Searcharxiv⌕ Search

arXiv subjects

Nikolaos Tzirakis

Publications and source records attributed to Nikolaos Tzirakis.

At least 19 recordsLinked to original sources
arXiv2023

Well-posedness for the Schrodinger-KdV system on the half-line

In this paper we obtain improved local well-posedness results for the Schrödinger-KdV system on the half-line. We employ the Laplace-Fourier method in conjunction with the restricted norm method of Bourgain appropriately modified in order to accommodate the bounded operators of the half-line problem. Our result extends the previous local results in [6], [7] and [21] matching the results that Wu, [28], obtained for the real line system. We also demonstrate the uniqueness for the full range of locally well-posed solutions. In addition we obtain global well-posedness on the half-line for the energy solutions with zero boundary data, along with polynomial-in-time bounds for higher order Sobolev norms for the Schrödinger part.

math.AP↗
arXiv2019

Sharp well--posedness for the generalized KdV of order three on the half line

In this paper we study the generalized Korteweg de Vries (KdV) equation with the nonlinear term of order three: $(u^{3+1})_x$. We prove sharp local well--posedness for the initial and boundary value problem posed on the right half line. We thus close the gap in the well--posedness theory of the generalized KdV which remained open after the seminal work of Colliander and Kenig in \cite{CK}.

math.AP↗
arXiv2016

Well-posedness and nonlinear smoothing for the "good" Boussinesq equation on the half-line

In this paper we study the regularity properties of the "good" Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. Moreover we prove that the nonlinear part of the solution on the half line is smoother than the initial data, obtaining half derivative smoothing of the nonlinear term in some cases. Our paper improves the result in [Himonas-Mantzavinos 2015], being the first result that constructs solutions for the initial and boundary value problem of the "good" Boussinesq equation below the $L^2$ space. Our theorems are sharp within the framework of the restricted norm method that we use and match the known results on the full line in [Kenig-Ponce-Vega 1996] and [Farah 2009].

math.AP↗
arXiv2016

Regularity properties of the Zakharov system on the half line

In this paper we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. Smoothing means that the nonlinear part of the solution on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic Zakharov and the Zakharov on the real line. Uniqueness is proved in the class of smooth solutions. When the boundary value of the Schrödinger part of the solution is zero, uniqueness can be extended to the full range of local solutions. Under the same assumptions on the initial data we also prove global-in-time existence and uniqueness of energy solutions. For more regular data we prove that all higher Sobolev norms grow at most polynomially-in-time.

math.AP↗
arXiv2016

Regularity properties of the cubic nonlinear Schrödinger equation on the half line

In this paper we study the local and global regularity properties of the cubic nonlinear Schrödinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. In particular, we prove that the nonlinear part of the cubic NLS on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic cubic NLS \cite{et2} and the cubic NLS on the line \cite{erin}. We also prove that in the defocusing case the norm of the solution grows at most polynomially-in-time while in the focusing case it grows exponentially-in-time. As a byproduct of our analysis we provide a different proof of an almost sharp local wellposedness in $H^s(\R^+)$. Sharp $L^2$ local wellposedness was obtained in \cite{holmer} and \cite{bonaetal}. Our methods simplify some ideas in the wellposedness theory of initial and boundary value problems that were developed in \cite{collianderkenig, holmer,holmer1,bonaetal}.

math.AP↗
arXiv2016

Multilinear Morawetz identities for the Gross-Pitaevskii hierarchy

This article consists of two parts. In the first part, we review the most recent proofs establishing quadratic Morawetz inequalities for the nonlinear Schrödinger equation (NLS). We also describe the applications of these estimates to the problem of quantum scattering. In the second part, we generalize some of the methods developed for the NLS by many authors to the case of Gross-Pitaevskii (GP) hierarchies. In particular, we prove both regular and interaction Morawetz identities for the GP hierarchy, which appear here for the first time in the literature.

math-ph↗
arXiv2015

The Structure of Global Attractors for Dissipative Zakharov Systems with Forcing on the Torus

The Zakharov system was originally proposed to study the propagation of Langmuir waves in an ionized plasma. In this paper, motivated by earlier work of the first and third authors, we numerically and analytically investigate the dynamics of the dissipative Zakharov system on the torus in 1 dimension. We find an interesting family of stable periodic orbits and fixed points, and explore bifurcations of those points as we take weaker and weaker dissipation.

math.AP↗
arXiv2014

Fractal solutions of linear and nonlinear dispersive partial differential equations

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schrödinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schrödinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schrödinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}.

math.AP↗
arXiv2012

Smoothing and Global Attractors for the Zakharov System on the Torus

In this paper we consider the Zakharov system with periodic boundary conditions in dimension one. In the first part of the paper, it is shown that for fixed initial data in a Sobolev space, the difference of the nonlinear and the linear evolution is in a smoother space for all times the solution exists. The smoothing index depends on a parameter distinguishing the resonant and nonresonant cases. As a corollary, we obtain polynomial-in-time bounds for the Sobolev norms with regularity above the energy level. In the second part of the paper, we consider the forced and damped Zakharov system and obtain analogous smoothing estimates. As a corollary we prove the existence and smoothness of global attractors in the energy space.

math.AP↗
arXiv2011

Long time dynamics for forced and weakly damped KdV on the torus

The forced and weakly damped Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. Starting from $L^2$ and mean-zero initial data we prove that the solution decomposes into two parts; a linear one which decays to zero as time goes to infinity and a nonlinear one which always belongs to a smoother space. As a corollary we prove that all solutions are attracted by a ball in $H^s$, $s\in(0,1)$, whose radius depends only on $s$, the $L^2$ norm of the forcing term and the damping parameter. This gives a new proof for the existence of a smooth global attractor and provides quantitative information on the size of the attractor set in $H^s$.

math.AP↗
arXiv2011

Global Smoothing for the Periodic KdV Evolution

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. It is shown that for $H^s$ initial data, $s>-1/2$, and for any $s_1<\min(3s+1,s+1)$, the difference of the nonlinear and linear evolutions is in $H^{s_1}$ for all times, with at most polynomially growing $H^{s_1}$ norm. The result also extends to KdV with a smooth, mean zero, time-dependent potential in the case $s\geq 0$. Our result and a theorem of Oskolkov for the Airy evolution imply that if one starts with continuous and bounded variation initial data then the solution of KdV (given by the $L^2$ theory of Bourgain) is a continuous function of space and time. In addition, we demonstrate smoothing for the modified KdV equation on the torus for $s>1/2$.

math.AP↗
arXiv2009

Asymptotically linear solutions in H^1 of the 2-d defocusing nonlinear Schroedinger and Hartree equations

In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schrödinger equation $i\partial_t v +Δv =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schrödinger (NLS) equation $i\partial_t u + Δu -|u|^{p-1}u=0$ for nonlinear powers $2<p<3$ and the existence of an $H^1$ solution $u(t)$ to the defocusing Hartree equation $i\partial_t u + Δu -(|x|^{-γ}\star|u|^{2})u=0$ for interaction powers $1<γ<2$, such that $\|u(t)-v(t)\|_{H^1} \to 0$ as $t\to +\infty$. This is a partial result toward the existence of well-defined continuous wave operators $H^1 \to H^1$ for these equations. For NLS in 2-d, such wave operators are known to exist for $p\geq 3$, while for $p\leq 2$ it is known that they cannot exist. The Hartree equation in 2-d only makes sense for $0<γ<2$, and it was previously known that wave operators cannot exist for $0<γ\leq 1$, while no result was previously known in the range $1<γ<2$. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis (2008) to a strategy devised by Nakanishi (2001). For the Hartree equation, we prove a new correlation estimate following the method of Colliander-Grillakis-Tzirakis (2008).

math.AP↗
arXiv2009

Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies

We consider solutions of the focusing cubic and quintic Gross-Pitaevskii (GP) hierarchies. We identify an observable corresponding to the average energy per particle, and we prove that it is a conserved quantity. We prove that all solutions to the focusing GP hierarchy at the $L^2$-critical or $L^2$-supercritical level blow up in finite time if the energy per particle in the initial condition is negative. Our results do not assume any factorization of the initial data.

math-ph↗
arXiv2008

Tensor products and Correlation Estimates with applications to Nonlinear Schrödinger equations

We prove new interaction Morawetz type (correlation) estimates in one and two dimensions. In dimension two the estimate corresponds to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. For the 2d case we provide a proof in two different ways. First, we follow the original approach of Lin and Strauss but applied to tensor products of solutions. We then demonstrate the proof using commutator vector operators acting on the conservation laws of the equation. This method can be generalized to obtain correlation estimates in all dimensions. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. We then apply the 2d estimate to nonlinear Schrödinger equations and derive a direct proof of Nakanishi's $H^1$ scattering result for every $L^{2}$-supercritical nonlinearity. We also prove scattering below the energy space for a certain class of $L^{2}$-supercritical equations.

math.AP↗
arXiv2007

Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$

We prove global well-posedness for low regularity data for the one dimensional quintic defocusing nonlinear Schrödinger equation. Precisely we show that a unique and global solution exists for initial data in the Sobolev space $H^{s}(\mathbb R)$ for any $s>{1/3}$. This improves the result in \cite{tz}, where global well-posedness was established for any $s>{4/9}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient in our proof is an interaction Morawetz estimate for the smoothed out solution $Iu$. As a byproduct of our proof we also obtain that the $H^{s}$ norm of the solution obeys polynomial-in-time bounds.

math.AP↗
arXiv2006

Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in One Dimension

We consider the $L^{2}$-critical quintic focusing nonlinear Schrödinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$ spherically symmetric blow-up solutions of the $L^{2}$-critical focusing NLS, there is a minimal amount of concentration of the $L^{2}$-norm (the mass of the ground state) at the origin. In this paper we prove the existence of a similar phenomenon for the one-dimensional case and rougher initial data, $(u_{0}\in H^{s}, s<1)$, without any additional assumption.

math.AP↗