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N. Tzirakis

Publications and source records attributed to N. Tzirakis.

14 recordsLinked to original sources
arXiv2025

The Zakharov System on the Upper Half-Plane

In this paper we study the Zakharov system on the upper half--plane $U=\{(x ,y)\in \R^2: y>0\}$ with non-homogenous boundary conditions. In particular we obtain low regularity local well--posedness using the restricted norm method of Bourgain and the Fourier--Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data. To our knowledge this is the first paper which establishes low regularity results for the 2d initial-boundary value Zakharov system.

math.AP↗
arXiv2024

Smoothing estimates for weakly nonlinear internal waves in a rotating ocean

In this paper we study the effect of rotation on nonlinear wave phenomena in weakly dispersive media modeled by the Korteweg-de Vries equation on the real line. It is well known that smoothing in the case of the KdV equation with periodic boundary conditions is a result of the presence of high frequency waves that weaken the nonlinearity through time averaging, \cite{bit}, \cite{et1}. High frequency interactions through normal form transformations can be thought of as adding a fast rotating term in certain toy PDE models that arrest the development of singularities, \cite{bit}. It is crucial for this phenomena that the zero Fourier mode can be removed due to the conservation of the mean. On the real line this mechanism breaks down as the resonance sets close to zero frequency are sizable and normal form transformations are not useful, \cite{imt}, and hence smoothing fails. The model we study is a perturbation of the KdV equation on a rotating frame of reference. A combination of a suitable normal form transformation along with the restricted norm method of Bourgain \cite{Bou2} implies smoothing on the real line.

math.AP↗
arXiv2019

The Fifth Order KP--II Equation on the Upper Half--plane

In this paper we study the fifth order Kadomtsev--Petviashvili II (KP--II) equation on the upper half-plane $U=\{(x,y)\in \R^2: y>0\}$. In particular we obtain low regularity local well-posedness using the restricted norm method of Bourgain and the Fourier-Laplace method of solving initial and boundary value problems. Moreover we prove that the nonlinear part of the solution is in a smoother space than the initial data.

math.AP↗
arXiv2018

Low-regularity global well-posedness for the Klein-Gordon-Schrödinger system on $\mathbb R^{+}$

In this paper we establish an almost optimal well-posedness and regularity theory for the Klein-Gordon-Schrödinger system on the half line. In particular we prove local-in-time well-posedness for rough initial data in Sobolev spaces of negative indices. Our results are consistent with the sharp well-posedness results that exist in the full line case and in this sense appear to be sharp. Finally we prove a global well-posedness result by combining the $L^2$ conservation law of the Schrödinger part with a careful iteration of the rough wave part in lower order Sobolev norms.

math.AP↗
arXiv2017

The derivative nonlinear Schrödinger equation on the half line

We study the initial-boundary value problem for the derivative nonlinear Schrödinger (DNLS) equation. More precisely we study the wellposedness theory and the regularity properties of the DNLS equation on the half line. We prove almost sharp local wellposedness, nonlinear smoothing, and small data global wellposedness in the energy space. One of the obstructions is that the crucial gauge transformation we use replaces the boundary condition with a nonlocal one. We resolve this issue by running an additional fixed point argument. Our method also implies almost sharp local and small energy global wellposedness, and an improved smoothing estimate for the quintic Schrödinger equation on the half line. In the last part of the paper we consider the DNLS equation on $\R$ and prove smoothing estimates by combining the restricted norm method with a normal form transformation.

math.AP↗
arXiv2017

Smoothing for the fractional Schrodinger equation on the torus and the real line

In this paper we study the cubic fractional nonlinear Schrodinger equation (NLS) on the torus and on the real line. Combining the normal form and the restricted norm methods we prove that the nonlinear part of the solution is smoother than the initial data. Our method applies to both focusing and defocusing nonlinearities. In the case of full dispersion (NLS) and on the torus, the gain is a full derivative, while on the real line we get a derivative smoothing with an $ε$ loss. Our result lowers the regularity requirement of a recent theorem of Kappeler et al. on the periodic defocusing cubic NLS, and extends it to the focusing case and to the real line. We also obtain estimates on the higher order Sobolev norms of the global smooth solutions in the defocusing case.

math.AP↗
arXiv2013

Existence and Uniqueness theory for the fractional Schrödinger equation on the torus

We study the Cauchy problem for the $1$-d periodic fractional Schrödinger equation with cubic nonlinearity. In particular we prove local well-posedness in Sobolev spaces, for solutions evolving from rough initial data. In addition we show the existence of global-in-time infinite energy solutions. Our tools include a new Strichartz estimate on the torus along with ideas that Bourgain developed in studying the periodic cubic NLS.

math.AP↗
arXiv2013

Talbot effect for the cubic nonlinear Schrödinger equation on the torus

We study the evolution of the one dimensional periodic cubic Schrödinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. For rational times the solution is a linear combination of finitely many translates of the initial data. Such a dichotomy was first observed by Talbot in an optical experiment performed in 1836. In this paper we prove that a similar phenomenon occurs in the case of the NLS equation.

math.AP↗
arXiv2011

High frequency perturbation of cnoidal waves in KdV

The Korteweg-de Vries (KdV) equation with periodic boundary conditions is considered. The interaction of a periodic solitary wave (cnoidal wave) with high frequency radiation of finite energy ($L^2$-norm) is studied. It is proved that the interaction of low frequency component (cnoidal wave) and high frequency radiation is weak for finite time in the following sense: the radiation approximately satisfies Airy equation.

math.AP↗
arXiv2010

Nearly linear dynamics of nonlinear dispersive waves

Dispersive averaging effects are used to show that KdV equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly. Numerical simulations are presented which indicate high accuracy of this approximation. Furthermore, this result is applied to shallow water wave dynamics in the limit of KdV approximation, which is obtained by asymptotic analysis in combination with numerical simulations of KdV.

math-ph↗
arXiv2010

Near-linear Dynamics for Shallow Water Waves

It is shown that spatially periodic one-dimensional surface waves in shallow water behave almost linearly, provided large part of the energy is contained in sufficiently high frequencies. The amplitude is not required to be small (apart from the shallow water approximation assumption) and the near-linear behavior occurs on a much longer time scale than might be anticipated based on the amplitude size. Heuristically speaking, this effect is due to the nonlinearity getting averaged by the dispersive action. This result is obtained by an averaging procedure, which is briefly outlined, and is also confirmed by numerical simulations.

physics.flu-dyn↗
arXiv2009

Remarks on global a priori estimates for the nonlinear Schrödinger equation

We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted $L_t^4L_γ^4$ Strichartz norm of the solution on any curve $γ$ in $\mathbb R^2$. For the specific case of a straight line we upgrade this estimate to a weighted Strichartz estimate valid in the full plane.

math.AP↗
arXiv2007

Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\R^2$

We prove global well-posedness for low regularity data for the $L^2-critical$ defocusing nonlinear Schrödinger equation (NLS) in 2d. More precisely we show that a global solution exists for initial data in the Sobolev space $H^{s}(\mathbb R^2)$ and any $s>{2/5}$. This improves the previous result of Fang and Grillakis where global well-posedness was established for any $s \geq {1/2}$. We use the $I$-method to take advantage of the conservation laws of the equation. The new ingredient is an interaction Morawetz estimate similar to one that has been used to obtain global well-posedness and scattering for the cubic NLS in 3d. The derivation of the estimate in our case is technical since the smoothed out version of the solution $Iu$ introduces error terms in the interaction Morawetz inequality. A byproduct of the method is that the $H^{s}$ norm of the solution obeys polynomial-in-time bounds.

math.AP↗