The flux ratio serves as a valuable characteristic for assessing the influence of permanent charge on the fluxes of different ion species. Our work analyzes flux ratios and their bifurcations in ionic flows using Poisson–Nernst–Planck models. Unlike Ussing's and Hodgkin–Keynes's ratios, our flux-bifurcation analysis captures the emergence and disappearance of multiple flux-ratio solutions as boundary conditions and permanent charge vary. We identify bifurcation points for flux ratios and characterize their dependence on system parameters, such as boundary conditions and permanent charge. Specifically, we examine the qualitative behavior of fluxes and flux ratios with respect to current, boundary concentrations, and boundary electric potentials at bifurcation points. This study can contribute to a deeper understanding of the underlying mechanisms governing ionic flows and could offer valuable insights for the design and optimization of ion channels in practical applications.
{"title":"Analysis of Flux-Ratio Bifurcation in Ionic Flows via Classical Poisson–Nernst–Planck Models","authors":"Hamid Mofidi, Fazel Hadadifard, Mingji Zhang","doi":"10.1111/sapm.70087","DOIUrl":"https://doi.org/10.1111/sapm.70087","url":null,"abstract":"<div>\u0000 \u0000 <p>The flux ratio serves as a valuable characteristic for assessing the influence of permanent charge on the fluxes of different ion species. Our work analyzes flux ratios and their bifurcations in ionic flows using Poisson–Nernst–Planck models. Unlike Ussing's and Hodgkin–Keynes's ratios, our flux-bifurcation analysis captures the emergence and disappearance of multiple flux-ratio solutions as boundary conditions and permanent charge vary. We identify bifurcation points for flux ratios and characterize their dependence on system parameters, such as boundary conditions and permanent charge. Specifically, we examine the qualitative behavior of fluxes and flux ratios with respect to current, boundary concentrations, and boundary electric potentials at bifurcation points. This study can contribute to a deeper understanding of the underlying mechanisms governing ionic flows and could offer valuable insights for the design and optimization of ion channels in practical applications.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144782725","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}
As part of the efforts aimed at extending Painlevé and Gambier's work on second-order equations in one variable to first-order ones in two, in 1981, Bureau classified the systems of ordinary quadratic differential equations in two variables which are free of movable critical points (which have the Painlevé Property). We revisit this classification, which we complete by adding some cases overlooked by Bureau, and by correcting some of his arguments. We also simplify the canonical forms of some systems, bring the natural symmetries of others into their study, and investigate the birational equivalence among some of the systems in the class. Lastly, we study the birational geometry of Okamoto's space of initial conditions for Bureau's system VIII, in order to establish the sufficiency of some necessary conditions for the absence of movable critical points.
{"title":"On Bureau's Classification of Quadratic Differential Equations in Two Variables Free of Movable Critical Points","authors":"Adolfo Guillot","doi":"10.1111/sapm.70083","DOIUrl":"https://doi.org/10.1111/sapm.70083","url":null,"abstract":"<p>As part of the efforts aimed at extending Painlevé and Gambier's work on second-order equations in one variable to first-order ones in two, in 1981, Bureau classified the systems of ordinary quadratic differential equations in two variables which are free of movable critical points (which have the Painlevé Property). We revisit this classification, which we complete by adding some cases overlooked by Bureau, and by correcting some of his arguments. We also simplify the canonical forms of some systems, bring the natural symmetries of others into their study, and investigate the birational equivalence among some of the systems in the class. Lastly, we study the birational geometry of Okamoto's space of initial conditions for Bureau's system VIII, in order to establish the sufficiency of some necessary conditions for the absence of movable critical points.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70083","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144782639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Camassa–Holm (CH) equation, which serves as the dual to the Korteweg–de Vries (KdV) equation, is a completely integrable equation that admits peaked solitons. In this paper, we establish the KAM theory for quasi-linear Hamiltonian perturbations of the dispersive CH equation over the circle. The existence and linear stability of Cantor families of small-amplitude quasi-periodic solutions of this model are proved. Our proof generalizes the arguments for the Degasperis–Procesi equation and makes use of the Birkhoff normal form technique, a Nash–Moser iterative scheme in Sobolev scales and a reduction procedure. Both the Hamiltonian and reversible structures of the equation are fully utilized in our approach. In the reversible case, some new properties of the Birkhoff maps are explored to set up the reversibility of the linearized operator. In addition, a new technique on a presupposed hypothesis of the relation between the perturbed and unperturbed frequencies is proposed to tackle a parameter-independent quasi-linear equation with reversible structure.
{"title":"KAM for Quasi-Linear Hamiltonian Perturbations of the Dispersive Camassa–Holm Equation","authors":"Xiaoping Wu, Ying Fu, Changzheng Qu","doi":"10.1111/sapm.70088","DOIUrl":"https://doi.org/10.1111/sapm.70088","url":null,"abstract":"<div>\u0000 \u0000 <p>The Camassa–Holm (CH) equation, which serves as the dual to the Korteweg–de Vries (KdV) equation, is a completely integrable equation that admits peaked solitons. In this paper, we establish the KAM theory for quasi-linear Hamiltonian perturbations of the dispersive CH equation over the circle. The existence and linear stability of Cantor families of small-amplitude quasi-periodic solutions of this model are proved. Our proof generalizes the arguments for the Degasperis–Procesi equation and makes use of the Birkhoff normal form technique, a Nash–Moser iterative scheme in Sobolev scales and a reduction procedure. Both the Hamiltonian and reversible structures of the equation are fully utilized in our approach. In the reversible case, some new properties of the Birkhoff maps are explored to set up the reversibility of the linearized operator. In addition, a new technique on a presupposed hypothesis of the relation between the perturbed and unperturbed frequencies is proposed to tackle a parameter-independent quasi-linear equation with reversible structure.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144782638","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}
J. L. Cieśliński, A. V. Mikhailov, M. Nieszporski, F. W. Nijhoff
We discuss an integrable discretization of the principal chiral field models equations and its involutive reduction. We present a Darboux transformation and general construction of soliton solutions for these discrete equations.
{"title":"Discrete Integrable Principal Chiral Field Model and Its Involutive Reduction","authors":"J. L. Cieśliński, A. V. Mikhailov, M. Nieszporski, F. W. Nijhoff","doi":"10.1111/sapm.70084","DOIUrl":"https://doi.org/10.1111/sapm.70084","url":null,"abstract":"<p>We discuss an integrable discretization of the principal chiral field models equations and its involutive reduction. We present a Darboux transformation and general construction of soliton solutions for these discrete equations.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 2","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70084","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144773621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work aims to study some dynamical aspects of the nonlinear logarithmic Schrödinger equation (NLS-log) on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex. By considering Neumann–Kirchhoff boundary conditions at the junction, we show the existence and the orbital stability of standing wave solutions with a profile determined by a positive single-lobe state. Via a splitting-eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around a positive single-lobe state. To our knowledge, the results contained in this paper are the first to study the (NLS-log) on tadpole graphs. In particular, our approach has the prospect of being extended to study stability properties of other bound states for the (NLS-log) on a tadpole graph or other non-compact metric graph such as a looping-edge graphs.
{"title":"Existence and Orbital Stability of Standing-Wave Solutions of the Nonlinear Logarithmic Schrödinger Equation On a Tadpole Graph","authors":"Jaime Angulo Pava, Andrés Gerardo Pérez Yépez","doi":"10.1111/sapm.70085","DOIUrl":"https://doi.org/10.1111/sapm.70085","url":null,"abstract":"<p>This work aims to study some dynamical aspects of the nonlinear logarithmic Schrödinger equation (NLS-log) on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex. By considering Neumann–Kirchhoff boundary conditions at the junction, we show the existence and the orbital stability of standing wave solutions with a profile determined by a positive single-lobe state. Via a splitting-eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around a positive single-lobe state. To our knowledge, the results contained in this paper are the first to study the (NLS-log) on tadpole graphs. In particular, our approach has the prospect of being extended to study stability properties of other bound states for the (NLS-log) on a tadpole graph or other non-compact metric graph such as a looping-edge graphs.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70085","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144714826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the 3D inhomogeneous incompressible micropolar system with the initial density being discontinuous and the initial velocity possessing critical regularity. Assuming that is close to a positive constant, we obtain the global existence and uniqueness of the solution if is small in
{"title":"Global Solvability for 3D Incompressible Inhomogeneous Micropolar System in Critical Spaces","authors":"Yelei Guo, Chenyin Qian, Ting Zhang, Xiaole Zheng","doi":"10.1111/sapm.70086","DOIUrl":"https://doi.org/10.1111/sapm.70086","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we investigate the 3D inhomogeneous incompressible micropolar system with the initial density <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ρ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$rho _0$</annotation>\u0000 </semantics></math> being discontinuous and the initial velocity <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>u</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(u_0,omega _0)$</annotation>\u0000 </semantics></math> possessing critical regularity. Assuming that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ρ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$rho _0$</annotation>\u0000 </semantics></math> is close to a positive constant, we obtain the global existence and uniqueness of the solution if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>u</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(u_0,omega _0)$</annotation>\u0000 </semantics></math> is small in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mover>\u0000 <mi>B</mi>\u0000 <mo>̇</mo>\u0000 </mover>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <mo>/</mo>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>R</mi>\u0000 <mspace></mspace>\u0000 </mrow>\u0000 <mn>3</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo><</mo>\u0000 <mi>p</mi>\u0000 <mo>&","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144705635","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}
We study the three-dimensional, divergence-free, incompressible Euler equations in the -plane approximation for off-equatorial oceanic flows of constant vorticity, where the fluid domain is bounded by a free surface and a flat bed. The major difference, compared to related earlier works, is that we refrain from any global restrictions on solutions with respect to the latitudinal coordinate , which we justify by the -plane approximation's locality. The resulting flows are necessarily steady (despite the time dependence of the governing equations), zonal, independent of the zonal coordinate , and fully explicit; the corresponding free surface exhibits a nontrivial parabolic structure in . We also provide an application to the Antarctic Circumpolar Current (ACC) for which we compare the sea surface height predicted by our constant vorticity model with satellite altimetry measurements available in the literature.
{"title":"On Ocean Currents with Constant Vorticity: Explicit Solutions and an Application to the ACC","authors":"Anna Geyer, Ronald Quirchmayr","doi":"10.1111/sapm.70081","DOIUrl":"https://doi.org/10.1111/sapm.70081","url":null,"abstract":"<p>We study the three-dimensional, divergence-free, incompressible Euler equations in the <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math>-plane approximation for off-equatorial oceanic flows of constant vorticity, where the fluid domain is bounded by a free surface and a flat bed. The major difference, compared to related earlier works, is that we refrain from any global restrictions on solutions with respect to the latitudinal coordinate <span></span><math>\u0000 <semantics>\u0000 <mi>y</mi>\u0000 <annotation>$y$</annotation>\u0000 </semantics></math>, which we justify by the <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math>-plane approximation's locality. The resulting flows are necessarily steady (despite the time dependence of the governing equations), zonal, independent of the zonal coordinate <span></span><math>\u0000 <semantics>\u0000 <mi>x</mi>\u0000 <annotation>$x$</annotation>\u0000 </semantics></math>, and fully explicit; the corresponding free surface exhibits a nontrivial parabolic structure in <span></span><math>\u0000 <semantics>\u0000 <mi>y</mi>\u0000 <annotation>$y$</annotation>\u0000 </semantics></math>. We also provide an application to the Antarctic Circumpolar Current (ACC) for which we compare the sea surface height predicted by our constant vorticity model with satellite altimetry measurements available in the literature.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70081","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144606551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Preface: Modeling Wave Propagation: Mathematical Theory and Numerical Analysis, in Memory of Prof. Vassilios Dougalis","authors":"Georgios Akrivis, Jerry Bona, Angel Durán, Ohannes Karakashian, Dimitrios Mitsotakis, Beatrice Pelloni, Jean-Claude Saut","doi":"10.1111/sapm.70082","DOIUrl":"https://doi.org/10.1111/sapm.70082","url":null,"abstract":"","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144589897","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 this paper, we establish the existence of a weak solution to the Riccati equation via the canonical Hamiltonian formulation and Ekeland variational principle and present an application in the closed-loop control. It is well-known that the general Riccati equation admits no classical solution due to its blow-up behavior. Nevertheless, by introducing a residual through the combined application of the canonical Hamiltonian formulation and the Ekeland variational principle, we observe that under appropriate conditions, the weak solution to the Riccati equation exists. Initially, we derive the Hamilton–Jacobi equation from the Riccati equation incorporating the residual , utilizing the canonical Hamiltonian formalism. Subsequently, we elucidate the relationship between the viscosity solution of the Hamilton–Jacobi equation and the residual , thereby justifying the introduction of and establishing the existence of the weak solution. Finally, we present the application of the Riccati equation with the residual in a closed-loop control setting, thereby further substantiating the existence of the weak solution.
{"title":"Weak Solutions to the Riccati Equation and the Application in the Closed-Loop Control","authors":"Deqin Su, Xiaoying Wang, Yong Li","doi":"10.1111/sapm.70078","DOIUrl":"https://doi.org/10.1111/sapm.70078","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we establish the existence of a weak solution to the Riccati equation via the canonical Hamiltonian formulation and Ekeland variational principle and present an application in the closed-loop control. It is well-known that the general Riccati equation admits no classical solution due to its blow-up behavior. Nevertheless, by introducing a residual <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math> through the combined application of the canonical Hamiltonian formulation and the Ekeland variational principle, we observe that under appropriate conditions, the weak solution to the Riccati equation exists. Initially, we derive the Hamilton–Jacobi equation from the Riccati equation incorporating the residual <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math>, utilizing the canonical Hamiltonian formalism. Subsequently, we elucidate the relationship between the viscosity solution <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <annotation>$mathcal {S}^*$</annotation>\u0000 </semantics></math> of the Hamilton–Jacobi equation and the residual <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math>, thereby justifying the introduction of <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math> and establishing the existence of the weak solution. Finally, we present the application of the Riccati equation with the residual <span></span><math>\u0000 <semantics>\u0000 <mi>μ</mi>\u0000 <annotation>$mu$</annotation>\u0000 </semantics></math> in a closed-loop control setting, thereby further substantiating the existence of the weak solution.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144550863","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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