We address the nonlinear Schrodinger equation with intensity-dependent dispersion which was recently proposed in the context of nonlinear optical systems. Contrary to the previous findings, we prove that no solitary wave solutions exist if the sign of the intensity-dependent dispersion coincides with the sign of the constant dispersion, whereas a continuous family of such solutions exists in the case of the opposite signs. The family includes two particular solutions, namely cusped and bell-shaped solitons, where the former represents the lowest energy state in the family and the latter is a limit of solitary waves in a regularized system. We further analyze the delicate analytical properties of these solitary waves such as the asymptotic behavior near singularities, the spectral stability, and the convergence of the fixed-point iterations near such solutions. The analytical theory is corroborated by means of numerical approximations.
{"title":"Localization in optical systems with an intensity-dependent dispersion","authors":"R. M. Ross, P. Kevrekidis, Dmitry E. Pelinovsky","doi":"10.1090/QAM/1596","DOIUrl":"https://doi.org/10.1090/QAM/1596","url":null,"abstract":"We address the nonlinear Schrodinger equation with intensity-dependent dispersion which was recently proposed in the context of nonlinear optical systems. Contrary to the previous findings, we prove that no solitary wave solutions exist if the sign of the intensity-dependent dispersion coincides with the sign of the constant dispersion, whereas a continuous family of such solutions exists in the case of the opposite signs. The family includes two particular solutions, namely cusped and bell-shaped solitons, where the former represents the lowest energy state in the family and the latter is a limit of solitary waves in a regularized system. We further analyze the delicate analytical properties of these solitary waves such as the asymptotic behavior near singularities, the spectral stability, and the convergence of the fixed-point iterations near such solutions. The analytical theory is corroborated by means of numerical approximations.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41607968","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}
Seung‐Yeal Ha, Myeongju Kang, Hansol Park, T. Ruggeri, Woojoo Shim
We consider the thermodynamic Kuramoto model proposed in cite{H-P-R-S}. For each oscillator in thermodynamic Kuramoto model, there is a coupling effect between the phase and the temperature field. For such a model, we study a uniform stability and uniform-in-time mean-field limit to the corresponding kinetic equation. For this, we first derive a uniform ℓ p ell ^p -stability of the thermodynamic Kuramoto model with respect to initial data by directly estimating the temporal evolution of ℓ p ell ^p -distance between two admissible solutions to the particle thermodynamic Kuramoto model. In a large-oscillator limit, the Vlasov type mean-field equation can be rigorously derived using the BBGKY hierarchy, uniform stability estimate, and particle-in-cell method. We construct unique global-in-time measure-valued solutions to the derived kinetic equation and also derive a uniform-in-time stability estimate and emergent estimates.
{"title":"Uniform stability and uniform-in-time mean-field limit of the thermodynamic Kuramoto model","authors":"Seung‐Yeal Ha, Myeongju Kang, Hansol Park, T. Ruggeri, Woojoo Shim","doi":"10.1090/QAM/1588","DOIUrl":"https://doi.org/10.1090/QAM/1588","url":null,"abstract":"We consider the thermodynamic Kuramoto model proposed in cite{H-P-R-S}. For each oscillator in thermodynamic Kuramoto model, there is a coupling effect between the phase and the temperature field. For such a model, we study a uniform stability and uniform-in-time mean-field limit to the corresponding kinetic equation. For this, we first derive a uniform \u0000\u0000 \u0000 \u0000 ℓ\u0000 p\u0000 \u0000 ell ^p\u0000 \u0000\u0000-stability of the thermodynamic Kuramoto model with respect to initial data by directly estimating the temporal evolution of \u0000\u0000 \u0000 \u0000 ℓ\u0000 p\u0000 \u0000 ell ^p\u0000 \u0000\u0000-distance between two admissible solutions to the particle thermodynamic Kuramoto model. In a large-oscillator limit, the Vlasov type mean-field equation can be rigorously derived using the BBGKY hierarchy, uniform stability estimate, and particle-in-cell method. We construct unique global-in-time measure-valued solutions to the derived kinetic equation and also derive a uniform-in-time stability estimate and emergent estimates.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45920539","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 article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping. They provide an alternate description of the metastable behavior in such lattice systems which agrees with previous predictions for the evolution of metastable states while providing more accurate approximations to these states. We analyze the stability of these breathers, finding a very small positive eigenvalue whose eigenvector lies almost tangent to the surface of the cylinder formed by the family of breathers. This causes solutions to slide along the cylinder without leaving its neighborhood for very long times.
{"title":"Damped and driven breathers and metastability","authors":"Daniel A. Caballero, C. E. Wayne","doi":"10.1090/qam/1650","DOIUrl":"https://doi.org/10.1090/qam/1650","url":null,"abstract":"In this article we prove the existence of a new family of periodic solutions for discrete, nonlinear Schrödinger equations subject to spatially localized driving and damping. They provide an alternate description of the metastable behavior in such lattice systems which agrees with previous predictions for the evolution of metastable states while providing more accurate approximations to these states. We analyze the stability of these breathers, finding a very small positive eigenvalue whose eigenvector lies almost tangent to the surface of the cylinder formed by the family of breathers. This causes solutions to slide along the cylinder without leaving its neighborhood for very long times.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48883034","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 a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.
{"title":"Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential","authors":"Dai-Ni Hsieh, S. Arguillère, N. Charon, L. Younes","doi":"10.1090/QAM/1600","DOIUrl":"https://doi.org/10.1090/QAM/1600","url":null,"abstract":"This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48040666","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}
A layered medium is excited by N N external or internal point sources. Boundary-value problems for the generated acoustic waves are formulated. General scattering and optical theorems are established relating the involved fields and far-field patterns due to groups of sources. Interaction scattering cross sections are defined and associated physical bounds are derived. The large- N N behavior of these cross sections is also investigated. Numerical results are presented demonstrating the variations of the interaction cross sections and their physical bounds.
{"title":"Excitation of a layered medium by 𝑁 sources: Scattering relations, interaction cross sections and physical bounds","authors":"Andreas Kalogeropoulos, N. Tsitsas","doi":"10.1090/qam/1581","DOIUrl":"https://doi.org/10.1090/qam/1581","url":null,"abstract":"A layered medium is excited by \u0000\u0000 \u0000 N\u0000 N\u0000 \u0000\u0000 external or internal point sources. Boundary-value problems for the generated acoustic waves are formulated. General scattering and optical theorems are established relating the involved fields and far-field patterns due to groups of sources. Interaction scattering cross sections are defined and associated physical bounds are derived. The large-\u0000\u0000 \u0000 N\u0000 N\u0000 \u0000\u0000 behavior of these cross sections is also investigated. Numerical results are presented demonstrating the variations of the interaction cross sections and their physical bounds.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46816003","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 article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid’s absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy’s law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation). Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation.
{"title":"A macro-micro elasticity-diffusion system modeling absorption-induced swelling in rubber foams: Proof of the strong solvability","authors":"T. Aiki, N. Kröger, A. Muntean","doi":"10.1090/QAM/1592","DOIUrl":"https://doi.org/10.1090/QAM/1592","url":null,"abstract":"In this article, we propose a macro-micro (two-scale) mathematical model for describing the macroscopic swelling of a rubber foam caused by the microscopic absorption of some liquid. In our modeling approach, we suppose that the material occupies a one-dimensional domain which swells as described by the standard beam equation including an additional term determined by the liquid pressure. As special feature of our model, the absorption takes place inside the rubber foam via a lower length scale, which is assumed to be inherently present in such a structured material. The liquid’s absorption and transport inside the material is modeled by means of a nonlinear parabolic equation derived from Darcy’s law posed in a non-cylindrical domain defined by the macroscopic deformation (which is a solution of the beam equation).\u0000\u0000Under suitable assumptions, we establish the existence and uniqueness of a suitable class of solutions to our evolution system coupling the nonlinear parabolic equation posed on the microscopic non-cylindrical domain with the beam equation posed on the macroscopic cylindrical domain. In order to guarantee the regularity of the non-cylindrical domain, we impose a singularity to the elastic response function appearing in the beam equation.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48160336","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 consider a class of elliptic partial differential equations (PDE) that can be understood as the Euler–Lagrange equations of an associated convex optimization problem. Discretizing this optimization problem, we present a strategy for a numerical solution that is based on the popular primal-dual hybrid gradients (PDHG) approach: we reformulate the optimization as a saddle-point problem with a dual variable addressing the quadratic term, introduce the PDHG optimization steps, and analytically eliminate the dual variable. The resulting scheme resembles explicit gradient descent; however, the eliminated dual variable shows up as a boosting term that substantially accelerates the scheme. We introduce the proposed strategy for a simple Laplace problem and then illustrate the technique on a variety of more complicated and relevant PDE, both on Cartesian domains and graphs. The proposed numerical method is easily implementable, computationally efficient, and applicable to relevant computing tasks across science and engineering.
{"title":"A primal-dual optimization strategy for elliptic partial differential equations","authors":"Dominique Zosso, B. Osting","doi":"10.1090/qam/1576","DOIUrl":"https://doi.org/10.1090/qam/1576","url":null,"abstract":"We consider a class of elliptic partial differential equations (PDE) that can be understood as the Euler–Lagrange equations of an associated convex optimization problem. Discretizing this optimization problem, we present a strategy for a numerical solution that is based on the popular primal-dual hybrid gradients (PDHG) approach: we reformulate the optimization as a saddle-point problem with a dual variable addressing the quadratic term, introduce the PDHG optimization steps, and analytically eliminate the dual variable. The resulting scheme resembles explicit gradient descent; however, the eliminated dual variable shows up as a boosting term that substantially accelerates the scheme. We introduce the proposed strategy for a simple Laplace problem and then illustrate the technique on a variety of more complicated and relevant PDE, both on Cartesian domains and graphs. The proposed numerical method is easily implementable, computationally efficient, and applicable to relevant computing tasks across science and engineering.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/qam/1576","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44025791","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}
Our aim in this paper is to prove the existence of solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a reaction-diffusion equation for the tumor density and of a Cahn–Hilliard type equation for the oxygen concentration. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering a modified equation and taking a logarithmic nonlinear term in the Cahn–Hilliard equation.
{"title":"A coupled Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells","authors":"Lu Li, A. Miranville, R. Guillevin","doi":"10.1090/qam/1585","DOIUrl":"https://doi.org/10.1090/qam/1585","url":null,"abstract":"Our aim in this paper is to prove the existence of solutions for a model for the proliferative-to-invasive transition of hypoxic glioma cells. The equations consist of the coupling of a reaction-diffusion equation for the tumor density and of a Cahn–Hilliard type equation for the oxygen concentration. The main difficulty is to prove the existence of a biologically relevant solution. This is achieved by considering a modified equation and taking a logarithmic nonlinear term in the Cahn–Hilliard equation.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/qam/1585","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43383405","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}
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":null,"pages":null},"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":null,"pages":null},"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}
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