Long-time Asymptotic Behavior Research Articles

The strong convergence and stability of explicit approximations for nonlinear stochastic delay differential equations

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under less restrictive conditions, the truncated Euler-Maruyama (TEM) schemes for SDDEs are proposed, which numerical solutions are bounded in the q th moment for q ≥ 2 and converge to the exact solutions strongly in any finite interval. The 1/2 order convergence rate is yielded. Furthermore, the long-time asymptotic behaviors of numerical solutions, such as stability in mean square and $\mathbb {P}-1$ , are examined. Several numerical experiments are carried out to illustrate our results.

Global well-posedness of logarithmic Keller-Segel type systems

We consider a class of logarithmic Keller-Segel type systems modeling the spatio-temporal behavior of either chemotactic cells or criminal activities in spatial dimensions two and higher. Under certain assumptions on parameter values and given functions, the existence of classical solutions is established globally in time, provided that initial data are sufficiently regular. In particular, we enlarge the range of chemotactic sensitivity χ, compared to known results, in case that spatial dimensions are between two and eight. In addition, we provide new type of small initial data to obtain global classical solution, which is also applicable to the urban crime model. We discuss long-time asymptotic behaviors of solutions as well.

Long-Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions in the Presence of a Discrete Spectrum

The long-time asymptotic behavior of solutions to the focusing nonlinear Schrödinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the xt-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.

Nonlinear dynamics of inertial particles in the ocean: from drifters and floats to marine debris and Sargassum

Buoyant, finite-size or inertial particle motion is fundamentally unlike neutrally buoyant, infinitesimally small or Lagrangian particle motion. The de-jure fluid mechanics framework for the description of inertial particle dynamics is provided by the Maxey-Riley equation. Derived from first principles - a result of over a century of research since the pioneering work by Sir George Stokes - the Maxey-Riley equation is a Newton-type-law with several forces including (mainly) flow, added mass, shear-induced lift, and drag forces. In this paper we present an overview of recent efforts to port the Maxey-Riley framework to oceanography. These involved: 1) including the Coriolis force, which was found to explain behavior of submerged floats near mesoscale eddies; 2) accounting for the combined effects of ocean current and wind drag on inertial particles floating at the air-sea interface, which helped understand the formation of great garbage patches and the role of anticyclonic eddies as plastic debris traps; and 3) incorporating elastic forces, which are needed to simulate the drift of pelagic Sargassum. Insight on the nonlinear dynamics of inertial particles in every case was possible to be achieved by investigating long-time asymptotic behavior in the various Maxey-Riley equation forms, which represent singular perturbation problems involving slow and fast variables.

Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations

We consider a mathematical model for the evolutionary dynamics of tumour cells in vascularised tumours under chemotherapy. The model comprises a system of coupled partial integro-differential equations for the phenotypic distribution of tumour cells, the concentration of oxygen and the concentration of a chemotherapeutic agent. In order to disentangle the impact of different evolutionary parameters on the emergence of intra-tumour phenotypic heterogeneity and the development of resistance to chemotherapy, we construct explicit solutions to the equation for the phenotypic distribution of tumour cells and provide a detailed quantitative characterisation of the long-time asymptotic behaviour of such solutions. Analytical results are integrated with numerical simulations of a calibrated version of the model based on biologically consistent parameter values. The results obtained provide a theoretical explanation for the observation that the phenotypic properties of tumour cells in vascularised tumours vary with the distance from the blood vessels. Moreover, we demonstrate that lower oxygen levels may correlate with higher levels of phenotypic variability, which suggests that the presence of hypoxic regions supports intra-tumour phenotypic heterogeneity. Finally, the results of our analysis put on a rigorous mathematical basis the idea, previously suggested by formal asymptotic results and numerical simulations, that hypoxia favours the selection for chemoresistant phenotypic variants prior to treatment. Consequently, this facilitates the development of resistance following chemotherapy.

Asymptotic stability in a chemotaxis-competition system with indirect signal production

This paper deals with a fully parabolic inter-species chemotaxis-competition system with indirect signal production \begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_{t} = \text{div}(d_{u}\nabla u+\chi u\nabla w)+\mu_{1}u(1-u-a_{1}v), &(x,t)\in \Omega\times (0,\infty), \\ &v_{t} = d_{v}\Delta v+\mu_{2}v(1-v-a_{2}u), &(x,t)\in \Omega\times (0,\infty), \\ & w_{t} = d_{w}\Delta w-\lambda w+\alpha v, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $\end{document} under zero Neumann boundary conditions in a smooth bounded domain $ \Omega\subset \mathbb{R}^{N} $ ($ N\geq 1 $), where $ d_{u}>0, d_{v}>0 $ and $ d_{w}>0 $ are the diffusion coefficients, $ \chi\in \mathbb{R} $ is the chemotactic coefficient, $ \mu_{1}>0 $ and $ \mu_{2}>0 $ are the population growth rates, $ a_{1}>0, a_{2}>0 $ denote the strength coefficients of competition, and $ \lambda $ and $ \alpha $ describe the rates of signal degradation and production, respectively. Global boundedness of solutions to the above system with $ \chi>0 $ was established by Tello and Wrzosek in [J. Math. Anal. Appl. 459 (2018) 1233-1250]. The main purpose of the paper is further to give the long-time asymptotic behavior of global bounded solutions, which could not be derived in the previous work.

Asymptotic dynamics of a system of conservation laws from chemotaxis

This paper is devoted to the analytical study of the long-time asymptotic behavior of solutions to the Cauchy problem of a system of conservation laws in one space dimension, which is derived from a repulsive chemotaxis model with singular sensitivity and nonlinear chemical production rate. Assuming the \begin{document}$ H^2 $\end{document} -norm of the initial perturbation around a constant ground state is finite and using energy methods, we show that there exists a unique global-in-time solution to the Cauchy problem, and the constant ground state is globally asymptotically stable. In addition, the explicit decay rates of the solutions to the chemically diffusive and non-diffusive models are identified under different exponent ranges of the nonlinear chemical production function.

Heavy-tailed branching random walks on multidimensional lattices. A moment approach

We study a continuous-time branching random walk (BRW) on the lattice ℤd, d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point βc > 0 is found for every d ≥ 1. In particular, if β > βc the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β > βc called supercritical. Classification of the BRW treated as subcritical (β < βc) or critical (β = βc) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤd and of the particle population on ℤd according to the ratio d/α.

Long-time asymptotic behavior for the complex short pulse equation

In this paper, we consider the initial value problem for the complex short pulse equation with a Wadati-Konno-Ichikawa type Lax pair. We show that the solution to the initial value problem has a parametric expression in terms of the solution of 2×2-matrix Riemann-Hilbert problem, from which an implicit one-soliton solution is obtained on the discrete spectrum. While on the continuous spectrum we further establish the explicit long-time asymptotic behavior of the non-soliton solution by using Deift-Zhou nonlinear steepest descent method.

Nonlinear economic growth dynamics in the context of a military arms race

In the present contribution, we propose and analyze a dynamical economic growth model for two rival countries that engage an arms race. Under natural assumptions, we prove that global solutions exist and discuss their asymptotic long-time behavior. The results of our stability analysis support the recurring hypothesis in Cold War political science that engaging in an arms race with a technologically superior and hence faster growing adversary has damaging economic consequences. Numerical findings illustrate and support our claims.

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form H=−L+V; the free operator L is the generator of a symmetric Lévy process in Rd, d>1 (with non-degenerate jump measure) and V is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp. decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.

Accurate closed-form solution of the SIR epidemic model

An accurate closed-form solution is obtained to the SIR Epidemic Model through the use of Asymptotic Approximants (Barlow et al., 2017). The solution is created by analytically continuing the divergent power series solution such that it matches the long-time asymptotic behavior of the epidemic model. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.

Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrodinger equation on the line with both zero boundary condition and nonzero boundary conditions at infinity. For zero boundary condition, the associated Riemann–Hilbert problem is related to two cases of scattering data: N simple poles and one Nth-order pole, which allows us to find the exact formulae of soliton solutions. In the case of nonzero boundary conditions and initial data that allow for the presence of discrete spectrum, the pure one-soliton solution and rogue waves are presented. The important advantage of this method is that one can study the long-time asymptotic behavior of the solutions and the infinite order rogue waves based on the associated Riemann–Hilbert problems.

Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

On the Long-Time Asymptotic Behavior of the Modified Korteweg--de Vries Equation with Step-like Initial Data

We study the long time asymptotic behaviour of the solution $q(x,t) $, to the modified Korteweg de Vries equation (MKDV) $q_t+6q^2q_x+q_{xxx}=0$ with step-like initial datum q(x,t=0)->c_- for x->-infinity and q(x,t=0)->c_+ for x-> +infinity. For the exact shock initial data q(x,t=0)=c_- for x<0 and q(x,t=0)=c_+ for x>0 the solution develops an oscillatory regions called dispersive shock wave that connects the two constant regions c_+ and c_-. We show that the dispersive shock wave is described by a modulated periodic travelling wave solution of the MKDV equation where the modulation parameters evolve according to the Whitham modulation equation. The oscillatory region is expanding within a cone in the $(x,t) plane defined as -6c_{-}^2+12c_{+}^2<x/t<4c_{-}^2+2c_{+}^2, with t\gg 1. For step like initial data we show that the solution decomposes for long times in three main regions: - a region where solitons and breathers travel with positive velocities on a constant background c_+; - an expanding oscillatory region that can contain generically breathers; - a region of breathers travelling with negative velocities on the constant background c_-. When the oscillatory region does not contain breathers, it coincides up to a phase shift with the dispersive shock wave solution obtained for the exact step initial data. The phase shift depends on the solitons, breathers and the radiation of the initial data. This shows that the dispersive shock wave is a coherent structure that interacts in an elastic way with solitons, breathers and radiation.

Long-time asymptotics for the generalized Sasa-Satsuma equation

In this paper, we study the long-time asymptotic behavior of the solution of the Cauchy problem for the generalized Sasa-Satsuma equation. Starting with the 3 × 3 Lax pair related to the generalized Sasa-Satsuma equation, we construct a Rieman-Hilbert problem, by which the solution of the generalized Sasa-Satsuma equation is converted into the solution of the corresponding RiemanHilbert problem. Using the nonlinear steepest decent method for the Riemann-Hilbert problem, we obtain the leading-order asymptotics of the solution of the Cauchy problem for the generalized SasaSatsuma equation through several transformations of the Riemann-Hilbert problem and with the aid of the parabolic cylinder function.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains.

How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena, such as heterogeneous catalysis, permeation through semipermeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behaviors of this random variable for both bounded and unbounded domains. This analysis provides complementary insights onto the dynamics of diffusing molecules near partially reactive boundaries.

The Kardar–Parisi–Zhang model of a random kinetic growth: effects of a randomly moving medium

The effects of a randomly moving environment on a randomly growing interface are studied by the field theoretic renormalization group. The kinetic roughening of an interface is described by the Kardar–Parisi–Zhang (KPZ) stochastic differential equation while the velocity field of the moving medium is modelled by the Navier–Stokes equation with an external random force. It is found that the large-scale, long-time (infrared) asymptotic behavior of the system is described by four non-equilibrium universality classes associated with possible fixed points of the renormalization group equations. In addition to the previously known regimes of asymptotic behavior (ordinary diffusion, kinetic growth, and passively advected scalar field), a new nontrivial regime (non-equilibrium universality class) is found. That regime corresponds to a process in which the motion of the environment and the nonlinearity of the KPZ equation are important simultaneously. The fixed points coordinates, their regions of stability and the critical exponents are calculated to the first order of the expansion in (one-loop approximation). However, the new fixed point is either infrared repulsive (d > 2) or corresponds to imaginary coupling constant (d < 2). Possible physical interpretation in terms of mapping to certain reaction-diffusion models and Bose systems is discussed.

Long-Time Asymptotic Behavior for the Discrete Defocusing mKdV Equation

In this article, we apply Deift-Zhou nonlinear steepest descent method to analyze the long-time asymptotic behavior of the solution for the discrete defocusing mKdV equation. This equation was proposed by Ablowitz and Ladik.

On gradient flows with obstacles and Euler’s elastica

We examine a steepest energy descent flow with obstacle constraint in higher order energy frameworks where the maximum principle is not available. We construct the flow under general assumptions using De Giorgi’s minimizing movement scheme. Our main application will be the elastic flow of graph curves with Navier boundary conditions for which we study long-time existence and asymptotic behavior.