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This paper concerns the asymptotics of certain parabolic–elliptic chemotaxis-consumption systems with logistic growth and constant concentration of chemoattractant on the boundary. First we prove that in two dimensional bounded domains there exists a unique global classical solution which is uniformly bounded in time, and then, we show that if the concentration of chemoattractant on the boundary is sufficiently low, then the solution converges to the positive steady state as time goes to infinity. [ABSTRACT FROM AUTHOR]

Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix 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, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads ∂ t u − F (u) (− Δ) s / 2 u + (− Δ) s / 2 (u F (u)) = 0 , x ∈ T d , with s ∈ (0 , 1 ] . We are interested in solutions stemming from periodic positive bounded initial data. The given function F ∈ C ∞ (R +) must satisfy F ′ > 0 a.e. on (0 , + ∞) . For instance, all the functions F (u) = u n with n ∈ N ∗ are admissible non-linearities. The local theory can also be developed on the whole space, however the most complete well-posedness result requires the periodic setting. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in L ∞ . We show that any weak solution is instantaneously regularized into C ∞ . We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations, in particular (Ann. Fac. Sci. Toulouse, Math. 25(4):723–758, 2016; Ann. Fac. Sci. Toulouse, Math. 27(4):667–677, 2018). [ABSTRACT FROM AUTHOR]

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point. [ABSTRACT FROM AUTHOR]

In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the ϑ(λi) (i = 0, 1) would like to be imaginary, which results in the δ λ i 0 contains an increasing t ± I m ϑ (λ i) 2 , and then the asymptotic behavior for nonlocal Hirota equation becomes differently. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa–Holm equation m t = b u x + 1 2 m | u | 2 - u x 2 x - 1 2 m u u ¯ x - u x u ¯ , m = u - u xx , where b > 0 is an arbitrary positive real constant. Long-time asymptotics of the equation is obtained through the ∂ ¯ -steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed via solving the corresponding Riemann–Hilbert problem. Then, we present different long-time asymptotic expansions of the solution u(y, t) in different space-time solitonic regions of ξ = y / t . The half-plane (y , t) : - ∞ < y < ∞ , t > 0 is divided into four asymptotic regions: ξ ∈ (- ∞ , - 1) , ξ ∈ (- 1 , 0) , ξ ∈ (0 , 1 8) and ξ ∈ (1 8 , + ∞) . When ξ falls in (- ∞ , - 1) ∪ (1 8 , + ∞) , no stationary phase point of the phase function θ (z) exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an N (Λ) -solitons with diverse residual error order O (t - 1 + 2 ε) . There are four stationary phase points and eight stationary phase points on the jump curve as ξ ∈ (- 1 , 0) and ξ ∈ (0 , 1 8) , respectively. The corresponding asymptotic form is accompanied by a residual error order O (t - 3 4 ) . [ABSTRACT FROM AUTHOR]

With ∂ ¯ -generalization of the Deift-Zhou steepest descent method, we investigate the long-time asymptotics of the solution to the Cauchy problem for the Hunter-Saxton (HS) equation u t x x − 2 ω u x + 2 u x u x x + u u x x x = 0 , x ∈ R , t > 0 , u (x , 0) = u 0 (x) , where u 0 ∈ H 3 , 4 (R) and ω > 0 is a constant. Via a series of deformations to a Riemann-Hilbert problem associated with the Cauchy problem, we obtain the long-time asymptotic approximations of the solution u (x , t) in two kinds of space-time regions under a new scale (y , t). The solution of the HS equation decays as a speed of O (t − 1 / 2) in the region y / t > 0 ; while in the region y / t < 0 , the solution of the HS equation is depicted by the solution of a parabolic cylinder model with an residual error order O (t − 1 + 1 2 p ) with 2 < p < ∞. • We extend Monvel's result to the solution under a weighted Sobolev initial data u 0 ∈ H 3 , 4 (R). • We establish a scattering map between initial data and reflection coefficient. • We obtain a complete long-time asymptotics for HS equation on whole (x , t) -plane. [ABSTRACT FROM AUTHOR]

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data for which (i) no solitons are present, (ii) the spectral functions have generic behavior near ± 1, and (iii) the solution exists globally. In a recent work, we identified 10 main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term. In this paper, we give a proof for the formula corresponding to the sector ... . [ABSTRACT FROM AUTHOR]

The asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time t$t$ asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location x$x$, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments. [ABSTRACT FROM AUTHOR]

In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems. [ABSTRACT FROM AUTHOR]

In this paper, we apply ...-steepest descent method to analyze the long-time asymptotics of complex mKdV equation with the initial value belonging to weighted Sobolev spaces. Firstly, 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 data. Then the long-time asymptotics of complex mKdV equation is obtained by studying the solution of the Riemann-Hilbert problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the main work is to study the long‐time asymptotics of the defocusing generalized nonlinear Schrödinger equation with the decaying initial value. The Riemann‐Hilbert method and the nonlinear steepest descent method by Deift‐Zhou have made great contributions to obtain it. Starting from the Lax pair of the defocusing generalized nonlinear Schrödinger equation, the associated oscillatory Riemann‐Hilbert problem can be obtained. Then, via the stationary point, the steepest decent contours, and the trigonometric decomposition of jump matrix, we get the solvable Riemann‐Hilbert problem from the associated oscillatory Riemann‐Hilbert problem. Based on the decaying initial value in Schwartz space, the Weber equation, and the standard parabolic cylinder function, the expression of the solution for the generalized nonlinear Schrödinger equation can be given. [ABSTRACT FROM AUTHOR]

In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in [4,20,21,22]. 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.

We study the Cauchy problem of the Harry Dym-type equation associated with the 3 × 3 matrix spectral problem by establishing the corresponding Riemann-Hilbert problem with the initial value lies in Schwartz space. Based on the nonlinear steepest descent method, we give the detailed contour deformation process to reduce the basic Riemann-Hilbert problem to a model Riemann-Hilbert problem, by which the long-time asymptotics for the Harry Dym-type equation is obtained with the aid of a series of precise and uniform error estimates. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The influence of oscillatory perturbations on autonomous strongly nonlinear systems in the plane is investigated. It is assumed that the intensity of perturbations decays with time, and their frequency increases according to a power law. The long-term behaviour of perturbed trajectories is discussed. It is shown that, depending on the structure and the parameters of perturbations, there are at least two different asymptotic regimes: a phase locking and a phase drifting. In the case of phase locking, resonant solutions with an unlimitedly growing energy occur. The stability and asymptotics at infinity of such solutions are investigated. The proposed analysis is based on a combination of the averaging technique and the method of Lyapunov functions. [ABSTRACT FROM AUTHOR]

In this article, we study the long-time asymptotic properties of a non-linear and non-local equation of diffusive type which describes the rock–paper–scissors game in an interconnected population. We fully characterize the self-similar solution and then prove that the solution of the initial–boundary value problem converges to the self-similar profile with an algebraic rate. [ABSTRACT FROM AUTHOR]

In these notes we review some recent results on the homoenergetic solutions for the Boltzmann equation obtained in [4,20-22]. 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. [ABSTRACT FROM AUTHOR]

We consider the long-time asymptotics for the defocusing Hirota equation with Schwartz Cauchy data in the transition region. On the basis of direct and inverse scattering transform of the Lax pair of Hirota equations, we first express the solution of the Cauchy problem in terms of the solution of a Riemann--Hilbert problem. Further, we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region |x/t -- (α²/3β)|t2/3 ≤M, where M is a positive constant. Our result shows that the long-time asymptotics of the Hirota equation can be expressed in terms of the solution of the Painlevé II equation. [ABSTRACT FROM AUTHOR]

We study the long-time asymptotics of the nonlocal Kundu–nonlinear-Schrödinger equation with a decaying initial value. The long-time asymptotics of the solution follow from the nonlinear steepest descent method proposed by Deift–Zhou and the Riemann–Hilbert method. [ABSTRACT FROM AUTHOR]

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space R n , where n ≥ 2 , assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as | x | → ∞ . Under suitable smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. Examples are given which show that the profile can be a rather general Gaussian-like function, and that the approach to the self-similar solution can be arbitrarily slow depending on the continuity and coercivity properties of the asymptotic matrix. The proof of our results relies on appropriate energy estimates for the diffusion equation in self-similar variables. The new ingredient consists in estimating not only the difference w between the solution and the self-similar profile, but also an antiderivative W obtained by solving a linear elliptic problem which involves w as a source term. Hence, a good part of our analysis is devoted to the study of linear elliptic equations whose coefficients are homogeneous of degree zero. [ABSTRACT FROM AUTHOR]

In this paper, the authors apply steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data where . Based on the spectral analysis of the Lax pair, they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region with ∣ξ + 2∣ < 1, the long time asymptotic is given by in which the leading term is N(I) solitons, the second term is a residual error from a equation. For the region ∣ξ + 2∣ > 1, the long time asymptotic is given by in which the leading term is N(I) solitons, the second order term is soliton-radiation interactions and the third term is a residual error from a equation. These results are verification of the soliton resolution conjecture for the derivative Schrödinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively. [ABSTRACT FROM AUTHOR]

On the basis of the spectral analysis of the 4×4 matrix Lax pair, the initial value problem of the coherently-coupled nonlinear Schrödinger system is transformed into a 4×4 matrix Riemann-Hilbert problem. By using the nonlinear steepest decent method, the long-time asymptotics of the solution of the initial value problem for the coherently-coupled nonlinear Schrödinger system is obtained through deforming the Riemann-Hilbert problem into a solvable model one. [ABSTRACT FROM AUTHOR]

Gas filtration governed by the Darcy law at constant thermodynamic potential in three-dimensional porous media is considered. We study the real gas described by the Landau–Lifshitz state equations and consider filtration with constant specific entropy, specific enthalpy and specific Gibbs free energy. Filtration equation can be written in one variable only due to constant potential. The initial and the first terms of long-time asymptotics of filtration equations are taken into consideration. The equations for the initial and the first terms are the Laplace and the Poisson ones, respectively. In addition, we consider phase transitions of methane, which is described by the Landau–Lifshitz equations of state. It is shown that phase transitions do not occur at constant specific entropy and specific enthalpy in the case of methane. [ABSTRACT FROM AUTHOR]

We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Cauchy problem for the modified Korteweg--de Vries hierarchy on the real line with decaying initial data. Using the Riemann--Hilbert formulation and the nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of t-1/(2n+1) with smooth coefficients of the variable (1)n+1x((2n + 1)t)-1/(2n+1) in the self-similarity region for the solution of the nth member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlevé II hierarchy, which generalize the classical Ablowitz--Segur solution for the Painlevé II equation and appear in a variety of random matrix and statistical physics models. We establish the connection formulas for this family of solutions. In the special case that the reflection coefficient vanishes at the origin, the solutions of the Painlevé II hierarchy in the leading coefficient vanishes as well, and the leading and subleading terms in the asymptotic expansion are instead given explicitly in terms of derivatives of the generalized Airy function. [ABSTRACT FROM AUTHOR]

The Cauchy problem of the three-component derivative nonlinear Schrödinger equation is turned into a 4 × 4 matrix Riemann–Hilbert problem by utilizing the spectral analysis. Through a transformation of the spectral parameters, a reduced Riemann–Hilbert problem is derived. Two distinct factorizations of the jump matrix for the reduced Riemann–Hilbert problem and a decomposition of the vector spectral function are deduced. The leading-order asymptotics of the solution for the Cauchy problem of the three-component derivative nonlinear Schrödinger equation is obtained with the aid of the Deift–Zhou nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak selection. The continuum limit, formulated as a reaction-advection-diffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys to the blowup into Dirac-delta singularities and hence brings great challenges to both analytical and numerical studies. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Dirac-delta singularities for genetic segregation on the one hand and preserve several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties and to demonstrate the spatiotemporal dynamics of random genetic drift. [ABSTRACT FROM AUTHOR]

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times. [ABSTRACT FROM AUTHOR]

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.

Fluid films spreading on hydrophobic solid surfaces exhibit complicated dynamics that describe transitions leading the films to break up into droplets. For viscous fluids coating hydrophobic solids this process is called “dewetting”. These dynamics can be represented by a lubrication model consisting of a fourth-order nonlinear degenerate parabolic partial differential equation (PDE) for the evolution of the film height. Analysis of the PDE model and its regimes of dynamics have yielded rich and interesting research bringing together a wide array of different mathematical approaches. The early stages of dewetting involve stability analysis and pattern formation from small perturbations and self-similar dynamics for finite-time rupture from larger amplitude perturbations. The intermediate dynamics describes further instabilities yielding topological transitions in the solutions producing sets of slowly-evolving near-equilibrium droplets. The long-time behavior can be reduced to a finite-dimensional dynamical system for the evolution of the droplets as interacting quasi-steady localized structures. This system yields coarsening, the successive re-arrangement and merging of smaller drops into fewer larger drops. To describe macro-scale applications, mean-field models can be constructed for the evolution of the number of droplets and the distribution of droplet sizes. We present an overview of the mathematical challenges and open questions that arise from the stages of dewetting and how they relate to issues in multi-scale modeling and singularity formation that could be applied to other problems in PDEs and materials science.

We study the long-time asymptotic behavior of oscillatory Riemann–Hilbert problems (RHPs) arising in the mKdV hierarchy (reducing from the AKNS hierarchy). Our analysis is based on the idea of ∂ ¯ -steepest descent. We consider RHPs generated from the inverse scattering transform of the AKNS hierarchy with weighted Sobolev initial data. The asymptotic formula for three regions of the spatial- and temporal-dependent variables is presented in details. [ABSTRACT FROM AUTHOR]

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier–Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights. [ABSTRACT FROM AUTHOR]

We study the defocusing nonlinear Schrödinger equation in the quarter-plane with decaying initial datum and Dirichlet and Neumann boundary values approaching periodic single exponentials at large times. By applying Deift-Zhou steepest descent arguments to an associated Riemann-Hilbert problem, we construct solutions and obtain detailed formulas for their long-time asymptotics. [ABSTRACT FROM AUTHOR]

Recently, we have established the l 2 bijectivity for the defocusing Ablowitz-Ladik system in the discrete weighted space l 2 , k with k ∈ N + by the inverse spectral method. Based on these results, our study is to investigate the long-time asymptotics for the initial-value problem of the defocusing Ablowitz-Ladik system with initial potential in lower regularity without rapid decay. In this case, since the reflection coefficient is not smooth enough, it is impossible to directly apply the nonlinear steepest descent method. Our main idea is to transform the corresponding Riemann-Hilbert problem on the unit circle as the jump contour into a ∂ ¯ -Riemann-Hilbert problem to be handled. As a result, we show that the solution admits the Zakharov-Manakov type formula in the region | n 2 t | ≤ V 0 < 1 , while the solution decays fast to zero in the region | n 2 t | ≥ V 0 > 1. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems and the Dbar approach, the long‐time asymptotic behavior of solutions to the fifth‐order modified KdV (Korteweg–de Vries) equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of the I‐method, we give its global well‐posedness in lower regularity Sobolev spaces and then obtain the asymptotic behavior in these spaces with weights. [ABSTRACT FROM AUTHOR]

The large-time behavior of solutions to a fifth-order modified Korteweg–de Vries equation in the quarter plane is established. Our approach uses the unified transform method of Fokas and the nonlinear steepest descent method of Deift and Zhou. [ABSTRACT FROM AUTHOR]

We are concerned with the long-time asymptotic behavior of the solution for the focusing Hirota equation (also called third-order nonlinear Schrödinger equation) with symmetric, non-zero boundary conditions (NZBCs) at infinity. Firstly, based on the Lax pair with NZBCs, the direct and inverse scattering problems are used to establish the oscillatory Riemann-Hilbert (RH) problem with distinct jump curves. Secondly, the Deift-Zhou nonlinear steepest-descent method is employed to analyze the oscillatory RH problem such that the long-time asymptotic solutions are proposed in two distinct domains of space-time plane (i.e., the plane-wave and modulated elliptic-wave domains), respectively. Finally, the modulation instability of the considered Hirota equation is also investigated. [ABSTRACT FROM AUTHOR]

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of an extended modified KdV equation with decaying initial data in two transition regions, completing previous results by Liu et al. in [18]. It turns out that in first region the asymptotics is expressed in terms of second Painlevé transcendents, in second region the asymptotics can be given by the sum of the cosine oscillation function and the solution of a Painlevé II equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line. We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3 × 3 matrix Riemann-Hilbert problem formulated in the complex k-plane. The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k) and S(k) that depend on the initial data and boundary values, respectively. Then, applying nonlinear steepest descent techniques to the associated 3 × 3 matrix-valued Riemann-Hilbert problem, we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]