Alexander Pankov

G-convergence and homogenization of nonlinear partial differential operators

Preface. Notations. 1. G-Convergence of Abstract Operators. 2. Strong G-Convergence of Nonlinear Elliptic Operators. 3. Homogenization of Elliptic Operators. 4. Nonlinear Parabolic Operators. A: Homogenization of Nonlinear Difference Schemes. B: Open Problems. References. Index.

Gap solitons in periodic discrete nonlinear Schrödinger equations

It is shown that the periodic discrete nonlinear Schrodinger equation, with cubic nonlinearity, possesses gap solutions, i.e. standing waves, with the frequency in a spectral gap, that are exponentially localized in the spatial variable. The proof is based on the linking theorem in combination with periodic approximations.

Standing wave solutions of the discrete non-linear Schrödinger equations with unbounded potentials, II

In this article, we prove the existence of infinitely many non-trivial standing wave solutions of the discrete non-linear Schrödinger equation with the unbounded potential at infinity by using the linking theorem.

Standing waves for discrete nonlinear Schrödinger equations: sign-changing nonlinearities

We consider the discrete nonlinear Schrödinger equation with infinitely growing potential and sign-changing power nonlinearity. Making use of critical point theory, we prove an existence and multiplicity result for standing wave solutions.

Periodic and solitary traveling wave solutions for the generalized Kadomtsev–Petviashvili equation

This paper is concerned with traveling waves for the generalized Kadomtsev-Petviashvili equation (w t +w ξξξ +f(w) ξ ) ξ =w yy , (ξ,y) ∈ R 2 , t∈ R, i.e. solutions of the form w(t,ξ,y) = u(ξ - et, y). We study both. solutions periodic in x = ξ - ct and solitary waves, which are decaying in x, and their interrelations In particular, we prove the existence of a sequence of k-periodic solutions, k ∈ N, which is uniformly bounded in norm and converges to a solitary wave in a suitable topology. This result also holds for the corresponding ground states, i.e. solutions with minimal energy.

Multivalued elliptic operators with nonstandard growth

Abstract The paper is devoted to the Dirichlet problem for monotone, in general multivalued, elliptic equations with nonstandard growth condition. The growth conditions are more general than the well-known p ⁢ ( x ) {p(x)} growth. Moreover, we allow the presence of the so-called Lavrentiev phenomenon. As consequence, at least two types of variational settings of Dirichlet problem are available. We prove results on the existence of solutions in both of these settings. Then we obtain several results on the convergence of certain types of approximate solutions to an exact solution.

Global well-posedness for discrete non-linear Schrödinger equation

We consider the initial value problem for the discrete non-linear Schrödinger equation in weighted l 2-spaces. Under quite general assumptions we prove that the problem is globally well-posed. The proof is based on simple general facts on abstract evolution equations.

Exponential decay of eigenfunctions of Schrödinger operators on infinite metric graphs

Abstract This paper concerns Schrödinger operators on infinite metric graphs. We show that, under natural assumptions, eigenfunctions corresponding to isolated eigenvalues of finite multiplicity decay at infinity exponentially fast.

Schrödinger operators with locally integrable potentials on infinite metric graphs

The paper is devoted to Schrödinger operators on infinite metric graphs. We suppose that the potential is locally integrable and its negative part is bounded in certain integral sense. First, we obtain a description of the bottom of the essential spectrum. Then we prove theorems on the discreteness of the negative part of the spectrum and of the whole spectrum that extend some classical results for one dimensional Schrödinger operators.

Almost Periodic Elliptic Equations: Sub- and Super-solutions

The method of sub- and super-solutions is a classical tool in the theory of second-order differential equations. It is known that this method does not have a direct extension to almost periodic equations. We show that if an almost periodic second-order semi-linear elliptic equation possesses an ordered pair of almost periodic sub- and super-solutions, then very many equations in the envelope have either almost automorphic solutions, or Besicovitch almost periodic solutions. In addition, we provide an application to almost periodically forced pendulum equations.