Related papers: Diophantine approximation and the solubility of th…
Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to $p$-Laplace double obstacle problems involving the Schr\"odinger term: $-\Delta_p u + \mathbb{V}|u|^{p-2}u$ with bound constraints $\psi_1…
Motivated by questions in cryptography, we look for diophantine equations that are hard to solve but for which determining the number of solutions is easy.
We consider the free linear Schr\"odinger equation on a torus $\mathbb T^d$, perturbed by a hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$ u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u +…
We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schroedinger equation (DNLS) with cubic nonlinearity, near the continuous limit.We construct a family of solutions…
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi = |\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an…
We prove sharp estimates in a shrinking target problem for the action of an arbitrary subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ on the 2-torus. This can also be viewed as a non-commutative Diophantine approximation problem. The methods…
As a first step in exploring time-periodic solutions of the Einstein equations with a negative cosmological constant, we study the cubic conformal wave equation on the Einstein cylinder. Using a combination of numerical and perturbative…
We prove the existence of a 2-parameter family of small quasi-periodic in time solutions of discrete nonlinear Schr\"odinger equation (DNLS). We further show that all small solutions of DNLS decouples to this quasi-periodic solution and…
In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not $C^\infty$-smooth.
Perturbation approaches developed so far for the dark soliton solutions of the (fully integrable) defocusing nonlinear Schroedinger equation cannot describe the dynamics resulting from dissipative perturbations of the Ginzburg-Landau type.…
We study the large-time behavior of the solutions to the Schr\"odinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The…
We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
In this paper we consider stationary solutions to the nonlinear one-dimensional Schroedinger equation with a periodic potential and a Stark-type perturbation. In the limit of large periodic potential the Stark-Wannier ladders of the linear…
In this paper, we obtain the Nth-order rational solutions for the defocusing nonlocal nonlinear Schrodinger equation by the Darboux transformation and some limit technique. Then, via an improved asymptotic analysis method relying on the…
We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, aiming to provide new and simpler proofs of some key $L^\infty$ bounds and $L^p$ \emph{\textit{a priori}}…
We study the nonlinear Schr\"odinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove…
We study numerically stabilized solutions of the two-dimensional Schrodinger equation with a cubic nonlinearity. We discuss in detail the numerical scheme used and explain why choosing the right numerical strategy is very important to avoid…
We investigate the approximation of $d$-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness $s>0$ on the $d$-dimensional torus, where the approximation error is measured in the $L_2-$norm. In other…
For a certain class of solutions of the cubic nonlinear Sch\"odinger equation we prove non-existence in the generic case. In the nongeneric case we present a two-parameter set of solutions, bounded or unbounded, depending on corresponding…