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We study a system of equations, involving a large parameter, arising from the study of stellar pulsation for which a combination of procedures is used to approximate a fun damental solution. We present a combination of singular and non-singular perturbation methods which, aided by symbolic computation, may be of multi-disciplinary interest for the analysis as well as a astrophysics application. Example software is presented in the Wolfram Language (Mathematica version 13.2). [ABSTRACT FROM AUTHOR]

In this paper, we propose a new integrable fractional Fokas--Lenells equation by using the completeness of the squared eigenfunctions, dispersion relation, and inverse scattering transform. To solve this equation, we employ the Riemann--Hilbert approach. Specifically, we focus on the case of the reflectionless potential with a simple pole for the zero boundary condition. And we provide the fractional N-soliton solution in determinant form. In addition, we prove the fractional one-soliton solution rigorously. Notably, we demonstrate that as |t| → ∞, the fractional N-soliton solution can be expressed as a linear combination of N fractional single-soliton solutions. [ABSTRACT FROM AUTHOR]

The main aim of this article is to study the Poisson type problem for anisotropic p-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo p-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension. [ABSTRACT FROM AUTHOR]

The creation of hairpin or lambda vortices, typical for the early stages of the laminar-turbulent transition process in various boundary layer flows, in some sense may be associated with blow-up solutions of the Fisher--Kolmogorov--Petrovsky--Piskunov equation. In contrast to the usual applications of this nonlinear evolution equation of the reaction-diffusion type, the solution quantity in the present context needs to stay neither bounded nor positive. We focus on the solution behavior beyond a finite-time point blow-up event, which consists of two moving singularities (representing the cores of the vortex legs) propagating in opposite directions, and their initial motion is determined with the method of matched asymptotic expansions. After resolving subtleties concerning the transition between logarithmic and algebraic expansion terms regarding asymptotic layers, we find that the internal singularity structure resembles a combination of second- and first-order poles in the form of a singular traveling wave with a time-dependent speed imprinted through the characteristics of the preceding blow-up event. [ABSTRACT FROM AUTHOR]

We study kinetic models for swarming. The interaction between individuals is given by self-propelling and friction forces, alignment, and noise. We consider that each individual relaxes its velocity toward some average velocity, such that the total momentum does not change. We concentrate on fluid models obtained when the time and space scales become very large. We derive first and second order approximations. [ABSTRACT FROM AUTHOR]

For a specific set of parameters, we analyze the stability of a one-spike equilibrium solution to the one-dimensional Gierer-Meinhardt reaction-diffusion model with delay in the components of the reaction-kinetics terms. Assuming slow activator diffusivity, we consider instabilities due to Hopf bifurcation in both the spike position and the spike profile for increasing values of the time-delay parameter T. Using method of matched asymptotic expansions it is shown that the model can be reduced to a system of ordinary differential equations representing the position of the slowly evolving spike solution. The reduced evolution equations for the one-spike solution undergoes a Hopf bifurcation in the spike position in two cases: when the negative feedback of the activator equation is delayed, and when delay is in both the negative feedback of the activator equation and the non-linear production term of the inhibitor equation. Instabilities in the spike profile are also considered, and it is shown that the spike solution is unstable as T is increased beyond a critical Hopf bifurcation value TH, and this occurs for the same cases as in the spike position analysis. In all cases, the instability in the profile is triggered before the positional instability. If however the degradation of activator is delayed, we find stable positional oscillations can occur in this system. [ABSTRACT FROM AUTHOR]

We mainly construct and analyze the multi elliptic‐localized solutions under the background of elliptic function solutions for the focusing modified Korteweg‐de Vries (mKdV) equation. Based on the Darboux–Bäcklund transformation, we provide a uniform expression for these solutions by the Jacobi theta functions. The asymptotic behaviors of multi elliptic‐localized solutions are provided directly in two categories. By the consistent asymptotic expression of those solutions, we obtain that the collisions between the elliptic‐breathers/solitons are elastic. Moreover, a sufficient condition of the strictly elastic collision between the solitons and breathers has been given by the symmetric analysis. In addition, as k→0+$k\rightarrow 0^{+}$, the multi elliptic‐localized solutions degenerate into solitons, breathers, or soliton‐breather solutions, which implies that we extend the solutions from the constant and vanishing backgrounds to the periodic solutions backgrounds. Moreover, we illustrate figures of the multi elliptic‐localized solutions to visualize the above analysis. [ABSTRACT FROM AUTHOR]

The Cahn--Hilliard equation with degenerate mobility is used in several areas including the modeling of living tissues, following the theory of mixtures. We are interested in quantifying the pressure jump at the interface between phases in the case of incompressible flows. To do so, we depart from the spherically symmetric dynamical compressible model and include an external force. We prove existence of stationary states as limits of the parabolic problems. Then we prove the incompressible limit and characterize compactly supported stationary solutions. This allows us to compute the pressure jump in the small dispersion regime and in particular the force dependent curvature effect. [ABSTRACT FROM AUTHOR]

The Fourier transform, F, on RN (N > 1) transforms the Cauchy problem for a strongly damped beam equation with structural damping utt - Δut + α(Δ²)u - Δu = 0; α > 0, to an ordinary differential equation in time. With u(t; x) being the weak solution of the problem given by the Fourier transform, the goal of the paper is to determine the asymptotic expansion of the squared L²-norm of u(t; x) as t - ∞. With suitable, additional assumptions on the initial data u(0; x) and ut(0; x), we establish the behavior of the squared L²-norm of u(t; x) as t - ∞. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the solutions to an initial boundary value problem of the Hunter-Saxton equation in the half space when the viscosity tends to zero. By means of the asymptotic analysis with multiple scales, we first formally derive the equations for boundary layer profiles. Next, we study the well-posedness of the equations for the boundary layer profiles by using the compactness argument. Moreover, we construct an accurate approximate solution and use the energy method to obtain the convergence results of the vanishing viscosity limit. [ABSTRACT FROM AUTHOR]

Using a multiscale analysis, we derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation with a cutoff diffusion mobility M(u) that is degenerate for u ≤ 0 and a continuously differentiable double-well potential W(u). We show that the bilayer interface does not move in the t = O(1) time scale. The interface motion occurs in the t = O(ε−1 ) time scale and is determined by porous medium diffusion processes in both phases with no jumps on the interface. In the longer O(ε−2 ) time scale, the interface motion is a complex combination of porous medium diffusion processes in both phases and the property of mass conservation. [ABSTRACT FROM AUTHOR]

This paper presents asymptotic formulas in the case of the following two problems for the Pucci's extremal operators M ± . It is considered the solution u ε (x) of − ε 2 M ± ∇ 2 u ε + u ε = 0 in Ω such that u ε = 1 on Γ. Here, Ω ⊂ R N is a domain (not necessarily bounded) and Γ is its boundary. It is also considered v (x , t) the solution of v t − M ± ∇ 2 v = 0 in Ω × (0 , ∞) , v = 1 on Γ × (0 , ∞) and v = 0 on Ω × { 0 }. In the spirit of their previous works [Berti D, Magnanini R. Asymptotics for the resolvent equation associated to the game-theoretic p-laplacian. Appl Anal. 2019;98(10):1827–1842.; Berti D, Magnanini R. Short-time behavior for game-theoretic p-caloric functions. J Math Pures Appl (9). 2019;(126):249–272.], the authors establish the profiles as ϵ or t → 0 + of the values of u ε (x) and v (x , t) as well as of those of their q-means on balls touching Γ. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime. [ABSTRACT FROM AUTHOR]

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to 3 × 3 matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known 3 × 3 Lax pair of this equation, the discrete generalized (m , 3 N − m) -fold Darboux transformation was constructed for the first time and extended from the 2 × 2 Lax pair to the 3 × 3 Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants. [ABSTRACT FROM AUTHOR]

We study the concentration phenomenon of the Lane-Emden equation with vanishing potentials { − Δ u n = W n (x) u n p n , u n > 0 , in Ω , u n = 0 , on ∂ Ω , ∫ Ω p n W n (x) u n p n d x ≤ C , where Ω is a smooth bounded domain in R 2 , W n (x) ≥ 0 are bounded functions with zeros in Ω, and p n → ∞ as n → ∞. A typical example is W n (x) = | x | 2 α with 0 ∈ Ω , i.e. the equation turns to be the well-known Hénon equation. The asymptotic behavior for α = 0 has been well studied in the literature. While for α > 0 , the problem becomes much more complicated since a singular Liouville equation appears as a limit problem. In this paper, we study the case α > 0 and prove a quantization property (suppose 0 is a concentration point) p n | x | 2 α u n (x) p n − 1 + t → 8 π e t 2 ∑ i = 1 k δ a i + 8 π (1 + α) e t 2 c t δ 0 , t = 0 , 1 , 2 , for some k ≥ 0 , a i ∈ Ω ∖ { 0 } and some c ≥ 1. Moreover, for α ∉ N , we show that the blow up must be simple, i.e. c = 1. As applications, we also obtain the complete asymptotic behavior of ground state solutions for the Hénon equation. [ABSTRACT FROM AUTHOR]

In this work, we explore the rogue wave patterns in the coupled Fokas–Lenells equation by using the Darboux transformation. We demonstrate that when one of the internal parameters is large enough, the general high-order rogue wave solutions generated at a branch point of multiplicity three can be decomposed into some first-order outer rogue waves and a lower-order inner rogue wave. Remarkably, the positions and the orders of these outer and inner rogue waves are intimately related to Okamoto polynomial hierarchies. • A compact determinant formula of rogue wave solutions is constructed. • The rogue wave patterns are first derived with the aid of the Darboux transformation. • Some examples of rogue wave patterns are exhibited. [ABSTRACT FROM AUTHOR]

In this paper we study the following nonlinear Choquard equation − Δ u + u = ln 1 | x | ∗ F (u) f (u) , in R 2 , where f ∈ C 1 (R , R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H 1 (R 2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11]. [ABSTRACT FROM AUTHOR]

We discuss conditions for Lipschitz and C 1 regularity of solutions for a uniformly elliptic equation in divergence form. We focus on coefficients having the form that was introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz or C 1 regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the positive solutions of the Wolfftype integral system where n ≥ 1, γ > 1, β > 0, βγ ̸= n and Ci(x) (i = 1, · · ·, m) are double bounded functions. Discussed in three situations of the integral system, a special iteration scheme in integral form as well as in differential form are applied. By some critical asymptotic analysis with Wolff potential integral estimates, we study the existence and nonexistence of the positive solutions, especially the radial solutions. Furthermore we obtain the asymptotic rates and the integrability of positive solutions. [ABSTRACT FROM AUTHOR]

Homogenization of an elliptic PDE with periodic oscillating coefficients and associated optimal control problems with energy type cost functional is considered. The domain is a 3-dimensional region (method applies to any n dimensional region) with oscillating boundary, where the base of the oscillation is curved and it is given by a Lipschitz function. Further, we consider general elliptic PDE with oscillating coefficients. We also include very general type functional of Dirichlet type given with oscillating coefficients which can be different from the coefficient matrix of the equation. We introduce appropriate unfolding operators and approximate unfolded domain to study the limiting analysis. The present article is new in this generality. [ABSTRACT FROM AUTHOR]

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin‐Bona‐Mahony (BBM) equation ut+uux=uxxt are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg‐de Vries equation ut+uux+uxxx=0. The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two‐phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two‐phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self‐similar solution of the BBM equation whose limit as t→∞ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schrödinger equations. The complex interplay between nonlocality, nonlinearity, and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem. [ABSTRACT FROM AUTHOR]

We consider a family of linearly viscoelastic shells with thickness 2 ε , all having the same middle surface S = θ (ω ¯) ⊂ I R 3 , where ω ⊂ I R 2 is a bounded and connected open set with a Lipschitz-continuous boundary γ and θ ∈ C 3 (ω ¯ ; I R 3) . The shells are clamped on a portion of their lateral face, whose middle line is θ (γ 0) , where γ 0 is a non-empty portion of γ . The aim of this work is to show that the viscoelastic Koiter's model is the most accurate two-dimensional approach in order to solve the displacements problem of a viscoelastic shell. Furthermore, the solution of the Koiter's model, ξ K ε = (ξ K , i ε) , is in H 1 (0 , T ; V K (ω)) , with ξ K , i ε : [ 0 , T ] × ω ¯ → R the covariant components of the displacements field ξ K , i ε a i of the points of the middle surface S and where V K (ω) : = { η = (η i) ∈ H 1 (ω) × H 1 (ω) × H 2 (ω) ; η i = ∂ ν η 3 = 0 in γ 0 } , with ∂ ν denoting the outer normal derivative along γ . Under the same assumptions as for the viscoelastic elliptic membranes problem, we show that the displacement field, ξ K , i ε a i , converges to ξ i a i (the solution of the two-dimensional problem for a viscoelastic elliptic membrane) in H 1 (0 , T ; H 1 (ω)) for the tangential components, and in H 1 (0 , T ; L 2 (ω)) for the normal component, as ε → 0 . Under the same assumptions as in the viscoelastic flexural shell problem, we show that the displacement field, ξ K , i ε a i , converges to ξ i a i (the solution of the two-dimensional problem for a viscoelastic flexural shell) in H 1 (0 , T ; H 1 (ω)) for the tangential components, and in H 1 (0 , T ; H 2 (ω)) for the normal component, as ε → 0 . Also, we obtain analogous results assuming the same assumptions as in the viscoelastic generalized membranes problem. Therefore, we justify the two-dimensional viscoelastic model of Koiter for all kind of viscoelastic shells. [ABSTRACT FROM AUTHOR]

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of N -fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal P T -symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals. [ABSTRACT FROM AUTHOR]

We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected media with Dirichlet conditions outside the domain. After showing the existence and uniqueness of a non-trivial bounded stationary state n + , we prove that it invades the unstable state zero exponentially fast in time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the monodromy and, in particular, the isomonodromy sets of accessory parameters for the Heun class equations. We show that the Heun class equations can be obtained as limits of the linear systems associated with the Painlevé equations when the Painlevé transcendents go to one of the actual singular points of the linear systems. The isomonodromy sets of accessory parameters for the Heun class equations are described by Taylor or Laurent coefficients of the corresponding Painlevé functions, or the associated tau functions, at the positions of the critical values. As an application of these results, we derive some asymptotic approximations for the isomonodromy sets of accessory parameters in the Heun class equations, including the confluent Heun equation, the doubly‐confluent Heun equation, and the reduced biconfluent Heun equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior as 𝜀 → 0 + of solutions u 𝜀 to the nonlocal stationary Fisher-KPP type equation 1 𝜀 m ∫ ℝ N J 𝜀 (x − y) (u 𝜀 (y) − u 𝜀 (x)) d y + u 𝜀 (x) (a (x) − u 𝜀 (x)) = 0  in  ℝ N , where 𝜀 > 0 and 0 ≤ m < 2. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution u 𝜀 and that u 𝜀 → a + as 𝜀 → 0 + where a + = max { 0 , a }. This generalizes the previously known results and answers an open question raised by Berestycki et al. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fourier-type condition on internal micro-surfaces. The first contribution of this work is the construction of a reliable linearization scheme that allows us, by a suitable choice of scaling arguments and stabilization constants, to prove the weak solvability of the microscopic model. Asymptotic behaviors of the microscopic solution with respect to the microscale parameter are thoroughly investigated in the second theme, based upon several cases of scaling. In particular, the variable scaling illuminates the trivial and non-trivial limits at the macroscale, confirmed by certain rates of convergence. Relying on classical results for homogenization of multiscale elliptic problems, we design a modified two-scale asymptotic expansion to derive the corresponding macroscopic equation, when the scaling choices are compatible. Moreover, we prove the high-order corrector estimates for the homogenization limit in the energy space , using a large amount of energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis. [ABSTRACT FROM AUTHOR]

The Cahn–Hilliard equation is one of the most common models to describe phase segregation processes in binary mixtures. Various dynamic boundary conditions have already been introduced in the literature to model interactions of the materials with the boundary more precisely. To take long-range interactions into account, we propose a new model consisting of a nonlocal Cahn–Hilliard equation with a nonlocal dynamic boundary condition comprising an additional boundary penalization term. We rigorously derive our model as the gradient flow of a nonlocal free energy with respect to a suitable inner product of order H − 1 containing both bulk and surface contributions. In the main model, the chemical potentials are coupled by a Robin type boundary condition depending on a specific relaxation parameter. We prove weak and strong well-posedness of this system, and we investigate the singular limits attained when this relaxation parameter tends to zero or infinity. [ABSTRACT FROM AUTHOR]

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the dynamics of dispersive optical solitons modeled via the Schrödinger–Hirota equation. The modulation instability of solutions is firstly studied in the presence of a small perturbation. With symbolic computation, the one-, two-, and three-soliton solutions are obtained through the Hirota bilinear method. The propagation and interaction of the solitons are simulated, and it is found the collision is elastic and the solitons enjoy the particle-like interaction properties. In the end, the asymptotic behavior is analyzed for the three-soliton solutions. [ABSTRACT FROM AUTHOR]

This paper studies the wave-breaking mechanism and dynamical behavior of solutions near the explicit self-similar singularity for the two component Camassa–Holm equations, which can be regarded as a model for shallow water dynamics and arising from the approximation of the Hamiltonian for Euler's equation in the shallow water regime. [ABSTRACT FROM AUTHOR]

This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e., low-Mach-number–slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a Gauss–Legendre collocation in space and an α-method in time. [ABSTRACT FROM AUTHOR]

We introduce a new family of paraxial asymptotic models that approximate the Vlasov-Maxwell equations in non-relativistic cases. This formulation is nth order accurate in a parameter n, which denotes the ratio between the characteristic velocity of the beam and the speed of light. This family of models is interesting, first because it is simpler than the complete Vlasov-Maxwell equation and then because it allows us to choose the model complexity according to the expected accuracy. [ABSTRACT FROM AUTHOR]

Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling solutions; cf. Healey, Li, and Cheng [J. Nonlinear Sci., 23 (2013), pp. 777-805]. These correspond to arbitrary phase shifts of the wrinkled pattern in the transverse direction. While such behavior is normally associated with problems in the presence of a continuous symmetry group, an unloaded rectangular sheet possesses only a finite symmetry group. In order to understand this phenomenon, we consider a simpler problem more amenable to analysis-a finite-length beam on a nonlinear softening foundation under axial compression. We first obtain asymptotic results via amplitude equations that are valid as a certain nondimensional beam length becomes sufficiently large. We deduce that any two phase shifts of a solution differ from one another by exponentially small terms in that length. We validate this observation with numerical computations, indicating the presence of solution orbits for sufficiently long beams. We refer to this as "hidden asymptotic symmetry". [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic analysis of 1D compressible Navier-Stokes-Vlasov equations. By taking advantage of the one space dimension, we obtain the hydrodynamic limit for compressible Navier-Stokes-Vlasov equations with the pressure P(ρ) = Aργ (γ > 1). The proof relies on weak convergence method. [ABSTRACT FROM AUTHOR]

The equation describing the over-reection of the slow magneto-sonic waves in plasma with background uniform shear ow is derived and analyzed in detail both analytically and numeri- cally. Using the methods of asymptotic analysis, analytical expressions for reection and transmission coefficients of the waves are obtained for relatively small shear rates. [ABSTRACT FROM AUTHOR]

The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyperbolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently. We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system. [ABSTRACT FROM AUTHOR]

We investigate a minimization problem on the pseudo‐relativistic Hartree equation for boson stars, in which the energy functional is defined by Ev(ψ)=12ψ,(−△+m2−m)ψ+i2⟨ψ,(v·∇)ψ⟩−14∫R31|x|∗|ψ|2|ψ|2dx,where v∈R3 denotes travelling velocity. Two kinds of asymptotic analysis with respect to m and the L2‐norm N=∫R3|ψ|2dx will be given, respectively. [ABSTRACT FROM AUTHOR]

In this paper, by using the long wave limit method, we study lump solution and interactional solution of the (2 + 1)-dimensional generalized breaking soliton equation without using bilinear form. The moving properties of the lump solution, and the interactional properties of a lump and a solitary wave, are analyzed theoretically and graphically with asymptotic analysis. [ABSTRACT FROM AUTHOR]

We consider a family of linearly viscoelastic shells of thickness 2 ε , all with the same middle surface and fixed on the lateral boundary. By using asymptotic analysis, we find that for external forces of a particular order of ε , a two-dimensional viscoelastic flexural shell model is an accurate approximation of the three-dimensional quasistatic problem. Most noticeable is that the limit problem includes a long-term memory that takes into account the previous history of deformations. We provide convergence results which justify our asymptotic approach. [ABSTRACT FROM AUTHOR]

Our objective is to identify two-dimensional equations that model an obstacle problem for a linearly elastic elliptic membrane shell subjected to a confinement condition expressing that all the points of the admissible deformed configurations remain in a given half-space. To this end, we embed the shell into a family of linearly elastic elliptic membrane shells, all sharing the same middle surface θ (ω ¯) , where ω is a domain in R 2 and θ : ω ¯ → E 3 is a smooth enough immersion, all subjected to this confinement condition, and whose thickness 2 ε > 0 is considered as a "small" parameter approaching zero. We then identify, and justify by means of a rigorous asymptotic analysis as ε approaches zero, the corresponding "limit" two-dimensional variational problem. This problem takes the form of a set of variational inequalities posed over a convex subset of the space H 0 1 (ω) × H 0 1 (ω) × L 2 (ω). The confinement condition considered here considerably departs from the Signorini condition usually considered in the existing literature, where only the "lower face" of the shell is required to remain above the "horizontal" plane. Such a confinement condition renders the asymptotic analysis substantially more difficult, however, as the constraint now bears on a vector field, the displacement vector field of the reference configuration, instead of on only a single component of this field. [ABSTRACT FROM AUTHOR]

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero. [ABSTRACT FROM AUTHOR]

In this work, we characterize the solution of a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection and migration between two habitats. Assuming that the effects of the mutations are small but nonzero, we show that the population's phenotypical distribution has at most two peaks and we give explicit conditions under which the population will be monomorphic (unimodal distribution) or dimorphic (bimodal distribution). More importantly, we provide a general method to determine the dominant terms of the population's distribution in each case. Our work, which is based on Hamilton-Jacobi equations with constraint, goes further than previous works where such tools were used, for different problems from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. The main elements for the computation of the dominant terms of the population's distribution are the convergence of the logarithmic transform of the solution to the unique solution of a Hamilton-Jacobi equation and the computation of the correctors. This method allows indeed to go further than the Gaussian approximation commonly used by biologists and makes a connection between the theories of adaptive dynamics and quantitative genetics. Our work being motivated by biological questions, the objective of this paper is to provide the mathematical details which are necessary for our biological results [S. Mirrahimi and S. Gandon, The equilibrium between selection, mutation and migration in spatially heterogeneous environments, in preparation]. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Hirota–Satsuma coupled KdV equations that can be used to describe the interaction of two classes of long waves. By using the Hirota bilinear method, the 1, 2, 3-soliton solutions are obtained. On this basis, the asymptotic analysis of soliton solutions proves that the collisions between solitons are elastic, and a set of visual figure is given to illustrate the results. • Integral terms and mixed derivative terms are added to the equation to describe more complex practical problems. • Function expansions can get richer results with more general expansions. • Not only bright soliton solutions and dark soliton solutions, but also periodic soliton solutions are obtained. • A wealth of graphic material is provided for readers to understand. • The asymptotic behavior of 2-, 3- soliton solutions is studied. [ABSTRACT FROM AUTHOR]

In this paper, we apply the asymptotic expansion method to the mechanical problem of beam equilibrium, aiming to derive a new beam model. The asymptotic procedure will lead to a series of mechanical problems at different order, solved successively. For each order, new transverse (in-plane) deformation and warping (out of plane) deformation modes are determined, in function of the applied loads and the limits conditions of the problem. The presented method can be seen as a more simple and efficient alternative to beam model reduction techniques such as POD or PGD methods. At the end of the asymptotic expansion procedure, an enriched kinematic describing the displacement of the beam is obtained, and will be used for the formulation of an exact beam element by solving analytically the arising new equilibrium equations. A surprising result of this work, is that even for concentrated forces (Dirac delta function), we obtain a very good representation of the beam’s deformation with only few additional degrees of freedom. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an innovative and systematic technique to study delay differential equations via polynomials. First, we review an intrinsic relation between delay differential equations and polynomials. From this relation, we obtain long time behaviors of the solutions to delay differential equations via asymptotic analysis of the corresponding polynomials. Moreover, we derive asymptotic formulas and upper bounds for the intrinsic growth rate of delay differential equations, as well as a Gronwall-type inequality for delay differential inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we consider delay-optimal power control for an energy harvesting wireless system with finite energy storage. The wireless system is powered solely by a renewable energy source with bursty data arrivals, and is characterized by a data queue and an energy queue. We consider a delay-optimal power control problem and formulate an infinite horizon average cost Markov decision process (MDP). To deal with the curse of dimensionality, we introduce a virtual continuous time system and derive closed-form approximate priority functions for the discrete time MDP at various operating regimes. Based on the approximation, we obtain an online power control solution which is adaptive to the channel state information as well as the data and energy queue state information. The derived power control solution has a multi-level water-filling structure, where the water level is determined jointly by the data and energy queue lengths. In the simulations, we show that the proposed scheme has significant performance gain compared with various baselines. [ABSTRACT FROM PUBLISHER]

We study the integrable three-level coupled Maxwell–Bloch equations with the mixed focusing-defocusing case. The n -fold Darboux transformation is constructed on basis of a linear 3 × 3 matrix eigenvalue problem. As an application, the n -dark-dark soliton solution in a simple determinant form is presented and the asymptotic behavior of the n -dark-dark soliton solution is rigorously given. In particular, under the resonant mechanism, the unusual dark-dark soliton molecules which represent the soliton bound states are analytically demonstrated. The double- and triple-dark-dark soliton molecules that are composed of two and three solitons propagating with identical velocities are graphically shown, respectively. The elastic collisions between a double- or triple-dark-dark soliton molecule and a usual dark-dark soliton, and the elastic collision of two double-dark-dark soliton molecules are discussed via the standard asymptotic analysis method. • The n-dark-dark soliton solution is derived. • The unusual dark-dark soliton molecules are analytically demonstrated. • The elastic collisions are discussed via the asymptotic analysis method. [ABSTRACT FROM AUTHOR]