Modulated waves in a semiclassical continuum limit of an integrable NLS chain

A one‐dimensional integrable lattice system of ODEs for complex functions Q_n(τ) that exhibits dispersive phenomena in the phase is studied. We consider wave solutions of the local form Q_n(τ) ∼ qexp(i(kn + ωτ + c)), in which q, k, and ω modulate on long time and long space scales t = ετ and x = εn. Such solutions arise from initial data of the form Q_n(0) = q(nε) exp(iϕ(nε)/ε), the phase derivative ϕ′ 0 giving the local value of the phase difference k. Formal asymptotic analysis as ε → 0 yields a first‐order system of PDEs for q and ϕ′ as functions of x and t. A certain finite subchain of the discrete system is solvable by an inverse spectral transform. We propose formulae for the asymptotic spectral data and use them to study the limiting behavior of the solution in the case of initial data |Q_n| < 1, which yield hyperbolic PDEs in the formal limit. We show that the hyperbolic case is amenable to Lax‐Levermore theory. The associated maximization problem in the spectral domain is solved by means of a scalar Riemann‐Hilbert problem for a special class of data for all times before breaking of the formal PDEs. Under certain assumptions on asymptotic behaviors, the phase and amplitude modulation of the discrete systems is shown to be governed by the formal PDEs. Modulation equations after breaking time are not studied. Full details of the WKB theory and numerical results are left to a future exposition. © 2000 John Wiley & Sons, Inc.     

A generalized normal form and formal equivalence of two-dimensional systems with quadratic zero approximation: IV

We continue the study of invertible formal transformations of two-dimensional autonomous systems of differential equations with zero approximation represented by homogeneous polynomials of degree 2 and with perturbations in the form of power series without terms of order < 3. In the regular case, we consider systems that have the canonical form (αx ₁ ² ? sgnα x ₂ ² , x ₁ x ₂) with α ≠ 0 as the zero approximation. For such systems, we obtain resonance equations in closed form and use them to prove the theorem on the formal equivalence of systems and establish a generalized normal form to which any original system can be reduced by an invertible change of variables.     

A linear system of differential equations with small parameter at a part of derivatives,a deviation of the argument,and a turning point

For a system of differential equations with small parameter at a part of derivatives, a linear deviation of the argument, and a turning point, we obtained conditions, under which its solutions are solutions of a system of differential equations with small parameter at a part of derivatives such that its matrices possess the asymptotic expansions at |ε| ≤ ε₀ with the coefficients holomorphic at |x| ≤ x ₀ . The existence and the infinite differentiability of a solution of the system of differential equations with small parameter at a part of derivatives and with a linear deviation of the argument in the presence of a turning point are proved.     

Transseries for a class of nonlinear difference equations *

Given a nonlinear analytic difference equation of level 1 with a formal power series solution ? ₀ we associate with it a stable manifold of solutions with asymptotic expansion ? ₀. This manifold can be represented by means of Borel summable series. All solutions with asymptotic expansion ? ₀ in some sector can be written as certain exponential series which are called transseries. Some of their properties are investigated: are resurgence properties and Stokes transition. Analogous problems for differential equations have been studied by Costin in [7]     

A mixed problem for the Laplace operator in a domain with moderately close holes

We investigate the behavior of the solution of a mixed problem in a domain with two moderately close holes. We introduce a positive parameter ε and we define a perforated domain Ω_ε obtained by making two small perforations in an open set. Both the size and the distance of the cavities tend to 0 as ε → 0. For ε small, we denote by u_ε the solution of a mixed problem for the Laplace equation in Ω_ε. We describe what happens to u_ε as ε → 0 in terms of real analytic maps and we compute an asymptotic expansion.     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

A method of asymptotic constructions with improved accuracy for a quasilinear singularly perturbed parabolic convection-diffusion equation

The Dirichlet problem on an interval for quasilinear singularly perturbed parabolic convection-diffusion equation is considered. The higher order derivative of the equation is multiplied by a parameter ε that takes any values from the half-open interval (0, 1]. For this type of linear problems, the order of the ε-uniform convergence (with respect to x and t) for the well-known schemes is not higher than unity (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge ε-uniformly at the rate of O(N ^?2ln² N + N ^?2 ₀), where N + 1 and N ₀ + 1 are the numbers of the mesh points with respect to x and t, respectively. On the x axis, piecewise uniform meshes that condense in the boundary layer are used. If the parameter value is small compared to the effective step of the spatial grid, the domain decomposition method is used, which is motivated by “asymptotic constructions.” Monotone approximations of “auxiliary” subproblems describing the main terms of the asymptotic expansion of the solution outside a neighborhood of the boundary layer neighborhood are used. In the neighborhood of the boundary layer (of the width O(ε ln N)) the first derivative with respect to x is approximated by the central difference derivative. These subproblems are successively solved in the subdomains on uniform grids. If the parameter values are not sufficiently small (compared to the effective step of the mesh with respect to x), the classical implicit difference schemes approximating the first derivative with respect to x by the central difference derivative are applied. To improve the accuracy in t, the defect correction technique is used. Notice that the calculation of the solution of the constructed difference scheme (the scheme based on the method of asymptotic constructions) can be considerably simplified for sufficiently small values of the parameter ε.     

Solution of the operator equationi∈dy/dt=A(t)y on intervals containing turning points

We obtain formal solutions of the equation i∈ dy/dt=A(t)y in the form of complete asymptotic expansions as ∈→0 on intervals containing parabolic or hyperbolic turning points. The highest orders of the power series in ∈ for the formal solutions are studied in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 357–371, March, 2000     

Smooth solutions for a p-system of mixed elliptic-hyperbolic type

In this note we analyze smooth solutions of a p-system of the mixed, elliptic-hyperbolic type. A motivating example for this is a 2-components reduction of the Benney moments chain which appears to be connected to the theory of integrable systems. We don’t assume a-priori that the solutions in question are in the Hyperbolic region. Our main result states that the only smooth solutions of the system which are periodic in x are necessarily constants. As for the initial value problem, we prove that if the initial data are strictly hyperbolic and periodic in x, then the solution cannot extend to [t ₀;+∞) and shocks are necessarily created.     

Some spectral relations

We consider the zeta function of a second-order differential operator which has a secon-dorder turning point: $$Lu = \frac{{d^2 u}}{{dx^2 }} + [\lambda ^2 q(x) + R(x)]u,$$ where q(x)=x²q₁(x), q₁(x)≠0 and u (0)=u (1)=0. We construct an asymptotic series and calculate regularized traces for the eigenvalues of this operator.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution

The Fourier method is used to obtain a classical solution of an initial-boundary value problem for a first-order partial differential equation with involution in the function and its derivative. The series Σ produced by the Fourier method as a formal solution of the problem is represented as Σ = S ₀ + (Σ − Σ₀₎, where Σ₀ is the formal solution of a special reference problem for which the sum S ₀ can be explicitly calculated. Refined asymptotic formulas for the solution of the Dirac system are used to show that the series Σ − Σ₀ and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem.     

Asymptotic form and infinite product representation of solution of a second order initial value problem with a complex parameter and a finite number of turning points

The paper studies the differential equation * $y'' + (\rho ^2 \varphi ^2 (x) - q(x))y = 0$ on the interval I = [0, 1], containing a finite number of zeros 0 < x ₁ < x ₂ < ... < x _m < 1 of ? ², i.e. so-called turning points. Using asymptotic estimates from [6] for appropriate fundamental systems of solutions of (*) as |ρ| → ∞, it is proved that, if there is an asymptotic solution of the initial value problem generated by (*) in the interval [0, x ₁), then the asymptotic solutions in the remaining intervals can be obtained recursively. Furthermore, an infinite product representation of solutions of (*) is studied. The representations are useful in the study of inverse spectral problems for such equations.     

Linear differential equations and multiple zeta values. II. A generalization of the WKB method

For linear differential equations x(n)+a₁x(n−1)+?+anx=0 (and corresponding linear differential systems) with large complex parameter λ and meromorphic coefficients aj=aj(t;λ) we prove existence of analogues of Stokes matrices for the asymptotic WKB solutions. These matrices may depend on the parameter, but under some natural assumptions such dependence does not take place. We also discuss a generalization of the Hukuhara-Levelt-Turritin theorem about formal reduction of a linear differential system near an irregular singular point t=0 to a normal form with ramified change of time to the case of systems with large parameter. These results are applied to some hypergeometric equations related with generating functions for multiple zeta values.     

Singularly perturbed two-dimensional parabolic problem in the case of intersecting roots of the reduced equation

The singularly perturbed parabolic equation ?u _t + ε²Δu ? f(u, x, ε) = 0, x ∈ D ? ?², t > 0 with Robin conditions on the boundary of D is considered. The asymptotic stability as t → ∞ and the global domain of attraction are analyzed for the stationary solution whose limit as ε → 0 is a nonsmooth solution to the reduced equation f(u, x, 0) = 0 that consists of two intersecting roots of this equation.     

Mixed problem for the wave equation with integrable potential in the case of two-point boundary conditions of distinct orders

We study a mixed problem for the wave equation with integrable potential and with two-point boundary conditions of distinct orders for the case in which the corresponding spectral problem may have multiple spectrum. Based on the resolvent approach in the Fourier method and the Krylov convergence acceleration trick for Fourier series, we obtain a classical solution u(x, t) of this problem under minimal constraints on the initial condition u(x, 0) = ?(x). We use the Carleson–Hunt theorem to prove the convergence almost everywhere of the formal solution series in the limit case of ?(x) ∈ L _p[0, 1], p > 1, and show that the formal solution is a generalized solution of the problem.     

Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x=0 and x=-1. We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p, any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε→0) of the form ∑Y_n(x/ε^′)ε^′n, where ε^′p+1=ε, for x=ε^′X with X large enough, but independent of ε. In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation.     

Linear Differential Equations with Variable Coefficients in Regular and Some Singular Cases

In this paper the asymptotic properties as t → + ∞ for a single linear differential equation of the form x⁽ⁿ⁾ + a₁ (t)x^(n?1)+…. + a_n(t)x = 0, where the coefficients aj (z) are supposed to be of the power order of growth, are considered. The results obtained in the previous publications of the author were related to the so called regular case when a complete set of roots {λ,(t)}, j = 1, 2, …, n of the characteristic polynomial yⁿ + a₁ (t)y^n?1 + … + a_n(t) possesses the property of asymptotic separability. One of the main restrictions of the regular case consists of the demand that the roots of the set {λ,(t)} have not to be equivalent in pairs for t → + ∞. In this paper we consider the some more general case when the set of characteristic roots possesses the property of asymptotic independence which includes the case when the roots may be equivdent in pairs. But some restrictions on the asymptotic behaviour of their differences λ_i(t)→ λ_j(t) are preserved. This case demands more complicated technique of investigation. For this purpose the so called asymptotic spaces were introduced. The theory of asymptotic spaces is used for formal solution of an operator equation of the form x = A(x) and has the analogous meaning as the classical theory of solving this equation in Band spaces. For the considered differential equation, the main asymptotic terms of a fundamental system of solution is given in a simple explicit form and the asymptotic fundamental system is represented in the form of asymptotic Emits for several iterate sequences.     

Two linear equations in formal power series

The equations [gradφ(x)]^TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x₁,…,x_n)^T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian F_x(0).