ODE Theory Research Articles

The highly nonlinear shallow water equation: local well-posedness, wave breaking data and non-existence of sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^2$$\\end{document} solutions

In the context of the initial data and an amplitude parameter ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space Hk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^k$$\\end{document} as long as k>5/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k>5/2$$\\end{document}. Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of ε-1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon ^{-1},$$\\end{document} while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$sech^2$$\\end{document}.

The Cauchy Problem for Boltzmann Bi-linear Systems: The Mixing of Monatomic and Polyatomic Gases

From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate for p-binomial forms producing sharper estimates for the k-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a 2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^+$$\\end{document} moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation.

Odd periodic waves for some Klein–Gordon type equations: Existence and stability

In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein–Gordon type equations, which include the well known and models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of explicit solutions for the and models is presented using the abstract approach of instability for general abstract Hamiltonian systems.

Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

We are concerned with the Lane-Emden problem{−Δu=upinΩ,u>0inΩ,u=0on∂Ω, where Ω⊂R2 is a smooth bounded domain and p>1 is sufficiently large.Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE's theory, various local Pohozaev identities, blow-up analysis and the properties of Green's function.

A heat flow for the mean field equation on a finite graph

Inspired by works of Castéras (Pac J Math 276:321–345, 2015), Li and Zhu (Calc Var Partial Differ Equ 58:1–18, 2019), Sun and Zhu (Calc Var Partial Differ Equ 60:1–26, 2021), we propose a heat flow for the mean field equation on a connected finite graph $$G=(V,E)$$ . Namely $$\begin{aligned} \left\{ \begin{array}{l} \partial _t\phi (u)=\Delta u-Q+\rho \frac{e^u}{\int _Ve^ud\mu }\\ u(\cdot ,0)=u_0, \end{array}\right. \end{aligned}$$ where $$\Delta $$ is the standard graph Laplacian, $$\rho $$ is a real number, $$Q:V\rightarrow {\mathbb {R}}$$ is a function satisfying $$\int _VQd\mu =\rho $$ , and $$\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}$$ is one of certain smooth functions including $$\phi (s)=e^s$$ . We prove that for any initial data $$u_0$$ and any $$\rho \in {\mathbb {R}}$$ , there exists a unique solution $$u:V\times [0,+\infty )\rightarrow {\mathbb {R}}$$ of the above heat flow; moreover, u(x, t) converges to some function $$u_\infty :V\rightarrow {\mathbb {R}}$$ uniformly in $$x\in V$$ as $$t\rightarrow +\infty $$ , and $$u_\infty $$ is a solution of the mean field equation $$\begin{aligned} \Delta u_\infty -Q+\rho \frac{e^{u_\infty }}{\int _Ve^{u_\infty }d\mu }=0. \end{aligned}$$ Though G is a finite graph, this result is still unexpected, even in the special case $$Q\equiv 0$$ . Our approach reads as follows: the short time existence of the heat flow follows from the ODE theory; various integral estimates give its long time existence; moreover we establish a Lojasiewicz–Simon type inequality and use it to conclude the convergence of the heat flow.

Existence, uniqueness and an ODE approach to the L1-Karcher mean

We extend the domain of the Karcher mean Λ of positive operators on a Hilbert space to L1-Borel probability measures on the cone of positive operators equipped with the Thompson part metric. We establish existence and uniqueness of Λ as the solution of the Karcher equation and develop a nonlinear ODE theory for the relative operator entropy in the spirit of Crandall-Liggett, such that the solutions of the Karcher equation are stationary solutions of the ODE, and all generated solution curves enjoy the exponential contraction estimate. This is possible despite the facts that the Thompson metric is non-Euclidean, non-differentiable, non-commutative as a metric space as well as non-separable, and the positive cone is non-locally compact as a manifold. As further applications of the ODE approach, we prove the norm convergence conjecture of the power means of positive operators to the Karcher mean, and a Trotter-Kato product formula for the nonlinear semigroups explicitly expressed by compositions of two-variable geometric means. This can be regarded as a nonlinear continuous-time version of the law of large numbers.

Parameters and States Estimates of COVID-19 Model Using Lagrange Polynomial, Least Square Approximation and Kenya Quarantine Data

Aims/ Objectives: To develop a compartment based mathematical model, fit daily quarantine data from Ministry of Health of Kenya, estimate individuals in latency and infected in general community and predict dynamics of quarantine for the next 90 days.Study Design: Cross-sectional study.Place and Duration of Study: 13thMarch 2020 to 30th June 2020. 
 Methodology: The population based model was developed using status and characteristic of COVID-19 infection. Quarantine data up to 30/6/2020 was fitted using integrating and differentiating theory of odes and numerical differentiation polynomials. Parameter and state estimates was approximated using least square. Simulations were carried out using ode Matlab solver. Daily community estimates of individuals in latency and infected were obtained together with daily estimate of rate of enlisting individual to quarantine center and their proportions were summarized.Results: The results indicated that maximum infection rate was equal 0.892999 recorded on 28/6/2020, average infection rate was 0.019958 and minimum 0.00012 on 26/6/2020.Conclusion: Predictions based on parameters and state averages indicated that the number of individuals in quarantine are expected to rise exponentially up to about 26,855 individuals by 130th day and remain constant up to 190th day.

The Bloch equation for spin dynamics in electron storage rings: Computational and theoretical aspects

In this paper, we describe our work on spin polarization in high-energy electron storage rings which we base on the Full Bloch equation (FBE) for the polarization density and which aims towards the [Formula: see text] option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The FBE takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov–Ternov effect with its Baier–Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev–Kondratenko formulas. For our numerical and analytical studies of the FBE, we develop an approximation to the latter to obtain an effective FBE. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations (SDEs) underlying the FBE and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective FBE. This discretizes the phase space using spectral methods and discretizes time via the additive Runge–Kutta (ARK) method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

The mathieu differential equation and generalizations to infinite fractafolds

One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear equations for the Fourier coefficients. We study the asymptotic behavior of these Fourier coefficients and discuss the ways in which to numerically approximate solutions. We present both theoretical and numerical results pertaining to the stability of the Mathieu differential equation and the properties of solutions. Further, based on the idea of using Fourier series, we provide a method in which the Mathieu differential equation can be generalized to be defined on the infinite Sierpinski gasket. We discuss the stability of solutions to this fractal differential equation and describe further results concerning properties and behavior of these solutions.

Idempotent Geometry in Generic Algebras

Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, Honor of Wolfgang SprßigTrends Math, Birkhäuser/Springer Basel AG, Basel, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems

We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.

Lagrangian solutions to the Vlasov-Poisson system with a point charge

We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [ 17 ], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [ 8 ] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.

On bounding exact models of epidemic spread on networks

In this paper we use comparison theorems from classical ODE theory in order to rigorously show that closures or approximations at individual or node level lead to mean-field models that bound the exact stochastic process from above. This will be done in the context of modelling epidemic spread on networks and the proof of the result relies on the observation that the epidemic process is negatively correlated (in the sense that the probability of an edge being in the susceptible-infected state is smaller than the product of the probabilities of the nodes being in the susceptible and infected states, respectively). The results in the paper hold for Markovian epidemics and arbitrary weighted and directed networks. Furthermore, we cast the results in a more general framework where alternative closures, other than that assuming the independence of nodes connected by an edge, are possible and provide a succinct summary of the stability analysis of the resulting more general mean-field models. While deterministic initial conditions are key to obtain the negative correlation result we show that this condition can be relaxed as long as extra conditions on the edge weights are imposed.

Pengembangan Pembelajaran Kemahiran Berbicara Bahasa Arab Menggunakan Pendekatan Cross Cultural Understanding

Language and culture are the two components that can not be separated from human communication. As a tool to communicate with one and another, many discussion about the theory between language and culture. The relationship between the language and culture of linguists is the most common concern for those who have included language instruction for foreign language, understanding the language where different cultural complexities are needed in this study were a concern of the Arabic language. Moreover, many scholars emphasize the learning model of Arabic-speaking skills on the basis of theory through cultural understanding. The main aspect you should be interested in learning the Arabic language such as cross cultural knowledge, cross cultural awareness, cross-cultural sensitivity, cross cultural. The theory of Odes developing the intentional skill of speaking Arabic, devolution of educational materials, the development of professional teacher, the development of approaches and the development of the method, and the development of learning assessment. Each component had elements of mutual cultural understanding.

Regularization of Ill-Posed Point Neuron Models

Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous but steep firing rate function is employed, then standard ODE theory implies that such models are well-posed and can thus, approximately, be solved with finite precision arithmetic. We investigate whether the solution of this well-posed model converges to a solution of the ill-posed limit problem as the steepness parameter of the firing rate function tends to infinity. Our argument employs the Arzelà–Ascoli theorem and also yields the existence of a solution of the limit problem. However, we only obtain convergence of a subsequence of the regularized solutions. This is consistent with the fact that models with a Heaviside firing rate function can have several solutions, as we show. Our analysis assumes that the vector-valued limit function v, provided by the Arzelà–Ascoli theorem, is threshold simple: That is, the set containing the times when one or more of the component functions of v equal the threshold value for firing, has zero Lebesgue measure. If this assumption does not hold, we argue that the regularized solutions may not converge to a solution of the limit problem with a Heaviside firing function.

On uniqueness for some non-Lipschitz SDE

We study the uniqueness in the path-by-path sense (i.e. ω-by-ω) of solutions to stochastic differential equations with additive noise and non-Lipschitz autonomous drift. The notion of path-by-path solution involves considering a collection of ordinary differential equations and is, in principle, weaker than that of a strong solution, since no adaptability condition is required. We use results and ideas from the classical theory of ode's, together with probabilistic tools like Girsanov's theorem, to establish the uniqueness property for some classes of noises, including Brownian motion, and some drift functions not necessarily bounded nor continuous.

Stability by linear approximation for time scale dynamical systems

We study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods. In this paper we consider linear systems and their small nonlinear perturbations. In terms of time scales and of eigenvalues of matrices we formulate conditions, sufficient for stability by linear approximation. For non-periodic time scales we use techniques of central upper Lyapunov exponents (a common tool of the theory of linear ODEs) to study stability of solutions. Also, time scale versions of the famous Chetaev's theorem on conditional instability are proved. In a nutshell, we have developed a completely new technique in order to demonstrate that methods of non-autonomous linear ODE theory may work for time-scale dynamics.

Slow motion for the 1D Swift–Hohenberg equation

The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one-dimensional torus T. Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L1-close to a jump function v, remain close to v for large time. This can be achieved by regarding the equation as the L2-gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v.

Ill-Posed Point Neuron Models.

We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard ODE theory implies that point-neuron models are well posed. Nevertheless, in the steep firing rate regime, the problem may become close to ill posed, and the error amplification, in finite time, can be very large. This observation is illuminated by numerical experiments. We conclude that, if a steep firing rate function is employed, then minor round-off errors can have a devastating effect on simulations, unless proper error-control schemes are used.

On the lack of controllability of fractional in time ODE and PDE

This paper is devoted to the analysis of the problem of controllability of fractional (in time) ordinary and partial differential equations (ODE/PDE). The fractional time derivative introduces some memory effects on the system that need to be taken into account when defining the notion of control. In fact, in contrast with the classical ODE and PDE control theory, when driving these systems to rest, one is required not only to control the value of the state at the final time, but also the memory accumulated by the long-tail effects that the fractional derivative introduces. As a consequence, the notion of null controllability to equilibrium needs to take into account both the state and the memory term. The existing literature so far is only concerned with the problem of partial controllability in which the state is controlled, but the behavior of the memory term is ignored. In the present paper, we consider the full controllability problem and show that, due to the memory effects, even at the ODE level, controllability cannot be achieved in finite time. This negative result holds even for finite-dimensional systems in which the control is of full dimension. Consequently, the same negative results hold also for fractional PDE, regardless of their parabolic or hyperbolic nature. This negative result exhibits a completely opposite behavior with respect to the existing literature on classical ODE and PDE control where sharp sufficient conditions for null controllability are well known.