Long-time Asymptotics Research Articles

Deterministic and stochastic theory for a resonant triad

We study a simple model of three resonantly-interacting nonlinear waves. Linear stability analysis allows for an easy identification of stable and unstable initial conditions. Subsequently, a reformulation of the problem allows us to establish the onset of phase coherence, demonstrating its critical role in the growth of instabilities. Furthermore, we are able to provide explicit expressions for the time-varying moments of the modal amplitudes, as well as for the bispectrum which measures phase coherence. We provide direct insight into the long-time statistics of the system without Monte-Carlo simulation. This includes long-time asymptotics, which show that the system desynchronises, leading to the convergence of the second order moments and decay in the bispectrum.

Long-time asymptotic behavior for the matrix modified Korteweg–de Vries equation

The long-time asymptotics of solution to the integrable matrix modified Korteweg–de Vries equation with a 4 × 4 Lax pair on the line in the case of decaying initial data is established. The study makes crucial use of the inverse scattering transform in the form of an associated 4 × 4 matrix Riemann–Hilbert problem, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. It was shown that the (x,t)-plane mainly decomposes asymptotically in time into three types of regions: a left oscillating region, where to leading order the solution exhibits decaying (of the order O(t−1/2)) modulated oscillations, a central Painlevé region where the asymptotic behavior is expressed in terms of the solution of a coupled Painlevé II equation which is related to a 4 × 4 matrix Riemann–Hilbert problem, and a right fast decaying region.

The Hermitian symmetric space Fokas–Lenells equation: spectral analysis and long-time asymptotics

Abstract Based on the inverse scattering transformation, we carry out spectral analysis of the $4\times 4$ matrix spectral problems related to the Hermitian symmetric space Fokas–Lenells (FL) equation, by which the solution of the Cauchy problem of the Hermitian symmetric space FL equation is transformed into the solution of a Riemann–Hilbert problem. The nonlinear steepest descent method is extended to study the Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems and strict error estimates, we obtain explicitly the long-time asymptotics of the Cauchy problem of the Hermitian symmetric space FL equation with the aid of the parabolic cylinder function.

Long time asymptotics for the nonlocal mKdV equation with finite density initial data

We consider the Cauchy problem for an integrable real nonlocal (reverse space–time) mKdV equation with nonzero boundary conditions qt(x,t)−6σq(x,t)q(−x,−t)qx(x,t)+qxxx(x,t)=0,q(x,0)=q0(x),limx→±∞q0(x)=q±, where |q±|=1 and q+=δq−, σδ=−1. Based on the spectral analysis of the Lax pair, we express the solution of the Cauchy problem of the nonlocal mKdV equation in terms of a Riemann–Hilbert problem. In a fixed space–time solitonic region −6<x/t<6, we apply ∂̄-steepest descent method to analyze the long-time asymptotic behavior of the solution q(x,t). We find that the long time asymptotic behavior of q(x,t) can be characterized with an N(Λ)-soliton on discrete spectrum and leading order term O(t−1/2) on continuous spectrum up to a residual error order O(t−1).

Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential

Nous considérons le modèle parabolique d’Anderson (PAM) ∂tu=12Δu+ξu dans R2 avec un potentiel de bruit blanc ξ en espace. Nous prouvons que le comportement asymptotique presque sûr de la masse totale U(t) au temps t est donnée par logU(t)∼χtlogt pour t→∞, avec une constante déterministe χ que nous identifions à l’aide d’une formule variationnelle. Cette constante a déjà été utilisée, dans un travail antérieur de l’un des auteurs, pour décrire le comportement asymptotique λ1(Qt)∼χlogt de la valeur propre principale λ1(Qt) de l’opérateur d’Anderson muni de conditions aux limites de Dirichlet sur la boîte Qt=[−t2,t2]2.

Spectral Analysis and Long-Time Asymptotics of a Coupled Nonlinear Schrödinger System

Spectral Analysis and Long-Time Asymptotics of a Coupled Nonlinear Schrödinger System

Riemann–Hilbert approach and long-time asymptotics of the positive flow short-pulse equation

In this paper, we study the Cauchy problem of the positive flow short-pulse equation by using the Riemann–Hilbert approach, from which we obtain multi-soliton formulas of the Cauchy problem under the reflectionless case and the long-time asymptotic behavior of the solution of the Cauchy problem in the solitonless sector region. Starting with the spectral analysis of the Lax pair related to this equation, a Riemann–Hilbert problem is established by introducing some proper spectral function transformations and variable transformations, and the solution of the Cauchy problem for the positive flow short-pulse equation is transformed into the solution of the corresponding Riemann–Hilbert problem. Then we give the soliton classification of the positive flow short-pulse equation without reflection. Various Deift–Zhou contour deformations and the motivation behind them are given. We finally obtain the long-time asymptotics for the positive flow short-pulse equation in the solitonless sector region by resorting to the Deift–Zhou nonlinear steepest descent method.

Long-time asymptotics for the complex nonlinear transverse oscillation equation

In this paper, we study the long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation. Based on the corresponding Lax pair, the original Riemann–Hilbert problem is constructed by introducing some spectral function transformations and variable transformations, and the solution of the complex nonlinear transverse oscillation equation is transformed into the solution of the resulted Riemann–Hilbert problem. Various Deift–Zhou contour deformations and the motivation behind them are given, from which the original Riemann–Hilbert problem is further transformed into a solvable model problem. The long-time asymptotic behavior of the Cauchy problem for the complex nonlinear transverse oscillation equation is obtained by using the nonlinear steepest decent method.

Soliton asymptotics for the KdV shock problem via classical inverse scattering

We show how the inverse scattering transform can be used as a convenient tool to derive the long-time asymptotics of the Korteweg–de Vries (KdV) shock problem in the soliton region. In particular, we improve the results previously obtained via the nonlinear steepest decent approach both with respect to the decay of the initial conditions as well as the region where they are valid.

Riemann–Hilbert approach and long-time asymptotics for the three-component derivative nonlinear Schrödinger equation

Riemann–Hilbert approach and long-time asymptotics for the three-component derivative nonlinear Schrödinger equation

Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit

We consider the Keller--Rubinow model for Liesegang rings in one spatial dimension in the fast reaction limit as introduced by Hilhorst, van der Hout, Mimura, and Ohnishi in 2007. Numerical evidence suggests that solutions to this model converge, independent of the initial concentration, to a universal profile for large times in parabolic similarity coordinates. For the concentration function, the notion of convergence appears to be similar to attraction to a stable equilibrium point in phase space. The reaction term, however, is discontinuous so that it can only convergence in a much weaker, averaged sense. This also means that most of the traditional analytical tools for studying the long-time behavior fail on this problem. In this paper, we identify the candidate limit profile as the solution of a certain one-dimensional boundary value problem which can be solved explicitly. We distinguish two nontrivial regimes. In the first, the transitional regime, precipitation is restricted to a bounded region in space. We prove that the concentration converges to a single asymptotic profile. In the second, the supercritical regime, we show that the concentration converges to one of a one-parameter family of asymptotic profiles, selected by a solvability condition for the one-dimensional boundary value problem. Here, our convergence result is only conditional: we prove that if convergence happens, either pointwise for the concentration or in an averaged sense for the precipitation function, then the other field converges likewise; convergence in concentration is uniform, and the asymptotic profile is indeed the profile selected by the solvability condition. A careful numerical study suggests that the actual behavior of the equation is indeed the one suggested by the theorem.

Long-Time Asymptotics for the Modified Camassa–Holm Equation with Nonzero Boundary Conditions

Long-Time Asymptotics for the Modified Camassa–Holm Equation with Nonzero Boundary Conditions

On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regions

We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation in the solitonic regionsmt+(m(u2−ux2))x+κux=0,m=u−uxx,u(x,0)=u0(x)∈H4,2(R), where κ is a positive constant characterizing the effect of the linear dispersion. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale (y,t). Further using the ∂‾ generalization of the Deift-Zhou steepest descent method, we derive different long time asymptotic expansions of the solution u(y,t) in different space-time solitonic regions of ξ=y/t. We divide the half-plane {(y,t):−∞<y<∞,t>0} into four asymptotic regions: The phase function θ(z) has no stationary phase point on the jump contour in the space-time solitonic regions ξ∈(−∞,−1/4)∪(2,+∞), corresponding asymptotic approximations can be characterized with an N(Λ)-solitons with diverse residual error order O(t−1+2ρ); The phase function θ(z) has four phase points and eight phase points on the jump contour in the space-time solitonic regions ξ∈(0,2) and ξ∈(−1/4,0), respectively. The corresponding asymptotic approximations can be characterized with an N(Λ)-soliton as well as an interaction term between soliton solutions and the dispersion term with diverse residual error order O(t−3/4). Our results also confirm the soliton resolution conjecture and asymptotically stability of the N-soliton solutions for the mCH equation.

Spectral analysis and long-time asymptotics of complex mKdV equation

In this paper, we obtain the long-time asymptotics of the complex mKdV equation via the Defit–Zhou method (nonlinear steepest descent method). The Cauchy problem of the complex mKdV equation is transformed into the corresponding Riemann–Hilbert problem on the basis of the Lax pair and the scattering matrix. After that, Riemann–Hilbert problems are converted through a decomposition of the matrix-valued spectral function and factorizations of the jump matrix for Riemann–Hilbert problem. Finally, by solving the last model problem, the long-time asymptotics of complex mKdV equation are derived.

The Focusing NLS Equation with Step-Like Oscillating Background: The Genus 3 Sector

We consider the Cauchy problem for the focusing nonlinear Schrödinger equation with initial data approaching different plane waves $$A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}$$ , $$j=1,2$$ as $$x\rightarrow \pm \infty $$ . The goal is to determine the long-time asymptotics of the solution, according to the value of $$\xi =x/t$$ . The general situation is analyzed in a recent paper where we develop the Riemann–Hilbert approach and detect different asymptotic scenarios, depending on the relationships between the parameters $$A_1$$ , $$A_2$$ , $$B_1$$ , and $$B_2$$ . In particular, in the shock case $$B_1<B_2$$ , some scenarios include genus 3 sectors, i.e., ranges of values of $$\xi $$ where the leading term of the asymptotics is given in terms of hyperelliptic functions attached to a Riemann surface $$M(\xi )$$ of genus three. The present paper is devoted to the complete asymptotic analysis in such a sector.

Long-time asymptotics for the modified complex short pulse equation

&lt;p style='text-indent:20px;'&gt;Based on the spectral analysis and the inverse scattering method, by introducing some spectral function transformations and variable transformations, the initial value problem for the modified complex short pulse (mCSP) equation is transformed into a &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ 2\times2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrix Riemann-Hilbert problem. It is proved that the solution of the initial value problem for the mCSP equation has a parametric expression related to the solution of the matrix Riemann-Hilbert problem. Various Deift-Zhou contour deformations and the motivation behind them are given. Through several appropriate transformations and strict error estimates, the original matrix Riemann-Hilbert problem can be reduced to the model Riemann-Hilbert problem, whose solution can be solved explicitly in terms of the parabolic cylinder functions. Finally, the long-time asymptotics of the solution of the initial value problem for the mCSP equation is obtained by using the nonlinear steepest decent method.&lt;/p&gt;

Homoenergetic solutions of the Boltzmann equation: the case of simple-shear deformations

&lt;abstract&gt;&lt;p&gt;In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b4"&gt;4&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b20"&gt;20&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b21"&gt;21&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b22"&gt;22&lt;/xref&gt;]&lt;/sup&gt;. These solutions are a particular class of non-equilibrium solutions of the Boltzmann equation which are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression. Therefore, they do not behave asymptotically for long times as Maxwellian distributions, at least for all the choices of the collision kernels, and their behavior strongly depends on the homogeneity of the collision kernel and on the particular form of the hyperbolic terms which describe the deformation taking plance in the gas. We consider here the case of simple shear deformation and present different possible long-time asymptotics of these solutions. We discuss the current knowledge about the long-time behaviour of the homoenergetic solutions as well as some conjectures and open problems.&lt;/p&gt;&lt;/abstract&gt;

The "good" Boussinesq equation: a Riemann-Hilbert approach

We develop an inverse scattering transform formalism for the good Boussinesq equation on the line. Assuming that the solution exists, we show that it can be expressed in terms of the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. The Riemann-Hilbert problem is formulated in terms of two reflection coefficients whose definitions involve only the initial data, and it has a form which makes it suitable for the evaluation of long-time asymptotics via Deift-Zhou steepest descent arguments.

Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces

AbstractThe long-time behaviour of solutions to the defocussing modified Korteweg-de Vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift and Zhou and its reformulation by Dieng and McLaughlin through$\overline {\partial }$-derivatives. To extend the asymptotics to solutions with initial data in lower-regularity spaces, we apply a global approximation via PDE techniques.

Gated reactions in discrete time and space.

How much time does it take for two molecules to react? If a reaction occurs upon contact, the answer to this question boils down to the classic first-passage time problem: find the time it takes for the two molecules to meet. However, this is not always the case as molecules switch stochastically between reactive and non-reactive states. The reaction is then said to be "gated" by the internal states of the molecules involved, which could have a dramatic influence on kinetics. A unified, continuous-time, approach to gated reactions on networks was presented in a recent paper [Scher and Reuveni, Phys. Rev. Lett. 127, 018301 (2021)]. Here, we build on this recent advancement and develop an analogous discrete-time version of the theory. Similar to continuous-time, we employ a renewal approach to show that the gated reaction time can always be expressed in terms of the corresponding ungated first-passage and return times, which yields formulas for the generating function of the gated reaction-time distribution and its corresponding mean and variance. In cases where the mean reaction time diverges, we show that the long-time asymptotics of the gated problem is inherited from its ungated counterpart. However, when molecules spend most of their time non-reactive, an interim regime of slower power-law decay emerges prior to the terminal asymptotics. The discretization of time also gives rise to resonances and anti-resonances, which were absent from the continuous-time picture. These features are illustrated using two case studies that also demonstrate how the general approach presented herein greatly simplifies the analysis of gated reactions.