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.
{"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}
We offer a modern interpretation of Lagrangian hydrodynamics as employed in Lagrangian simulations of compressible fluid flow. Our main result is to show that artificial viscosity, traditionally viewed as a numerical artifice to control unphysical oscillations in flows with shocks, actually represents a physical process and is necessary to derive accurate simulations in any compressible flow. We begin by reviewing the origins of two numerical devices, artificial viscosity and finite‐volume methods. We proceed to construct a mathematical (PDE) model that incorporates those numerics and in which a new length scale, the observer, arises representing the discretization. Associated with that length scale, there are new inviscid fluxes that are the artificial viscosity as first formulated by Richtmyer and an artificial heat flux postulated by Noh but typically not included in Lagrangian codes. We discuss the connection of our results to bivelocity hydrodynamics. We conclude with some speculation as to the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.
{"title":"A modern concept of Lagrangian hydrodynamics","authors":"L. G. Margolin, J. M. Canfield","doi":"10.1111/sapm.12754","DOIUrl":"https://doi.org/10.1111/sapm.12754","url":null,"abstract":"We offer a modern interpretation of Lagrangian hydrodynamics as employed in Lagrangian simulations of compressible fluid flow. Our main result is to show that artificial viscosity, traditionally viewed as a numerical artifice to control unphysical oscillations in flows with shocks, actually represents a physical process and is necessary to derive accurate simulations in any compressible flow. We begin by reviewing the origins of two numerical devices, artificial viscosity and finite‐volume methods. We proceed to construct a mathematical (PDE) model that incorporates those numerics and in which a new length scale, the observer, arises representing the discretization. Associated with that length scale, there are new inviscid fluxes that are the artificial viscosity as first formulated by Richtmyer and an artificial heat flux postulated by Noh but typically not included in Lagrangian codes. We discuss the connection of our results to bivelocity hydrodynamics. We conclude with some speculation as to the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178333","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 spreading solutions of the nonlinear diffusion equation when the far‐field diffusivity is small. The method of strained coordinates is used to construct a uniform asymptotic correction to the similarity solution of the unperturbed problem. The equation provides a possible analogue to similar models of fluid jets and plumes.
{"title":"Weak long‐range diffusion","authors":"R. S. Garvey, A. C. Fowler","doi":"10.1111/sapm.12759","DOIUrl":"https://doi.org/10.1111/sapm.12759","url":null,"abstract":"We study the spreading solutions of the nonlinear diffusion equation when the far‐field diffusivity is small. The method of strained coordinates is used to construct a uniform asymptotic correction to the similarity solution of the unperturbed problem. The equation provides a possible analogue to similar models of fluid jets and plumes.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178332","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}
This paper investigates a class of planar Schrödinger–Poisson systems with critical exponential growth. The conditions for the local well‐posedness of the Cauchy problem in the energy space are defined. By introducing innovative ideas and relaxing some of the classical growth assumptions on nonlinearity, this study shows that such a system has at least two standing waves with a prescribed mass. One wave is a ground‐state standing wave with positive energy, and another one is a high‐energy standing wave with positive energy. In addition, with the help of the local well‐posedness, it is shown that the set of ground‐state standing waves is orbitally stable.
{"title":"Planar Schrödinger–Poisson system with exponential critical growth: Local well‐posedness and standing waves with prescribed mass","authors":"Juntao Sun, Shuai Yao, Jian Zhang","doi":"10.1111/sapm.12760","DOIUrl":"https://doi.org/10.1111/sapm.12760","url":null,"abstract":"This paper investigates a class of planar Schrödinger–Poisson systems with critical exponential growth. The conditions for the local well‐posedness of the Cauchy problem in the energy space are defined. By introducing innovative ideas and relaxing some of the classical growth assumptions on nonlinearity, this study shows that such a system has at least two standing waves with a prescribed mass. One wave is a ground‐state standing wave with positive energy, and another one is a high‐energy standing wave with positive energy. In addition, with the help of the local well‐posedness, it is shown that the set of ground‐state standing waves is orbitally stable.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178335","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 consider the motion of a “magnetic” soliton in two‐component condensates along a nonuniform and time‐dependent background in the framework of Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related to its inverse half‐width by the dispersion law for linear waves continued to the region of complex wave numbers. We obtain expressions for the canonical momentum and the Hamiltonian as functions of soliton's velocity and transform the Hamilton equations to a Newton‐like equation. The theory is illustrated by several examples of concrete soliton's dynamics.
{"title":"Hamiltonian mechanics of “magnetic” solitons in two‐component Bose–Einstein condensates","authors":"A. M. Kamchatnov","doi":"10.1111/sapm.12757","DOIUrl":"https://doi.org/10.1111/sapm.12757","url":null,"abstract":"We consider the motion of a “magnetic” soliton in two‐component condensates along a nonuniform and time‐dependent background in the framework of Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related to its inverse half‐width by the dispersion law for linear waves continued to the region of complex wave numbers. We obtain expressions for the canonical momentum and the Hamiltonian as functions of soliton's velocity and transform the Hamilton equations to a Newton‐like equation. The theory is illustrated by several examples of concrete soliton's dynamics.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178334","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}
Anna Ghazaryan, Stéphane Lafortune, Yuri Latushkin, Vahagn Manukian
We consider a diffusive Rosenzweig–MacArthur predator–prey model in the situation when the prey diffuses at a rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher–KPP (Kolmogorov–Petrovski–Piskunov) equation and the fronts supported by the full system are small perturbations of the Fisher–KPP fronts. The existence proof is based on the application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We show that, for some parameter regime, the fronts are spectrally and asymptotically stable using energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher–KPP equation. Furthermore, we perform numerical computations of the Evans function to explicitly identify regions in the parameter space where the fronts are spectrally stable.
{"title":"Stability of fronts in the diffusive Rosenzweig–MacArthur model","authors":"Anna Ghazaryan, Stéphane Lafortune, Yuri Latushkin, Vahagn Manukian","doi":"10.1111/sapm.12755","DOIUrl":"https://doi.org/10.1111/sapm.12755","url":null,"abstract":"We consider a diffusive Rosenzweig–MacArthur predator–prey model in the situation when the prey diffuses at a rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher–KPP (Kolmogorov–Petrovski–Piskunov) equation and the fronts supported by the full system are small perturbations of the Fisher–KPP fronts. The existence proof is based on the application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We show that, for some parameter regime, the fronts are spectrally and asymptotically stable using energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher–KPP equation. Furthermore, we perform numerical computations of the Evans function to explicitly identify regions in the parameter space where the fronts are spectrally stable.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178336","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}
Dimitrios A. Mitsoudis, Michael Plexousakis, George N. Makrakis, Charalambos Makridakis
This work is devoted to the numerical solution of the Helmholtz equation with variable wave number and including a point source in appropriately truncated infinite domains. Motivated by a two‐dimensional model, we formulate a simplified one‐dimensional model. We study its well posedness via wave number explicit stability estimates and prove convergence of the finite element approximations. As a proof of concept, we present the outcome of some numerical experiments for various wave number configurations. Our experiments indicate that the introduction of the artificial boundary near the source and the associated boundary condition lead to an efficient model that accurately captures the wave propagation features.
{"title":"Approximations of the Helmholtz equation with variable wave number in one dimension","authors":"Dimitrios A. Mitsoudis, Michael Plexousakis, George N. Makrakis, Charalambos Makridakis","doi":"10.1111/sapm.12756","DOIUrl":"https://doi.org/10.1111/sapm.12756","url":null,"abstract":"This work is devoted to the numerical solution of the Helmholtz equation with variable wave number and including a point source in appropriately truncated infinite domains. Motivated by a two‐dimensional model, we formulate a simplified one‐dimensional model. We study its well posedness via wave number explicit stability estimates and prove convergence of the finite element approximations. As a proof of concept, we present the outcome of some numerical experiments for various wave number configurations. Our experiments indicate that the introduction of the artificial boundary near the source and the associated boundary condition lead to an efficient model that accurately captures the wave propagation features.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178338","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}
This work utilizes an analytic expression for a model of acoustic propagation in a two‐layered, inhomogeneous atmosphere developed by the authors. The model is used to study the atmospheres of Earth, Mars, and Titan. In particular, vertical wave propagation in these atmospheres is studied. The effect(s) of a two‐layered, inhomogeneous atmosphere on vertical, acoustic propagation due to a time‐harmonic, point source are examined. An adiabatic atmosphere is used for the bottom layer (troposphere) and an isothermal one for the top (stratosphere). The derived, analytic solution is expressed in terms of the acoustic pressure fluctuations. For the adiabatic layers, the solutions satisfy Bessel's equation for orders of , and for Earth, Mars, and Titan, respectively. The Bessel function's argument is , where and are dimensionless frequency and height, respectively. For the isothermal layer, the solution represents a damped, harmonic oscillator with a cutoff value of . Only values greater than are considered. The analysis and results are reported for combinations of single‐ and double‐layer atmospheres in the presence of a source on given boundaries. Acoustic propagation and transmission loss results are shown and discussed for all three planetary bodies: Earth, Mars, and Titan.
{"title":"A two‐layered, analytically‐tractable, atmospheric model applied to Earth, Mars, and Titan with sources","authors":"Edward J. Yoerger, Ashok Puri","doi":"10.1111/sapm.12753","DOIUrl":"https://doi.org/10.1111/sapm.12753","url":null,"abstract":"This work utilizes an analytic expression for a model of acoustic propagation in a two‐layered, inhomogeneous atmosphere developed by the authors. The model is used to study the atmospheres of Earth, Mars, and Titan. In particular, vertical wave propagation in these atmospheres is studied. The effect(s) of a two‐layered, inhomogeneous atmosphere on vertical, acoustic propagation due to a time‐harmonic, point source are examined. An adiabatic atmosphere is used for the bottom layer (troposphere) and an isothermal one for the top (stratosphere). The derived, analytic solution is expressed in terms of the acoustic pressure fluctuations. For the adiabatic layers, the solutions satisfy Bessel's equation for orders of , and for Earth, Mars, and Titan, respectively. The Bessel function's argument is , where and are dimensionless frequency and height, respectively. For the isothermal layer, the solution represents a damped, harmonic oscillator with a cutoff value of . Only values greater than are considered. The analysis and results are reported for combinations of single‐ and double‐layer atmospheres in the presence of a source on given boundaries. Acoustic propagation and transmission loss results are shown and discussed for all three planetary bodies: Earth, Mars, and Titan.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178337","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 investigate a reaction–diffusion–advection benthic–drift model, where the population is divided into two interacting groups: individuals dispersing in the drift zone and individuals living in the benthic zone. For different growth types of the benthic population, we present asymptotic profiles of positive steady states in three cases: (i) large advection; (ii) small diffusion of the drift population; and (iii) large diffusion of the drift population. We prove that in case (i) both the benthic and drift individuals concentrate only at the downstream end; in case (ii), both benthic and drift population reside inhomogeneously in , stay away from the upstream end , and concentrate only at the downstream ; and in case (iii), the drift species distributes evenly on the entire habitat and the benthic species distributes inhomogeneously throughout the habitat. The result supplements the dynamical behaviors of benthic–drift models developed in earlier works and is also of its own interest.
{"title":"Asymptotic profiles of positive steady states in a reaction–diffusion benthic–drift model","authors":"Anqi Qu, Jinfeng Wang","doi":"10.1111/sapm.12752","DOIUrl":"https://doi.org/10.1111/sapm.12752","url":null,"abstract":"In this paper, we investigate a reaction–diffusion–advection benthic–drift model, where the population is divided into two interacting groups: individuals dispersing in the drift zone and individuals living in the benthic zone. For different growth types of the benthic population, we present asymptotic profiles of positive steady states in three cases: (i) large advection; (ii) small diffusion of the drift population; and (iii) large diffusion of the drift population. We prove that in case (i) both the benthic and drift individuals concentrate only at the downstream end; in case (ii), both benthic and drift population reside inhomogeneously in , stay away from the upstream end , and concentrate only at the downstream ; and in case (iii), the drift species distributes evenly on the entire habitat and the benthic species distributes inhomogeneously throughout the habitat. The result supplements the dynamical behaviors of benthic–drift models developed in earlier works and is also of its own interest.","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142178339","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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