A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of the work by A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020).
{"title":"Diffraction by a Dirichlet right angle on a discrete planar lattice","authors":"A. Shanin, A. I. Korolkov","doi":"10.1090/qam/1612","DOIUrl":"https://doi.org/10.1090/qam/1612","url":null,"abstract":"A problem of scattering by a Dirichlet right angle on a discrete square lattice is studied. The waves are governed by a discrete Helmholtz equation. The solution is looked for in the form of the Sommerfeld integral. The Sommerfeld transformant of the field is built as an algebraic function. The paper is a continuation of the work by A. V. Shanin and A. I. Korolkov, Sommerfeld-type integrals for discrete diffraction problems, Wave Motion 97 (2020).","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49028008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs. The kinetic Kuramoto equation governs the temporal-phase dynamics of the one-oscillator distribution function for an infinite ensemble of Kuramoto oscillators. When random inputs are absent in the coupling strength and initial data, it is well known that the incoherent state is nonlinearly stable in a subscritical regime where the coupling strength is below the critical coupling strength which is determined by the geometric shape of the distribution function for natural frequency. More precisely, Kuramoto order parameter measuring the fluctuations around the incoherent state tends to zero asymptotically and its decay mode depends on the regularity(smoothness) of natural frequency distribution function and initial datum. This phenomenon is called as Landau damping in the Kuramoto model in analogy with Landau damping arising from plasma physics. Our analytical results show that Landau damping is structurally robust with respect to random inputs at least in subscritical regime. As in the deterministic setting, the decay mode for the derivatives of the order parameter in random space can be either algebraic or exponential depending on the regularities of the initial datum and natural frequency distribution, respectively, and the smoothness for the order parameter in random space is determined by the smoothness of the coupling strength
{"title":"A local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs","authors":"Zhiyan Ding, Seung‐Yeal Ha, Shi Jin","doi":"10.1090/qam/1578","DOIUrl":"https://doi.org/10.1090/qam/1578","url":null,"abstract":"We present a local sensitivity analysis in Landau damping for the kinetic Kuramoto equation with random inputs. The kinetic Kuramoto equation governs the temporal-phase dynamics of the one-oscillator distribution function for an infinite ensemble of Kuramoto oscillators. When random inputs are absent in the coupling strength and initial data, it is well known that the incoherent state is nonlinearly stable in a subscritical regime where the coupling strength is below the critical coupling strength which is determined by the geometric shape of the distribution function for natural frequency. More precisely, Kuramoto order parameter measuring the fluctuations around the incoherent state tends to zero asymptotically and its decay mode depends on the regularity(smoothness) of natural frequency distribution function and initial datum. This phenomenon is called as Landau damping in the Kuramoto model in analogy with Landau damping arising from plasma physics. Our analytical results show that Landau damping is structurally robust with respect to random inputs at least in subscritical regime. As in the deterministic setting, the decay mode for the derivatives of the order parameter in random space can be either algebraic or exponential depending on the regularities of the initial datum and natural frequency distribution, respectively, and the smoothness for the order parameter in random space is determined by the smoothness of the coupling strength","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"1 1","pages":"1"},"PeriodicalIF":0.8,"publicationDate":"2020-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45188948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies both existence and spectral stability properties of bounded spatially periodic traveling wave solutions to a large class of scalar viscous balance laws in one space dimension with a reaction function of monostable or Fisher-KPP type. Under suitable structural assumptions, it is shown that this class of equations underlies two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a Hopf bifurcation around a critical value of the wave speed. The second family pertains to arbitrarily large period waves which arise from a homoclinic bifurcation and tend to a limiting traveling (homoclinic) pulse when their fundamental period tends to infinity. For both families, it is shown that the Floquet (continuous) spectrum of the linearization around the periodic waves intersects the unstable half plane of complex values with positive real part, a property known as spectral instability. For that purpose, in the case of small-amplitude waves it is proved that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution and determined by a dispersion relation which intersects the unstable complex half plane. In the case of large period waves, we verify that the family satisfies the assumptions of the seminal result by Gardner (Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math. 491 (1997), 149–181) of convergence of periodic spectra in the infinite-period limit to that of the underlying homoclinic wave, which is unstable. A few examples are discussed.
{"title":"Existence and spectral instability of bounded spatially periodic traveling waves for scalar viscous balance laws","authors":"E. Alvarez, R. Plaza","doi":"10.1090/QAM/1591","DOIUrl":"https://doi.org/10.1090/QAM/1591","url":null,"abstract":"This paper studies both existence and spectral stability properties of bounded spatially periodic traveling wave solutions to a large class of scalar viscous balance laws in one space dimension with a reaction function of monostable or Fisher-KPP type. Under suitable structural assumptions, it is shown that this class of equations underlies two families of periodic waves. The first family consists of small amplitude waves with finite fundamental period which emerge from a Hopf bifurcation around a critical value of the wave speed. The second family pertains to arbitrarily large period waves which arise from a homoclinic bifurcation and tend to a limiting traveling (homoclinic) pulse when their fundamental period tends to infinity. For both families, it is shown that the Floquet (continuous) spectrum of the linearization around the periodic waves intersects the unstable half plane of complex values with positive real part, a property known as spectral instability. For that purpose, in the case of small-amplitude waves it is proved that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution and determined by a dispersion relation which intersects the unstable complex half plane. In the case of large period waves, we verify that the family satisfies the assumptions of the seminal result by Gardner (Spectral analysis of long wavelength periodic waves and applications, J. Reine Angew. Math. 491 (1997), 149–181) of convergence of periodic spectra in the infinite-period limit to that of the underlying homoclinic wave, which is unstable. A few examples are discussed.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48049711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.
{"title":"The relative entropy method for inhomogeneous systems of balance laws","authors":"C. Christoforou","doi":"10.1090/qam/1577","DOIUrl":"https://doi.org/10.1090/qam/1577","url":null,"abstract":"General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/qam/1577","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42283157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder's fixed point theorem and semigroup theory.
{"title":"Strong solutions to a nonlocal-in-time semilinear heat equation","authors":"Christoph Walker","doi":"10.1090/QAM/1579","DOIUrl":"https://doi.org/10.1090/QAM/1579","url":null,"abstract":"Existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder's fixed point theorem and semigroup theory.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":"1"},"PeriodicalIF":0.8,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48555586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1990, von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector field u u in a 3D bounded domain that is tangential to the boundary can be written as the curl of a vector field vanishing on the boundary of the domain. We extend this result to higher dimension and to Lipschitz boundaries in a form suitable for integration in flat space, showing that u u can be written as the divergence of an antisymmetric matrix field. We also demonstrate how obtaining a kernel for such a matrix field is dual to obtaining a Biot-Savart kernel for the domain.
{"title":"Stream functions for divergence-free vector fields","authors":"J. Kelliher","doi":"10.1090/qam/1575","DOIUrl":"https://doi.org/10.1090/qam/1575","url":null,"abstract":"In 1990, von Wahl and, independently, Borchers and Sohr showed that a divergence-free vector field \u0000\u0000 \u0000 u\u0000 u\u0000 \u0000\u0000 in a 3D bounded domain that is tangential to the boundary can be written as the curl of a vector field vanishing on the boundary of the domain. We extend this result to higher dimension and to Lipschitz boundaries in a form suitable for integration in flat space, showing that \u0000\u0000 \u0000 u\u0000 u\u0000 \u0000\u0000 can be written as the divergence of an antisymmetric matrix field. We also demonstrate how obtaining a kernel for such a matrix field is dual to obtaining a Biot-Savart kernel for the domain.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46122898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 are interested in a semilinear Mindlin-Timoshenko system with an irrotationality condition for the angle variables. Under the smallness assumption on the initial data, the decay rate of solutions is obtained by using both the multiplier method in the Fourier space and the fundamental solution method. Moreover, we also prove the global existence of solutions by the standard method.
{"title":"Decay estimate and global existence of a semilinear Mindlin-Timoshenko plate system with full frictional damping in the whole space","authors":"Kaiqiang Li, R. Xue","doi":"10.1090/qam/1569","DOIUrl":"https://doi.org/10.1090/qam/1569","url":null,"abstract":"In this paper, we are interested in a semilinear Mindlin-Timoshenko system with an irrotationality condition for the angle variables. Under the smallness assumption on the initial data, the decay rate of solutions is obtained by using both the multiplier method in the Fourier space and the fundamental solution method. Moreover, we also prove the global existence of solutions by the standard method.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"78 1","pages":"703-724"},"PeriodicalIF":0.8,"publicationDate":"2020-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/qam/1569","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47870827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The interaction of linear waves with periodic structures arises in a broad range of scientific and engineering applications. For such problems it is often mandatory that numerical simulations be rapid, robust, and highly accurate. With such qualities in mind High-Order Spectral methods are often utilized, and in this paper we describe and test a perturbative method which fits into this class. Here we view the inhomogeneous (but laterally periodic) permittivity as a perturbation of a constant value and pursue (regular) perturbation theory. We demonstrate that not only does this lead to a fast and accurate numerical method, but also that the expansion of the field in this geometric parameter is valid for large deformations (up to topological obstruction). Finally, we show that, if the permittivity deformation is spatially analytic, then so is the field scattered by it.
{"title":"A high-order perturbation of envelopes (HOPE) method for scattering by periodic inhomogeneous media","authors":"D. Nicholls","doi":"10.1090/qam/1568","DOIUrl":"https://doi.org/10.1090/qam/1568","url":null,"abstract":"The interaction of linear waves with periodic structures arises in a broad range of scientific and engineering applications. For such problems it is often mandatory that numerical simulations be rapid, robust, and highly accurate. With such qualities in mind High-Order Spectral methods are often utilized, and in this paper we describe and test a perturbative method which fits into this class. Here we view the inhomogeneous (but laterally periodic) permittivity as a perturbation of a constant value and pursue (regular) perturbation theory. We demonstrate that not only does this lead to a fast and accurate numerical method, but also that the expansion of the field in this geometric parameter is valid for large deformations (up to topological obstruction). Finally, we show that, if the permittivity deformation is spatially analytic, then so is the field scattered by it.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44153256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 a scheme for the active manipulation of electromagnetic fields in prescribed exterior regions using a surface source. We prove the existence of the necessary surface current (electric or magnetic) on a single source to approximate prescribed electromagnetic fields on given regions of space (bounded or possibly the far field). We provide two constructive schemes for the computation of the required surface currents: our first strategy makes use of the Debye representation results for the electromagnetic field and builds up on previous control results for scalar fields discussed in [J. Integral Equations Appl. 26 (2014), pp. 553–579]; the second strategy we propose makes use of integral electromagnetic representation results and follows theoretically from the first. We provide theoretical validation for both computational schemes and present supporting numerical simulations for the first strategy in several applied scenarios.
{"title":"Active manipulation of exterior electromagnetic fields by using surface sources","authors":"D. Onofrei, E. Platt, N. J. A. Egarguin","doi":"10.1090/qam/1567","DOIUrl":"https://doi.org/10.1090/qam/1567","url":null,"abstract":"In this paper, we establish a scheme for the active manipulation of electromagnetic fields in prescribed exterior regions using a surface source. We prove the existence of the necessary surface current (electric or magnetic) on a single source to approximate prescribed electromagnetic fields on given regions of space (bounded or possibly the far field). We provide two constructive schemes for the computation of the required surface currents: our first strategy makes use of the Debye representation results for the electromagnetic field and builds up on previous control results for scalar fields discussed in [J. Integral Equations Appl. 26 (2014), pp. 553–579]; the second strategy we propose makes use of integral electromagnetic representation results and follows theoretically from the first. We provide theoretical validation for both computational schemes and present supporting numerical simulations for the first strategy in several applied scenarios.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"78 1","pages":"641-670"},"PeriodicalIF":0.8,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/qam/1567","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45433224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with a nonlinear system of reaction-diffusion partial differential equations modelling the evolution of a prey-predator biological system with chemotaxis. The system is constituted by three coupled equations: a fully parabolic chemotaxis system describing the behavior of the active predators and preys and an ordinary equation, describing the dynamics of the dormant predators, coupled to it. Chemotaxis in this context affects the active predators so that they move towards the regions where the density of resting eggs (dormant predators) is higher. Under suitable assumptions on the initial data and the coefficients of the system, the global-in-time existence of classical solutions is proved in any space dimension. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system. The theoretical and numerical findings show that the model considered here can provide very interesting and complex dynamics.
{"title":"Uniform boundedness for a predator-prey system with chemotaxis and dormancy of predators","authors":"R. Dáger, Víctor Navarro, M. Negreanu","doi":"10.1090/qam/1583","DOIUrl":"https://doi.org/10.1090/qam/1583","url":null,"abstract":"This paper deals with a nonlinear system of reaction-diffusion partial differential equations modelling the evolution of a prey-predator biological system with chemotaxis. The system is constituted by three coupled equations: a fully parabolic chemotaxis system describing the behavior of the active predators and preys and an ordinary equation, describing the dynamics of the dormant predators, coupled to it. Chemotaxis in this context affects the active predators so that they move towards the regions where the density of resting eggs (dormant predators) is higher. Under suitable assumptions on the initial data and the coefficients of the system, the global-in-time existence of classical solutions is proved in any space dimension. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system. The theoretical and numerical findings show that the model considered here can provide very interesting and complex dynamics.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60584610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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