Institute of Mathematics with Computer Center

In this article, the authors report 3 experiments demonstrating that a simple booklet containing self-explanation training, designed to focus students' attention on logical relationships within a mathematical proof, can significantly improve their proof comprehension.

(u 2 xx − r(u)) + cux, r (5) = 0 (1) appeared (up to change u = p(ũ), ˙p 2 = r(p)) in [1] for the first time in connection with study of finite-gap solutions of the Kadomtsev-Petviashvili equation. The distinctive feature of the equation (1) is that, accordingly to [2], no differential substitution exists connecting it with other KdV-type equations. This property impedes the construction of the Bäcklund transformation (BT) which in other cases can be obtained by composition of two differential substitutions. Nevertheless, we demonstrate that (1) admits BT which connects it with other equation of the same form vt = vxxx − 3

We consider the Laplacian in a strip R×(0,d) with the boundary condition which is Dirichlet except at the segment of a length 2a of one of the boundaries where it is switched to Neumann. This operator is known to have a non-empty and simple discrete spectrum for any a>0. There is a sequence 0<a1<a2<⋯ of critical values at which new eigenvalues emerge from the continuum when the Neumann window expands. We find the asymptotic behavior of these eigenvalues around the thresholds showing that the gap is in the leading order proportional to (a−an)2 with an explicit coefficient expressed in terms of the corresponding threshold-energy resonance eigenfunction.

We consider an infinite planar straight strip perforated by small holes along a curve. In such a domain, we consider a general second-order elliptic operator subject to classical boundary conditions on the holes. Assuming that the perforation is non-periodic and satisfies rather weak assumptions, we describe all possible homogenized problems. Our main result is the norm-resolvent convergence of the perturbed operator to a homogenized one in various operator norms and the estimates for the rate of convergence. On the basis of the norm-resolvent convergence, we prove the convergence of the spectrum.

The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues.

This paper reveals an intrinsic feature of thermal transport underlying a long-range interacting Fermi-Pasta-Ulam chain, showing a high length-divergence of thermal conductivity. Its mechanism is related to a new heat diffusion process, responsible by the peculiar traveling discrete breathers' dynamics and their weak interactions, resulted from system's weaker chaotic property.

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.

In this paper with the help of the spectral method we obtain a criterion for the unique solvability of the inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain. This problem implies the search of the unknown right-hand side.

The analogue of the notion of the zero-curvature representation is given for equations of the discrete Toda lattice type on an arbitrary planar graph. Several examples are presented which generalize known integrable equations on 2.

This article is published in the journal Notices of the American Mathematical Society and the definitive version is available at \thttp://dx.doi.org/10.1090/noti1263.

We consider regular and singular perturbations of the Dirichlet and Neumann boundary value problems for the Helmholtz equation in n -dimensional cylinders. The existence of eigenvalues and their asymptotics are studied.

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacian, an indefinite nonlinearity, and depend on a real parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A special focus is given to the extreme value <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi> λ </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\lambda ^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Nehari manifold that determines the threshold of applicability of the Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where the parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi> λ </mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> lies above the threshold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi> λ </mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\lambda ^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is obtained.

Abstract We consider standing lattice solitons for discrete nonlinear Schrödinger equation with saturation (NLSS), where so-called transparent points were recently discovered. These transparent points are the values of the governing parameter (e.g. the lattice spacing) for which the Peierls–Nabarro barrier vanishes. In order to explain the existence of transparent points, we study a solitary wave solution in the continuous NLSS and analyse the singularities of its analytic continuation in the complex plane. The existence of a quadruplet of logarithmic singularities nearest to the real axis is proven and applied to two settings: (i) the fourth-order differential equation arising as the next-order continuum approximation of the discrete NLSS and (ii) the advance-delay version of the discrete NLSS. In the context of (i), the fourth-order differential equation generally does not have solitary wave solutions due to small oscillatory tails. Nevertheless, we show that solitary waves solutions exist for specific values of governing parameter that form an infinite sequence. We present an asymptotic formula for the distance between two subsequent elements of the sequence in terms of the small parameter of lattice spacing. To derive this formula, we used two different analytical techniques: the semi-classical limit of oscillatory integrals and the beyond-all-order asymptotic expansions. Both produced the same result that is in excellent agreement with our numerical data. In the context of (ii), we also derive an asymptotic formula for values of lattice spacing for which approximate standing lattice solitons can be constructed. The asymptotic formula is in excellent agreement with the numerical approximations of transparent points. However, we show that the asymptotic formulas for the cases (i) and (ii) are essentially different and that the transparent points do not generally imply existence of continuous standing lattice solitons in the advance-delay version of the discrete NLSS.

Characteristic Lie rings for Toda type 2+1 dimensional lattices are defined. Some proper-ties of these rings are studied. Infinite sequence of special kind modules are introduced. It is proved that for known integrable lattices these modules are finitely generated. Classification algorithm based on this observation is briefly discussed.

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.

We study the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables. By integrability, we mean the presence of reductions of a chain to a system of hyperbolic equations of an arbitrarily high order that are integrable in the Darboux sense. Darboux integrability admits a remarkable algebraic interpretation: the Lie—Rinehart algebras related to both characteristic directions corresponding to the reduced system of hyperbolic equations must have a finite dimension. We discuss a classification algorithm based on the properties of the characteristic algebra and present some classification results. We find new examples of integrable equations.

Generalized symmetry integrability test for discrete equations on the square lattice is studied. Integrability conditions are discussed. A method for searching higher symmetries (including non-autonomous ones) for quad graph equations is suggested based on characteristic vector fields.

We study a differential-difference equation of the form tx(n+1)=f(t(n),t(n+1),tx(n)) with unknown t=t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f(x,y,z)=z+d(x,y).

For a system of second-order ordinary differential equations conditions of linearizability to the form x'' = 0 are well known. However, an arbitrary linear system need not be equivalent via an invertible point transformation to this simple form. We provide the criteria for a system of two second-order equations to be mapped to the linear system of the general form. Necessary and sufficient conditions for linearization by means of a point transformation are given in terms of coefficients of the system. These results are illustrated with a number of examples.

We consider systems of two nonlinear nonautonomous differential equations on the real line which arise when averaging rapid nonlinear vibrations. We study the Lyapunov stability of solutions with infinitely increasing amplitude. Such solutions are related to the description of the initial stage of autoresonance or resonance trapping in oscillating nonlinear systems with a small perturbation. We obtain conditions on the coefficients of the equations under which the increasing solutions are stable or unstable. The problem is reduced to the analysis of an equilibrium. The stability of the equilibrium is studied by the Lyapunov second method. The construction of Lyapunov functions is based the presence of dissipative terms with coefficients moderately decaying in time.