the NIST does not rely on time-stepping to compute the solution. Instead, it directly solves the associated Riemann–Hilbert problem. This unique characteristic of the NIST eliminates convergence issues typically encountered in other numerical approaches and proves to be more effective, especially for long-time simulations.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1088/1361-6544/ad76f4
Liuyi Pan, Lei Wang, Lei Liu, Wenrong Sun and Xiaoxia Ren
We study the non-degenerate dynamics of localised waves beyond Manakov system and offer their new perspectives based on the wave component analysis. Our investigation is in the framework of the coupled Hirota (CH) equations. An exact multi-parameter family of solutions for the localised waves is derived within a new Lax pair which is necessary for producing the new types of solutions describing the non-degenerate localised waves, such as the non-degenerate general breathers, non-degenerate Akhmediev breathers, non-degenerate Kuznetsov-Ma solitons and non-degenerate rogue waves. Especially, the degenerate and non-degenerate solutions for rogue waves are different from previous ones, even within the context of the Manakov system. A new technique of wave mode analysis (or the characteristic line analysis) is provided to classify degenerate and non-degenerate solutions beyond the eigenvalue perspectives, namely the critical relative wave number. Such technique is suitable for both the CH equations as well as Manakov system. Hereby, we redefine the non-degenerate localised waves from a fully different view. We further prove that a transition between the non-degenerate localised waves to various types of solitons appears in the CH equations due to the higher-order effects and there is no analogue in Manakov system. In order to further understand such transition dynamics and physical properties of the non-degenerate solutions, the physical spectra are presented analytically. The higher-order terms take impacts on the spectra, for which the state transition solutions as well as a new type of breathers are found. Furthermore, we investigate the relation between non-degenerate modulation instability and higher-order effects. We also offer an exact initial condition to excite the degenerate and non-degenerate localised waves using the numerical simulation and test the stability for the excitation of such solutions by adding a weak perturbation. Since the CH equations can model a large number of physical phenomena in the deep ocean, in the birefringent fibre as well as in the nonlinear channel, our results may provide insights for the related experimental studies.
{"title":"Non-degenerate localised waves beyond Manakov system and their new perspectives","authors":"Liuyi Pan, Lei Wang, Lei Liu, Wenrong Sun and Xiaoxia Ren","doi":"10.1088/1361-6544/ad76f4","DOIUrl":"https://doi.org/10.1088/1361-6544/ad76f4","url":null,"abstract":"We study the non-degenerate dynamics of localised waves beyond Manakov system and offer their new perspectives based on the wave component analysis. Our investigation is in the framework of the coupled Hirota (CH) equations. An exact multi-parameter family of solutions for the localised waves is derived within a new Lax pair which is necessary for producing the new types of solutions describing the non-degenerate localised waves, such as the non-degenerate general breathers, non-degenerate Akhmediev breathers, non-degenerate Kuznetsov-Ma solitons and non-degenerate rogue waves. Especially, the degenerate and non-degenerate solutions for rogue waves are different from previous ones, even within the context of the Manakov system. A new technique of wave mode analysis (or the characteristic line analysis) is provided to classify degenerate and non-degenerate solutions beyond the eigenvalue perspectives, namely the critical relative wave number. Such technique is suitable for both the CH equations as well as Manakov system. Hereby, we redefine the non-degenerate localised waves from a fully different view. We further prove that a transition between the non-degenerate localised waves to various types of solitons appears in the CH equations due to the higher-order effects and there is no analogue in Manakov system. In order to further understand such transition dynamics and physical properties of the non-degenerate solutions, the physical spectra are presented analytically. The higher-order terms take impacts on the spectra, for which the state transition solutions as well as a new type of breathers are found. Furthermore, we investigate the relation between non-degenerate modulation instability and higher-order effects. We also offer an exact initial condition to excite the degenerate and non-degenerate localised waves using the numerical simulation and test the stability for the excitation of such solutions by adding a weak perturbation. Since the CH equations can model a large number of physical phenomena in the deep ocean, in the birefringent fibre as well as in the nonlinear channel, our results may provide insights for the related experimental studies.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-11DOI: 10.1088/1361-6544/ad7662
Majid Gazor and Nasrin Sadri
This paper explores the simplest truncated orbital and parametric normal forms of controlled Hopf zero singularities. We assume a quadratic generic condition and complete the remaining results on their simplest truncated orbital and parametric normal forms of Hopf-zero singularities. Different normal form styles are explored for their potential applications in bifurcation control. We obtain their associated universal asymptotic unfolding normal forms. We derive coefficient normal form formulas of the most generic cases and present the relations between the controller coefficients and asymptotic universal unfolding parameters. These play an important role in their potential applications in bifurcation control. Finally, the results are implemented on a controlled Chua circuit system to illustrate the applicability of our results.
{"title":"Orbital and parametric normal forms for families of Hopf-zero singularity","authors":"Majid Gazor and Nasrin Sadri","doi":"10.1088/1361-6544/ad7662","DOIUrl":"https://doi.org/10.1088/1361-6544/ad7662","url":null,"abstract":"This paper explores the simplest truncated orbital and parametric normal forms of controlled Hopf zero singularities. We assume a quadratic generic condition and complete the remaining results on their simplest truncated orbital and parametric normal forms of Hopf-zero singularities. Different normal form styles are explored for their potential applications in bifurcation control. We obtain their associated universal asymptotic unfolding normal forms. We derive coefficient normal form formulas of the most generic cases and present the relations between the controller coefficients and asymptotic universal unfolding parameters. These play an important role in their potential applications in bifurcation control. Finally, the results are implemented on a controlled Chua circuit system to illustrate the applicability of our results.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-10DOI: 10.1088/1361-6544/ad7450
Francesca De Marchis, Habib Fourti and Isabella Ianni
We consider the elliptic equation in a bounded, smooth domain subject to the nonlinear Neumann boundary condition on and study the asymptotic behaviour as the exponent of families of positive solutions up satisfying uniform energy bounds. We prove energy quantisation and characterise the boundary concentration. In particular we describe the local asymptotic profile of the solutions around each concentration point and get sharp convergence results for the -norm.
{"title":"Sharp boundary concentration for a two-dimensional nonlinear Neumann problem *","authors":"Francesca De Marchis, Habib Fourti and Isabella Ianni","doi":"10.1088/1361-6544/ad7450","DOIUrl":"https://doi.org/10.1088/1361-6544/ad7450","url":null,"abstract":"We consider the elliptic equation in a bounded, smooth domain subject to the nonlinear Neumann boundary condition on and study the asymptotic behaviour as the exponent of families of positive solutions up satisfying uniform energy bounds. We prove energy quantisation and characterise the boundary concentration. In particular we describe the local asymptotic profile of the solutions around each concentration point and get sharp convergence results for the -norm.","PeriodicalId":54715,"journal":{"name":"Nonlinearity","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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