Studies in Applied Mathematics最新文献

{"title":"Mixed Superposition Rules for Lie Systems, Compatible Geometric Structures, and Applications","authors":"Rutwig Campoamor-Stursberg,&nbsp;Oscar Carballal,&nbsp;Francisco J. Herranz,&nbsp;Javier de Lucas","doi":"10.1111/sapm.70297","DOIUrl":"https://doi.org/10.1111/sapm.70297","url":null,"abstract":"<div>\u0000 \u0000 <p>Mixed superposition rules are, in short, a method to describe the general solutions of a time-dependent system of first-order differential equations, a so-called Lie system, in terms of particular solutions of other ones. This article is concerned with the theory of mixed superposition rules and their connections with geometric structures. We provide methods to obtain mixed superposition rules for systems admitting an imprimitive finite-dimensional Lie algebra of vector fields or given by a semidirect sum. In particular, we develop a novel mixed coalgebra method for Lie systems that are Hamiltonian relative to a Dirac structure, which is quite general, although we restrict to symplectic and contact manifolds in applications. This provides us with practical methods to derive mixed superposition rules and extends the coalgebra method to a new field of application while solving minor technical issues of the known formalism. Throughout the paper, we apply our results to physical systems including Schrödinger Lie systems, Riccati systems, time-dependent Calogero–Moser systems with external forces, time-dependent harmonic oscillators, and time-dependent thermodynamical systems, where general solutions can be obtained from reduced system solutions. Our results are finally extended to Lie systems of partial differential equations and a new source of such PDE Lie systems, related to the formal construction of solutions close to a given particular solution of PDEs, is provided. An example based on the Tzitzéica equation and a related system illustrating the potential applicability of these methods is given.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 3","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148872106","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}

{"title":"Standing Waves of abcd-Systems With a Two-Dimensional Kernel","authors":"Yuhao Xie,&nbsp;Min Chen,&nbsp;Shenghao Li","doi":"10.1111/sapm.70294","DOIUrl":"https://doi.org/10.1111/sapm.70294","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we continue the study of bifurcation standing wave solutions of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 <mi>c</mi>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$abcd$</annotation>\u0000 </semantics></math>-systems started by Chen–Iooss based on Lyapunov–Schmidt method. Our study is to generalize their theories for multiple-dimensional kernels. More precisely, we will provide wave patterns of an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 <mi>c</mi>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$abcd$</annotation>\u0000 </semantics></math>-system with both one-dimensional kernel and two-dimensional kernels.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148848966","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}

{"title":"Correction to “Selective Convergence of Followers to Multiple Leaders With Repulsion and Cohesion on Riemannian Manifolds”","authors":"","doi":"10.1111/sapm.70288","DOIUrl":"https://doi.org/10.1111/sapm.70288","url":null,"abstract":"<p>H. Ahn, “Selective Convergence of Followers to Multiple Leaders With Repulsion and Cohesion on Riemannian Manifolds,” <i>Studies in Applied Mathematics</i> 156 (2026): e70252, https://doi.org/10.1111/sapm.70252.</p><p>An additional geometric estimate is required in the proof of Proposition 3.3 of [<span>1</span>]. More precisely, when differentiating <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>log</mi>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 </msub>\u0000 <msubsup>\u0000 <mi>x</mi>\u0000 <mi>j</mi>\u0000 <mo>∗</mo>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$log _{x_i}x_j^*$</annotation>\u0000 </semantics></math>, one must estimate not only the variation of the first argument <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <annotation>$x_i$</annotation>\u0000 </semantics></math>, but also the variation of the moving second argument <span></span><math>\u0000 <semantics>\u0000 <msubsup>\u0000 <mi>x</mi>\u0000 <mi>j</mi>\u0000 <mo>∗</mo>\u0000 </msubsup>\u0000 <annotation>$x_j^*$</annotation>\u0000 </semantics></math>. The latter term requires a lower bound for the differential of the exponential map.</p><p>\u0000 </p><p>\u0000 </p><p>For completeness, we also correct the following two minor typographical errors. In the third assertion of Theorem 3.1, “<span></span><math>\u0000 <semantics>\u0000 <mi>max</mi>\u0000 <annotation>$max$</annotation>\u0000 </semantics></math>” should be replaced by “<span></span><math>\u0000 <semantics>\u0000 <mi>min</mi>\u0000 <annotation>$min$</annotation>\u0000 </semantics></math>,” and in Step D of the proof of Theorem 3.2, “Lemma 2.1” should be replaced by “Theorem 2.1.”</p><p>With the above corrections, the proof of Proposition 3.3 is complete. All other statements, estimates, proofs, examples, numerical simulations, and conclusions of the original article remain unchanged.</p><p>We apologizes for these omissions.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70288","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148785050","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":"A Unified Framework From Boltzmann Transport to Proton Treatment Planning","authors":"Andreas E. Kyprianou,&nbsp;Aaron Pim,&nbsp;Tristan Pryer","doi":"10.1111/sapm.70285","DOIUrl":"https://doi.org/10.1111/sapm.70285","url":null,"abstract":"<p>We develop a unified stochastic–deterministic framework for proton transport in radiotherapy. The deterministic formulation is based on a Boltzmann–Fokker–Planck (BFP) equation with continuous slowing-down, angular diffusion, and scattering terms, posed in a kinetic variational setting. The stochastic formulation describes proton trajectories by a track-length parameterized Markov process with energy loss, angular diffusion, and jump scattering. The main result is an operator-theoretic connection between these two descriptions. We identify the backward generator of the stochastic process with the adjoint BFP operator and show that the associated resolvent represents the Green's function of the deterministic transport equation. This gives a rigorous interpretation of deterministic fluence as an expected occupation density of stochastic particle paths. We then apply this duality to dose computation and treatment planning. Dose is represented equivalently as a deterministic fluence functional and as an expected accumulated stopping-power functional along stochastic tracks. Finally, we formulate a hybrid SDE–PDE optimization framework in which stochastic forward simulations are combined with deterministic adjoint equations for gradient computation. We prove existence and differentiability of the resulting objective functional and derive the corresponding first-order optimality system.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70285","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148785079","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":"Numerical Study of the Peregrine–Boussinesq System in 2D","authors":"Christian Klein,&nbsp;Jean-Claude Saut","doi":"10.1111/sapm.70291","DOIUrl":"https://doi.org/10.1111/sapm.70291","url":null,"abstract":"<div>\u0000 \u0000 <p>A numerical study of the 2D Peregrine–Boussinesq or Amick–Schonbek system is presented. Numerical evidence is given for the transverse stability of the 1D solitary waves that are line solitary waves of the 2D equations. It is shown that initial data not satisfying the noncavitation condition can lead to the formation of a gradient catastrophe in finite time. The numerical propagation of localized smooth initial data does not lead to the formation of stable structures localized in both spatial directions. For initial data satisfying the noncavitation condition, smooth solutions appear to exist for all times.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148784557","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}

{"title":"Construction of Weyl Matrix for Schrödinger Operators on Carbon Nano-Structures","authors":"Dong-Jie Wu,&nbsp;Chuan-Fu Yang,&nbsp;Natalia P. Bondarenko,&nbsp;Rafael Abeldinov","doi":"10.1111/sapm.70289","DOIUrl":"https://doi.org/10.1111/sapm.70289","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we develop a method for synthesis of the Weyl matrix for Schrödinger operators on carbon nano-structures which are equivalent to the hexagonal lattice. We construct Weyl matrices for graphs that contain any number of hexagons by adding new edges and solving elementary systems of linear algebraic equations at each step. Our method can be applied in numerical simulations for studying inverse spectral problems for differential operators on hexagonal lattices.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783977","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}

{"title":"Solitary Waves of Moderate Amplitude and Dispersive Radiation in the Serre Equations: The Extended KdV–Whitham Approximation","authors":"Benjamin Martin,&nbsp;Dmitri Tseluiko,&nbsp;Karima Khusnutdinova","doi":"10.1111/sapm.70283","DOIUrl":"https://doi.org/10.1111/sapm.70283","url":null,"abstract":"<p>We consider the extended Korteweg–de Vries (eKdV) equation as a model for long moderately nonlinear surface water waves and use it to describe the evolution of initial conditions generating solitary waves with and without significant dispersive radiation, as well as cases of pure dispersive radiation without any solitary waves. In the slow time formulation for the modeled solutions, this equation also generates fast propagating resonant forward radiation due to the non-convexity of its linear dispersion curve, which is not present in the direct numerical simulations of the strongly nonlinear Serre parent system (also known as the Su–Gardner and Green–Naghdi equations). We show that the extended KdV–Whitham approximation and the slow space formulation of the eKdV equation are suitable regularizations of the eKdV equation in several cases of interest. Importantly, unlike the KdV-type equations, it can be used to model waves of moderate amplitude. Numerical comparisons are made between the Serre system and several respective reduced models, where simulations are initiated with an approximate soliton solution of the eKdV equation, constructed by use of Kodama–Fokas–Liu near-identity transformation to the KdV equation, as well as a generic localized initial condition.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70283","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783984","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":"A Direct Approach for Detection of Bottom Topography in Shallow Water","authors":"Noureddine Lamsahel,&nbsp;Carole Rosier","doi":"10.1111/sapm.70290","DOIUrl":"https://doi.org/10.1111/sapm.70290","url":null,"abstract":"<p>We propose a fast, stable, and direct analytic method to detect underwater channel topography from surface wave measurements, based on one-dimensional shallow water equations. The technique requires knowledge of the free surface and its first two time derivatives at a single instant <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>t</mi>\u0000 <mi>★</mi>\u0000 </msup>\u0000 <annotation>$t^{star }$</annotation>\u0000 </semantics></math> above the fixed, bounded open segment of the domain. We first restructure the forward shallow water equations to obtain an inverse model in which the bottom profile is the only unknown, and then discretize this model using a second-order finite-difference scheme to infer the floor topography. We demonstrate that the approach satisfies a Lipschitz stability estimate and is independent of the initial conditions of the forward problem. The well-posedness of this inverse model requires that, at the chosen measurement time <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>t</mi>\u0000 <mi>★</mi>\u0000 </msup>\u0000 <annotation>$t^{star }$</annotation>\u0000 </semantics></math>, the discharge be strictly positive across the fixed portion of the open channel, which is automatically satisfied for steady and supercritical flows. For unsteady subcritical and transcritical flows, we derive two empirically validated sufficient conditions ensuring strict positivity after a sufficiently large time. The proposed methodology is tested on a range of scenarios, including classical benchmarks and different types of inlet discharges and bathymetries. We find that this analytic approach yields high approximation accuracy and that the bed profile reconstruction is stable under noise. In addition, the sufficient conditions are met across all tests.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70290","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783976","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":"Bulk-Surface Coupled PDE With an Open Boundary","authors":"Charles L. Epstein,&nbsp;Yoichiro Mori,&nbsp;Han Zhou","doi":"10.1111/sapm.70286","DOIUrl":"https://doi.org/10.1111/sapm.70286","url":null,"abstract":"<p>We study a bulk-surface coupled Laplace system involving an embedded open boundary. The problem is reformulated as an integro-differential equation using boundary integral representations, for which we establish existence and uniqueness of the solution. A Wiener–Hopf technique is employed to study the solution regularity and derive asymptotic expressions for the edge singularity. Building on these results, we develop a finite element method that incorporates the singularity structure, and we provide a rigorous error analysis. Numerical experiments confirm the theoretical convergence rates.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70286","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783599","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":"Exact Normal-Shock Solutions at Intermediate Prandtl Numbers","authors":"Raimund Feodor Perschke","doi":"10.1111/sapm.70287","DOIUrl":"https://doi.org/10.1111/sapm.70287","url":null,"abstract":"<div>\u0000 \u0000 <p>Exact solutions for the steady normal shock in the Navier–Stokes–Fourier framework for calorically perfect Newtonian gases are presented. The governing equations are reduced to an Abel differential equation of the second kind for general local transport laws of the longitudinal viscosity and thermal conductivity that maintain a constant longitudinal Prandtl number, while shock profiles are obtained by subsequent quadrature. For certain combinations of the specific heat ratio <span></span><math>\u0000 <semantics>\u0000 <mi>γ</mi>\u0000 <annotation>$gamma$</annotation>\u0000 </semantics></math>, longitudinal Prandtl number <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Pr</mi>\u0000 <mi>ℓ</mi>\u0000 </msub>\u0000 <annotation>$mathit {Pr_{ell }}$</annotation>\u0000 </semantics></math> and upstream Mach number <span></span><math>\u0000 <semantics>\u0000 <mi>Ma</mi>\u0000 <annotation>$mathit {Ma}$</annotation>\u0000 </semantics></math>, exact solutions are obtained on one-parameter curves in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>γ</mi>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>Pr</mi>\u0000 <mi>ℓ</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mi>Ma</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(gamma,mathit {Pr_{ell }},mathit {Ma})$</annotation>\u0000 </semantics></math>-space. The resulting longitudinal Prandtl numbers range from values close to zero up to 36. Exact shock profiles are developed for the constant-coefficient case. Most solutions determine the fluid velocity implicitly, while one solution yields an explicit representation of the velocity as a function of the spatial coordinate.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"157 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2026-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148783600","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}