Studies in Applied Mathematics最新文献

{"title":"Exponentially localized interface eigenmodes in finite chains of resonators","authors":"Habib Ammari, Silvio Barandun, Bryn Davies, Erik Orvehed Hiltunen, Thea Kosche, Ping Liu","doi":"10.1111/sapm.12765","DOIUrl":"https://doi.org/10.1111/sapm.12765","url":null,"abstract":"This paper studies wave localization in chains of finitely many resonators. There is an extensive theory predicting the existence of localized modes induced by defects in infinitely periodic systems. This work extends these principles to finite‐sized systems. We consider one‐dimensional, finite systems of subwavelength resonators arranged in dimers that have a geometric defect in the structure. This is a classical wave analog of the Su–Schrieffer–Heeger model. We prove the existence of a spectral gap for defectless finite dimer structures and find a direct relationship between eigenvalues being within the spectral gap and the localization of their associated eigenmode. Then, for sufficiently large‐size systems, we show the existence and uniqueness of an eigenvalue in the gap in the defect structure, proving the existence of a unique localized interface mode. To the best of our knowledge, our method, based on Chebyshev polynomials, is the first to characterize quantitatively the localized interface modes in systems of finitely many resonators.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261639","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":"Inverse scattering transform for continuous and discrete space‐time‐shifted integrable equations","authors":"Mark J. Ablowitz, Ziad H. Musslimani, Nicholas J. Ossi","doi":"10.1111/sapm.12764","DOIUrl":"https://doi.org/10.1111/sapm.12764","url":null,"abstract":"Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space‐time‐shifted nonlinear Schrödinger (NLS) and space‐time‐shifted modified Korteweg‐de Vries equations are studied. Additionally, the semidiscrete IST is developed for the time, space, and space‐time‐shifted variants of the Ablowitz–Ladik integrable discretization of the NLS. One‐soliton solutions are constructed for all continuous and discrete cases.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261604","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":"Impact of mixed boundary conditions and nonsmooth data on layer‐originated nonpremixed combustion problems: Higher‐order convergence analysis","authors":"Shridhar Kumar, Ishwariya R, Pratibhamoy Das","doi":"10.1111/sapm.12763","DOIUrl":"https://doi.org/10.1111/sapm.12763","url":null,"abstract":"This work explores the theoretical and computational impacts of mixed‐type flux conditions and nonsmooth data on boundary/interior layer‐originated singularly perturbed semilinear reaction–diffusion problems. Such problems are prevalent in nonpremixed combustion models and catalytic reaction models. The inclusion of arbitrarily small diffusion terms results in boundary layers influenced by specific flux conditions normalized by perturbation parameters. We have demonstrated theoretically that the sharpness of the boundary layer is significantly reduced when this normalization is independent of the diffusion parameter. In addition, the presence of a nonsmooth source function gives rise to an interior layer in the current problem. We show that using upwind discretizations for mixed boundary fluxes achieves nearly second‐order accuracy if the first derivatives are not normalized concerning perturbation parameters. This outcome arises from the bounds of a prior derivative of the continuous solution. Furthermore, it is theoretically shown that nearly second‐order uniform accuracy can be attained for reaction‐dominated semilinear problems using a hybrid scheme at the discontinuity point. To ensure the uniform stability of the discrete solution, a transformation is necessary for the corresponding discrete problem. Theoretical results are supported by various experiments on nonlinear problems, illustrating the pointwise rates and highlighting both linear and higher‐order accuracy at specific points.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261605","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":"Predator–prey systems with a variable habitat for predators in advective environments","authors":"Baifeng Zhang, Xianning Liu, Yangjiang Wei","doi":"10.1111/sapm.12758","DOIUrl":"https://doi.org/10.1111/sapm.12758","url":null,"abstract":"Community composition in aquatic environments can be shaped by a broad array of factors, encompassing habitat conditions in addition to abiotic conditions and biotic interactions. This paper pertains to reaction–diffusion–advection predator–prey model featuring a variable predator habitat in advective environments, governed by a unidirectional flow. First, we establish the near‐complete global dynamics of the system. In instances where the functional response to predation conforms to Holling‐type I or II, we explore the uniqueness and stability of positive steady‐state solutions via the application of particular auxiliary techniques, the comparison principle for parabolic equations, and perturbation analysis. Furthermore, we obtain the critical positions at the upper and lower boundaries of the predator's habitat, which determine the survival of the prey irrespective of the predator's growth rate. Finally, we show how the location and length of the predator's habitat affect the persistence and extinction of predators and prey in the event of a small population loss rate at the downstream end. From the biological point of view, these results contribute to our deeper understanding of the effects of habitat on aquatic populations and may have applications in aquaculture and the establishment of protection zones for aquatic species.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142261606","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":"Two‐phase magma flow with phase exchange: Part II. 1.5D numerical simulations of a volcanic conduit","authors":"Alain Burgisser, Marielle Collombet, Gladys Narbona‐Reina, Didier Bresch","doi":"10.1111/sapm.12747","DOIUrl":"https://doi.org/10.1111/sapm.12747","url":null,"abstract":"In a review paper in this same volume, we present the state of the art on modeling of compressible viscous flows ranging from single‐phase to two‐phase systems. It focuses on mathematical properties related to weak stability because they are important for numerical resolution and on the homogenization process that leads from a microscopic description of two separate phases to an averaged two‐phase model. This review serves as the foundation for Parts I and II, which present averaged two‐phase models with phase exchange applicable to magma flow during volcanic eruptions. Part I establishes a two‐phase transient conduit flow model ensuring: (1) mass and volatile species conservation, (2) disequilibrium degassing considering both viscous relaxation and volatile diffusion, and (3) dissipation of total energy. The relaxation limit of this model is then used to obtain a drift‐flux system amenable to simplification. Here, in Part II, we summarize this model and propose a 1.5D simplification of it that alleviates three issues causing difficulties in its numerical implementation. We compare our model outputs to those of another steady‐state, equilibrium degassing, isothermal model under conditions typical of an effusive eruption at an andesitic volcano. Perfect equilibrium degassing is unreachable with a realistic water diffusion coefficient because conduit extremities always contain melt supersaturated with water. Such supersaturation has minor consequences on mass discharge rate. In contrast, releasing the isothermal assumption reduces significantly mass discharge rate by cooling due to gas expansion, which in turn increases liquid viscosity. We propose a simplified system using Darcy's law and omitting several processes such as shear heating and liquid inertia. This minimal system is not dissipative but approximates the steady‐state mass discharge rate of the full system within 10%. A regime diagram valid under a limited set of conditions indicates when this minimal system captures the ascent dynamics of effusive eruptions. Interestingly, the two novel aspects of the full model, diffusive degassing and heat balance, cannot be neglected. In some cases with high diffusion coefficients, a shallow region where porosity and velocities tend toward zero develops initially, possibly blocking an eventual steady state. This local porosity loss also occurs when a steady‐state solution is subjected to a change in shallow permeability. The resulting shallow porosity loss features many characteristics of a plug developing prior to a Vulcanian eruption.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178328","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":"Two‐phase magma flow with phase exchange: Part I. Physical modeling of a volcanic conduit","authors":"Gladys Narbona‐Reina, Didier Bresch, Alain Burgisser, Marielle Collombet","doi":"10.1111/sapm.12741","DOIUrl":"https://doi.org/10.1111/sapm.12741","url":null,"abstract":"In a review paper in this same volume, we present the state of the art on modeling of compressible viscous flows ranging from single‐phase to two‐phase systems. It focuses on mathematical properties related to weak stability because they are important for numerical resolution and on the homogenization process that leads from a microscopic description of two separate phases to an averaged two‐phase model. This review serves as the foundation for Parts I and II, which present averaged two‐phase models with phase exchange applicable to magma flow during volcanic eruptions. Here, in Part I, after introducing the physical processes occurring in a volcanic conduit, we detail the steps needed at both microscopic and macroscopic scales to obtain a two‐phase transient conduit flow model ensuring: (1) mass and volatile species conservation, (2) disequilibrium degassing considering both viscous relaxation and volatile diffusion, and (3) dissipation of total energy. The resulting compressible/incompressible system has eight transport equations on eight unknowns (gas volume fraction and density, dissolved water content, liquid pressure, and the velocity and temperature of both phases) as well as algebraic closures for gas pressure and bubble radius. We establish valid sets of boundary conditions such as imposing pressures and stress‐free conditions at the conduit outlet and either velocity or pressure at the inlet. This model is then used to obtain a drift‐flux system that isolates the effects of relative velocities, pressures, and temperatures. The dimensional analysis of this drift‐flux system suggests that relative velocities can be captured with a Darcy equation and that gas–liquid pressure differences partly control magma acceleration. Unlike the vanishing small gas–liquid temperature differences, bulk magma temperature is expected to vary because of gas expansion. Mass exchange being a major control of flow dynamics, we propose a limit case of mass exchange by establishing a relaxed system at chemical equilibrium. This single‐velocity, single‐temperature system is a generalization of an existing volcanic conduit flow model. Finally, we compare our full compressible/incompressible system to another existing volcanic conduit flow model where both phases are compressible. This comparison illustrates that different two‐phase systems may be obtained depending on the governing unknowns chosen. Part II presents a 1.5D version of the model established herein that is solved numerically. The numerical outputs are compared to those of another steady‐state, equilibrium degassing, isothermal model under conditions typical of an effusive eruption at an andesitic volcano.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178292","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":"Mathematical topics in compressible flows from single‐phase systems to two‐phase averaged systems","authors":"Didier Bresch, Gladys Narbona‐Reina, Alain Burgisser, Marielle Collombet","doi":"10.1111/sapm.12739","DOIUrl":"https://doi.org/10.1111/sapm.12739","url":null,"abstract":"We review the modeling and mathematical properties of compressible viscous flows, ranging from single‐phase systems to two‐phase systems, with a focus on the occurrence of oscillations and/or concentrations. We explain how establishing the existence of nonlinear weak stability ensures that no such instabilities occur in the solutions because of the system formulation. When oscillation/concentration are inherent to the nature of the physical situation modeled, we explain how the averaging procedure by homogenization helps to understand their effect on the averaged system. This review addresses systems of progressive complexity. We start by focusing on nonlinear weak stability—a crucial property for numerical simulations and well posedness—in single‐phase viscous systems. We then show how a two‐phase immiscible system may be rewritten as a single‐phase system. Conversely, we show then how to derive a two‐phase averaged system from a two‐phase immiscible system by homogenization. As in many homogenization problems, this is an example where physical oscillation/concentration occur. We then focus on two‐phase averaged viscous systems and present results on the nonlinear weak stability necessary for the convergence of numerical schemes. Finally, we review some singular limits frequently developed to obtain drift–flux systems. Additionally, the appendix provides a crash course on basic functional analysis tools for partial differential equation (PDE) and homogenization (averaging procedures) for readers unfamiliar with them. This review serves as the foundation for two subsequent papers (Part I and Part II in this same volume), which present averaged two‐phase models with phase exchange applicable to magma flow during volcanic eruptions. Part I introduces the physical processes occurring in a volcanic conduit and establishes a two‐phase transient conduit flow model ensuring: (1) mass and volatile species conservation, (2) disequilibrium degassing considering both viscous relaxation and volatile diffusion, and (3) dissipation of total energy. The relaxation limit of this model is then used to obtain a drift–flux system amenable to simplification. Part II revisits the model introduced in Part I and proposes a 1.5D simplification that addresses issues in its numerical implementation. Model outputs are compared to those of another well‐established model under conditions typical of an effusive eruption at an andesitic volcano.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178571","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":"General boundary conditions for a Boussinesq model with varying bathymetry","authors":"David Lannes, Mathieu Rigal","doi":"10.1111/sapm.12751","DOIUrl":"https://doi.org/10.1111/sapm.12751","url":null,"abstract":"This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178330","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":"Asymptotics of the partition function of the perturbed Gross–Witten–Wadia unitary matrix model","authors":"Yu Chen, Shuai‐Xia Xu, Yu‐Qiu Zhao","doi":"10.1111/sapm.12762","DOIUrl":"https://doi.org/10.1111/sapm.12762","url":null,"abstract":"We consider the asymptotics of the partition function of the extended Gross–Witten–Wadia unitary matrix model by introducing an extra logarithmic term in the potential. The partition function can be written as a Toeplitz determinant with entries expressed in terms of the modified Bessel functions of the first kind and furnishes a ‐function sequence of the Painlevé equation. We derive the asymptotic expansions of the Toeplitz determinant up to and including the constant terms as the size of the determinant tends to infinity. The constant terms therein are expressed in terms of the Riemann zeta‐function and the Barnes ‐function. A third‐order phase transition in the leading terms of the asymptotic expansions is also observed.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178331","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":"Bifurcation structure of steady states for a cooperative model with population flux by attractive transition","authors":"Masahiro Adachi, Kousuke Kuto","doi":"10.1111/sapm.12761","DOIUrl":"https://doi.org/10.1111/sapm.12761","url":null,"abstract":"This paper studies the steady states to a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. The first result gives many bifurcation points on the branch of the positive constant solution under the weak cooperative condition. The second result shows every steady state approaches a solution of the scalar field equation as the coefficients of the flux tend to infinity. Indeed, the numerical simulation using pde2path exhibits the global bifurcation branch of the cooperative model with large population flux is near that of the scalar field equation.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178329","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}