Long-time Asymptotics Research Articles

Solitons moving on background waves of the focusing nonlinear Schrödinger equation with step-like initial condition

This work concerns the long-time asymptotic behaviors of the focusing nonlinear Schrödinger equation with step-like initial condition in present of discrete spectrum. The exact step initial-value problem with non-vanishing boundary on one side has been solved, while the step-like initial-value problem with solitons emerging remains open. We study this problem and explore the solitons moving on background waves of the focusing nonlinear Schrödinger equation by classifying all possible locations of discrete spectrum associated with the spectral functions. It is shown that there are five kinds of zones for the discrete spectrum in complex plane, which are called dumbing zone, trapping zone, trapping/waking zone, transmitting/waking zone and transmitting zone, respectively. By means of Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems, the long-time asymptotics of the solution along with the locations of the solitons for each case are formulated. Numerical simulations match very well with the theoretical analysis.

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak shock solution and the steady strong shock solution, when a steady supersonic flow impinges upon the solid wedge – the half-angle of which is less than a critical angle (i.e., the detachment angle), and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled in a definitive manner. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the two-dimensional Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions of multidimensional hyperbolic systems of conservation laws. In this sense, we have to understand the reflection configurations in order to understand fully the global entropy solutions of two-dimensional hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for all the physical parameters up to the detachment angle.

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

Global Well-Posedness and Long-Time Asymptotics of a General Nonlinear Non-local Burgers Equation

The role of Allee effects for Gaussian and Lévy dispersals in an environmental niche.

In the study of biological populations, the Allee effect detects a critical density below which the population is severely endangered and at risk of extinction. This effect supersedes the classical logistic model, in which low densities are favorable due to lack of competition, and includes situations related to deficit of genetic pools, inbreeding depression, mate limitations, unavailability of collaborative strategies due to lack of conspecifics, etc. The goal of this paper is to provide a detailed mathematical analysis of the Allee effect. After recalling the ordinary differential equation related to the Allee effect, we will consider the situation of a diffusive population. The dispersal of this population is quite general and can include the classical Brownian motion, as well as a Lévy flight pattern, and also a "mixed" situation in which some individuals perform classical random walks and others adopt Lévy flights (which is also a case observed in nature). We study the existence and nonexistence of stationary solutions, which are an indication of the survival chance of a population at the equilibrium. We also analyze the associated evolution problem, in view of monotonicity in time of the total population, energy consideration, and long-time asymptotics. Furthermore, we also consider the case of an "inverse" Allee effect, in which low density populations may access additional benefits.

Long-time asymptotics for a complex cubic Camassa–Holm equation

Long-time asymptotics for a complex cubic Camassa–Holm equation

Long-time Asymptotics for the Reverse Space-time Nonlocal Hirota Equation with Decaying Initial Value Problem: without Solitons

Long-time Asymptotics for the Reverse Space-time Nonlocal Hirota Equation with Decaying Initial Value Problem: without Solitons

The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed ∂¯ -Riemann–Hilbert problem

In this paper, we investigate the Cauchy problem of the Sasa–Satsuma (SS) equation with initial data belonging to the Schwartz space. The SS equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3 × 3 Lax representation. With the aid of the ∂¯ -nonlinear steepest descent method of the mixed ∂¯ -Riemann–Hilbert problem, we give the soliton resolution and long-time asymptotics for the Cauchy problem of the SS equation with the existence of second-order discrete spectra in the space-time solitonic regions.

Long time asymptotics of small mass solutions for a chemotaxis-consumption system involving prescribed signal concentrations on the boundary

This paper investigates a parabolic–elliptic chemotaxis-consumption system with signal dependent sensitivity χ=χ(c) under no-flux/Dirichlet boundary conditions. For general χ which may allow singularities at c=0, the global existence and boundedness of radial large data solutions are established in dimensions d≥2. In particular, when χ(c)=1, we also find that the constructed solution converges asymptotically to a nonhomogeneous steady state if the initial mass is small. On the other hand, for the system with χ(c)=c, a Lyapunov-type inequality is derived. This inequality not only leads to a result on global existence of smooth solutions with non-radial large data in two dimensions but moreover, provides long-time asymptotics of non-radial (d=2) and radial (d≥2) solutions at suitably small mass levels.

A finite-volume scheme for fractional diffusion on bounded domains

Abstract We propose a new fractional Laplacian for bounded domains, expressed as a conservation law and thus particularly suited to finite-volume schemes. Our approach permits the direct prescription of no-flux boundary conditions. We first show the well-posedness theory for the fractional heat equation. We also develop a numerical scheme, which correctly captures the action of the fractional Laplacian and its anomalous diffusion effect. We benchmark numerical solutions for the Lévy–Fokker–Planck equation against known analytical solutions. We conclude by numerically exploring properties of these equations with respect to their stationary states and long-time asymptotics.

Space-like asymptotics of the thermal two-point functions of the XXZ spin-1/2 chain

This work proposes a closed formula for the leading term of the large-distance and long-time asymptotics in a cone of the space-like regime for the transverse dynamical two-point functions of the XXZ spin 1/2 chain at finite temperatures. The result follows from a simple analysis of the thermal form factor series for dynamical correlation functions. The obtained leading asymptotics are driven by the Bethe Ansatz data associated with the first sub-leading eigenvalue of the quantum transfer matrix.

On behavior analysis of solutions for the coupled higher-order WKI equation

A coupled higher-order WKI equation is proposed and its 4 × 4 matrix Lax pair is derived. On the basis of spectral analysis of the 4 × 4 matrix spectral problems related to the coupled higher-order WKI equation, the basic Riemann–Hilbert problem is established by using the inverse scattering transformation. Resorting to the nonlinear steepest descent method, the solution of the basic Riemann–Hilbert problem is transformed to the solution of the model Riemann–Hilbert problem after several contour deformation. Finally, we explicitly obtain the long-time asymptotics of the solution to the Cauchy problem of the coupled higher-order WKI equation in the solitonless sector region with the aid of the parabolic cylinder function.

Long-time asymptotics for the integrable nonlocal Lakshmanan–Porsezian–Daniel equation with decaying initial value data

In this work, we study the Cauchy problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation with rapid attenuation of initial data. The basic Riemann–Hilbert problem of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is constructed from Lax pair. Using Deift-Zhou nonlinear steepest descent method, the explicit long-time asymptotic formula of integrable nonlocal Lakshmanan-Porsezian-Daniel equation is derived, which is different from the local model. Besides, compared to the nonlocal nonlinear Schrödinger equation, since the increase of real stationary phase points, the long-time asymptotic formula for nonlocal Lakshmanan-Porsezian-Daniel equation becomes more complex.

Soliton resolution for the complex short-pulse positive flow with weighted Sobolev initial data in the space-time soliton regions

The Riemann Hilbert approach and ∂‾ steepest descent method are used to study the Cauchy problem of a new complex short-pulse positive flow, from which the long-time asymptotics and soliton resolution conjecture for the complex short-pulse positive flow with initial conditions is obtained in weighted Sobolev spaces. Resorting to the spectral analysis of Lax pair, the introduced transformations of field variables and transformations of independent variables, we establish the basic Riemann-Hilbert problem, and convert the solution of the Cauchy problem of the complex short-pulse positive flow into the corresponding solution of the Riemann Hilbert problem. The jump matrix then expands and deforms continuously over the initial contour. We finally obtain the long-time asymptotics and soliton resolution for the complex short-pulse positive flow in the soliton region by using the ∂‾ steepest descent method. The results also show that N-soliton solutions of the complex short-pulse positive flow are asymptotically stable.

Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4×4 matrix spectral problem. The spectral analysis of the 4×4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.

Long-time asymptotics for the coupled modified complex short-pulse equation

Long-time asymptotics for the coupled modified complex short-pulse equation

Long-time asymptotics of the Lifshitz–Slyozov equation with nucleation

Long-time asymptotics of the Lifshitz–Slyozov equation with nucleation

Long-time dynamics of classical Keller–Segel equation

In this article, we obtain global dynamics of 2D Keller–Segel equation (KS for short) in subcritical regime with no additional assumptions. With initial data small in some Triebel-Lizorkin space, we also characterize the long-time asymptotics of global mild solutions to the KS equation in higher dimensions where the spatial dimension n≥3.

Asymptotics for the infinite Brownian loop on noncompact symmetric spaces

The infinite Brownian loop on a Riemannian manifold is the limit in distribution of the Brownian bridge of length T around a fixed origin when T→+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T \\rightarrow +\\infty $$\\end{document}. The aim of this note is to study its long-time asymptotics on Riemannian symmetric spaces G/K of noncompact type and of general rank. This amounts to the behavior of solutions to the heat equation subject to the Doob transform induced by the ground spherical function. Unlike the standard Brownian motion, we observe in this case phenomena which are similar to the Euclidean setting, namely L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} asymptotic convergence without requiring bi-K-invariance for initial data, and strong L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} convergence.

Long-time asymptotics for coagulation equations with injection that do not have stationary solutions

In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity γ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma < 1$$\\end{document}, such that K(x, y) is approximately xγ+λy-λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x^{\\gamma + \\lambda } y^{-\\lambda }$$\\end{document}, when x is larger than y. We restrict the analysis to the case γ+2λ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma + 2 \\lambda \\ge 1 $$\\end{document}. In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case considered in Ferreira et al. (Annales de l’Institut Henri Poincaré C, Analyse Non Linéaire, 2023), where γ+2λ<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma + 2 \\lambda <1$$\\end{document}. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case γ+2λ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma +2\\lambda \\ge 1$$\\end{document}, the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document} for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.

Maxwell–Bloch equations without spectral broadening: the long-time asymptotics of an input pulse in a long two-level laser amplifier

We study the problem of propagation of an input electromagnetic pulse through a long two-level laser amplifier under trivial initial conditions. In this paper, we consider an unstable model described by the Maxwell–Bloch equations without spectral broadening. Previously, this model was studied by Manakov in (1982 Zh. Eksp. Teor. Fiz. 83 68–75) and together with Novokshenov in (1986 Teor. Mat. Fiz. 69 40–54). We consider this model in a more natural formulation as an initial-boundary (mixed) problem using a modern version of the inverse scattering transform method in the form of a suitable Riemann–Hilbert (RH) problem. The RH problem arises as a result of applying the Fokas–Its method of simultaneous analysis of the corresponding spectral problems for the Ablowitz–Kaup–Newell–Segur equations. This approach makes it possible to obtain rigorous asymptotic results at large times, which differ significantly from the previous ones. Differences take place both near the light cone and in the tail region, where a new type of solitons is found against an oscillating background. These solitons are physically relevant, their velocities are smaller than the speed of light. The number of such solitons can be either finite or infinite (in the latter case, the set of zeros has a condensation point at infinity). Such solitons cannot be reflectionless, they are generated by zeros of the reflection coefficient of the input pulse (and not by poles of the transmission coefficient). Thus our approach shows the presence of a new phenomenon in soliton theory, namely, the boundary condition (input pulse) of a mixed problem under trivial initial conditions can generate solitons due to the zeros of the reflection coefficient, while the poles of the transmission coefficient do not contribute to the asymptotics of the solution.