Localized Initial Data Research Articles

Numerical study of the Serre–Green–Naghdi equations in 2D *

A detailed numerical study of solutions to the Serre–Green–Naghdi (SGN) equations in 2D with vanishing curl of the velocity field is presented. The transverse stability of line solitary waves, 1D solitary waves being exact solutions of the 2D equations independent of the second variable, is established numerically. The study of localized initial data as well as crossing 1D solitary waves does not give an indication of existence of stable structures in SGN solutions localized in two spatial dimensions. For the numerical experiments, an approach based on a Fourier spectral method with a Krylov subspace technique is applied.

Global solutions of quasi-linear Hamiltonian mKdV equation

We study the initial value problem of quasi-linear Hamiltonian mKdV equations. Our goal is to prove the global-in-time existence of a solution given sufficiently smooth, localized, and small initial data. To achieve this, we utilize the bootstrap argument, Sobolev energy estimates, and the dispersive estimate. This proof relies on the space–time resonance method, as well as a bilinear estimate developed by Germain, Pusateri, and Rousset.

Numerical study of fractional Camassa–Holm equations

A numerical study of fractional Camassa–Holm equations is presented. Smooth solitary waves are constructed numerically. Their stability is studied as well as the long time behavior of solutions for general localized initial data from the Schwartz class of rapidly decreasing functions. The appearance of dispersive shock waves is explored.

Global solutions for 1D cubic defocusing dispersive equations: Part I

Abstract This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are both small and localized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data. In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for $L^2$ initial data which are small and nonlocalized. Our main structural assumption is that our nonlinearity is defocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global $L^6$ Strichartz estimates and bilinear $L^2$ bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.1 There, by scaling, our result also admits a large data counterpart.

Global existence and scattering for quadratic NLS with potential in three dimensions

In this article we study the asymptotic behavior of a quadratic NLS equation with small, time-dependent potential and small spatially localized initial data. We prove global existence and scattering of solutions. The two main ingredients of the proof are the space-time resonance method and the boundedness of wave operators for the linear Schr\"{o}dinger equation with potential.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

The Bramson delay in a Fisher–KPP equation with log-singular nonlinearity

We consider a class of reaction–diffusion equations of Fisher–KPP type in which the nonlinearity (reaction term) f is merely C1 at u=0 due to a logarithmic competition term. We first derive the asymptotic behavior of (minimal speed) traveling wave solutions that is, we obtain precise estimates on the decay to zero of the traveling wave profile at infinity. We then use this to characterize the Bramson shift between the traveling wave solutions and solutions of the Cauchy problem with localized initial data. We find a phase transition depending on how singular f is near u=0 with quite different behavior for more singular f. This is in contrast to the smooth case, that is, when f∈C1,δ, where these behaviors are completely determined by f′(0). In the singular case, several scales appear and require new techniques to understand.

On a nonlinear Schrödinger equation for nucleons in one space dimension

We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.

On the Forced Surface Quasi-Geostrophic Equation: Existence of Steady States and Sharp Relaxation Rates

We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all $L^1\cap L^\infty$ localized initial data produces global solutions of the forced SQG, which converge to the steady states in $L^p(\mathbf R^2), 1<p\leq 2$ as time goes to infinity. This establishes that the steady states serve as one point attracting set. Moreover, by employing the method of scaling variables, we compute the sharp relaxation rates, by requiring slightly more localized initial data.

Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity

<p style='text-indent:20px;'>We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

Начальные данные в задаче Коши мы называем локализованными, если они задаются функциями, сосредоточенными в окрестности подмногообразия положительной коразмерности, причeм размер окрестности зависит от малого параметра и вместе с ним стремится к нулю. Хотя решения линейных дифференциальных и псевдодифференциальных уравнений с локализованными начальными данными составляют относительно узкий подкласс множества всех решений, они очень важны с точки зрения физических приложений. Такие решения возникают во многих разделах математической физики. Они описывают распространение возмущений различной природы (будь то волны цунами, вызванные подводным землетрясением, или электромагнитные волны, излучаемые антеннами), и их исследованию (в том числе асимптотическому) посвящена обширная литература. Эффективными естественно называть асимптотики, позволяющие исследовать задачу достаточно быстро и с достаточно скромными вычислительными затратами. Понятие эффективности зависит от доступного вычислительного инструментария и значительно изменилось с появлением программных систем Wolfram Mathematica, MatLab и им подобных, обеспечивающих принципиально новые возможности оперативной реализации и визуализации математических построений, но и предъявляющих к конструкции асимптотик новые требования. В статье даeтся обзор современных методов построения эффективных асимптотик в задачах с локализованными начальными данными. Рассматриваемый класс уравнений и систем включает уравнения Шрeдингера и Дирака, уравнения Максвелла, линеаризованные уравнения газо- и гидродинамики, уравнения линейной теории волн на воде, теории упругости, акустики и т. д. Библиография: 109 названий.

Global existence and decay in multi-component reaction–diffusion–advection systems with different velocities: oscillations in time and frequency

It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e.~nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs $L^1$-$L^p$ estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.

Global dynamics of the generalized fifth-order KdV equation with quintic nonlinearity

We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.

Uniform Formulas for the Asymptotic Solution of a Linear Pseudodifferential Equation Describing Water Waves Generated by a Localized Source

We consider the Cauchy problem with localized initial data for a linear pseudodifferential equation describing water waves with dispersion taken into account and obtain a uniform asymptotic solution in a neighborhood of regular points of the wave front, the size of the neighborhood being independent of the small parameters occurring in the problem.

Nonstandard caustics for localized solutions of the 2d shallow water equations with applications to wave propagation and run-up on a shallow beach

The usual semiclassical approximation is not directly applicable to linearized two-dimensional shallow water equations with localized initial data describing long waves (for example, tsunami waves) in a bounded basin of variable depth, because the momentum variables on the geometric optics rays become infinite at the boundary of the region. To obtain asymptotic solutions, one needs to extend the phase space and introduce a modified Maslov canonical operator. The paper gives a brief survey of some results obtained in this direction by the authors and their colleagues.

Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part II: Generic nonlinearities

We consider the Cauchy problem where f is a C1 function on with and u0 is a nonnegative continuous function on whose limits at are equal to 0. Assuming that the solution u is bounded, we study its asymptotic behavior as In the first part of this study, we proved a general quasiconvergence result: as the solution approaches a set of steady states in the topology of In this paper, we show that under certain generic, explicitly formulated conditions on the nonlinearity f, the solution necessarily converges to a single steady state in Then, under the same conditions, we describe the global asymptotic shape of the solution: the graph of has a top part close to the graph of and two sides taking shapes of “terraces” moving in the opposite directions with precisely determined speeds.

Asymptotic Solutions of the Cauchy Problem with Localized Initial Data for a Finite-Difference Scheme Corresponding to the One-Dimensional Wave Equation

We pose the Cauchy problem with localized initial data that arises when passing from an explicit difference scheme for the wave equation to a pseudodifferential equation. The solution of the Cauchy problem for the difference scheme is compared with the asymptotics of the solution of the Cauchy problem for the pseudodifferential equation. We give a detailed study of the behavior of the asymptotic solution in the vicinity of the leading edge, where yet another version of the asymptotic solution is constructed based on vertical manifolds.

Global existence and decay in nonlinearly coupled reaction-diffusion-advection equations with different velocities

We develop techniques to capture the effect of transport on the long-term dynamics of small, localized initial data in nonlinearly coupled reaction-diffusion-advection equations on the real line. It is well-known that quadratic or cubic nonlinearities in such systems can lead to growth of small, localized initial data and even finite time blow-up. We show that, if the components exhibit different velocities, then quadratic or cubic mixed-terms, i.e. terms with nontrivial contributions from both components, are harmless. We establish global existence and diffusive Gaussian-like decay for exponentially and algebraically localized initial conditions allowing for quadratic and cubic mixed-terms. Our proof relies on a nonlinear iteration scheme that employs pointwise estimates. The situation becomes very delicate if other quadratic or cubic terms are present in the system. We provide an example where a quadratic mixed-term and a Burgers'-type coupling can compensate for a cubic term due to differences in velocities.

Spreading speeds for multidimensional reaction–diffusion systems of the prey–predator type

We investigate spreading properties of solutions of a large class of two-component reaction–diffusion systems, including prey–predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known—at least theoretically—about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution—one for the prey and the other for the predator—in some situations.

On the Stability of Type I Blow Up For the Energy Super Critical Heat Equation

We consider the energy super critical semilinear heat equation ∂tu = ∆u + u p , x ∈ R 3 , p > 5. We first revisit the construction of radially symmetric self similar solutions performed through an ode approach in [51], [2], and propose a bifurcation type argument suggested in [3] which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. We then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional non radial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self similar blow up in non radial energy super critical settings.