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Symmetric and antisymmetric terms have been obtained in the framework of the variational approach for two-dimensional (2D) Coulomb systems of symmetric trions XXY. Stability diagrams and certain anomalies arising in the 2D space are…
In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Schr"odinger equations in one space dimension. It turns out that for a system there exists a small solution of which asymptotic…
We prove a weak form of the soliton resolution conjecture of bounded solutions of high-dimensional fourth-order Schrodinger equations. The result relies upon two properties to be proved: the asymptotic frequency localization and the…
The purpose of this document is to describe the solution and implementation of the time-independent and time-dependent Schr\"odinger using pseudospectral methods. Currently, the description is for single particle systems interacting with a…
We consider the inverse problems of for the fractional Schr\"odinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal…
We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving…
We review the recent developments in the theory of the one-dimensional tight-binding Schr\"odinger equation for a class of deterministic ergodic potentials. In the typical examples the potentials are generated by substitutional sequences,…
We establish the existence of a nontrivial weak solution to strongly indefinite asymptotically linear and superlinear Schr\"odinger equations. The novelty is to identify the essential relation between the spectrum of the operator and the…
A class of semi-bounded solutions of the two-dimensional incompressible Euler equations satisfying either periodic or Dirichlet boundary conditions is examined. For smooth initial data, new blowup criteria in terms of the initial concavity…
We consider travelling periodical and quasiperiodical waves in single mode fibres, with weak birefringence and under the action of cross-phase modulation. The problem is reduced to the ``1:2:1" integrable case of the two-particle quartic…
Consider the Lienard system $ u'' + f(u) u' + g(u) = 0$ with a center at the origin 0. In the case where the period function $T$ is monotonic, we examine periodic solutions of the perturbed equation $ u'' + a(u)u' + f(u) = \epsilon h(t)$.…
Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian systems of PDEs, as well as certain nonlinear Schrodinger…
A system of two discrete nonlinear Schr\"odinger equations of the Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly periodic wave, whose long wave limit is also derived. As a by-product, several new solutions of…
Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi…
In this paper we consider a Schrodinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals…
We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic…
It is shown, that any sufficiently smooth periodic solution of the self-focusing Nonlinear Schr\"odinger equation can be approximated by periodic finite-gap ones with an arbitrary small error. As a corollary an analogous result for the…
The goal of this paper is to study the behavior of certain solutions to the Swift-Hohenberg equation on a one-dimensional torus $\mathbb{T}$. Combining results from $\Gamma$-convergence and ODE theory, it is shown that solutions…
We present several results concerning the semiclassical limit of the time dependent Schr\"odinger equation with potentials whose regularity doesn't guarantee the uniqueness of the underlying classical flow. Different topologies for the…
In this paper we obtain asymptotic formulas of arbitrary order for the Bloch eigenvalue and the Bloch function of the periodic Schrodinger operator of arbitrary dimension, when corresponding quasimomentum lies near a diffraction hyperplane.…