On the existence of ground states for nonlinear Schrödinger-Poisson equation

This paper is concerned with the existence of ground states for the Schrödinger-Poisson equation , where V(u) is a Hartree type nonlinearity, stemming from the coupling with the Poisson equation, which includes the so-called doping profile or impurities. By means of variational methods in the energy space we show that ground states exist and belong to the Schwartz space of rapidly decreasing functions whenever total charge not exceed some critical value, it is also shown that for values of the total charge greater than this critical value, energy is not bounded from below. In addition, we show that this critical value is the total charge given by the impurities.     

On the well-posedness of the Schrödinger-Korteweg-de Vries system

We prove that the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed for the initial data belonging to the Sobolev spaces L²(R)×H−3/4(R), and Hs(R)×H−3/4(R) (s>−1/16) for the resonant case. The new ingredient is that we use the -type space, introduced by the first author in Guo (2009) [10], to deal with the KdV part of the system and the coupling terms. In order to overcome the difficulty caused by the lack of scaling invariance, we prove uniform estimates for the multiplier. This result improves the previous one by Corcho and Linares (2007) [6].     

On the Cauchy problem for a coupled system of third-order nonlinear Schrödinger equations

We investigate some well-posedness issues for the initial value problem (IVP) associated with the system

     

The Cauchy problem for the Schrödinger-KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.     

Further regularity results for almost cubic nonlinear Schrödinger equation

We present three results related with the regularity of solutions of the almost cubic NLS. In the first one, following Ozawa’s idea, we establish mass and energy conservation for the solutions without regularizing the initial datum. Our second result is the H^s well-posedness for the Cauchy problem for 0<s<1. Finally, we show that the same solutions are also in some Bourgain spaces for possibly a smaller time interval. In all of our results, the non-local nonlinear term in the equation is shown to act like a cubic nonlinearity on the appropriate Sobolev and Besov spaces.     

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.     

The linear profile decomposition for the fourth order Schrödinger equation

In this paper, we establish the linear profile decomposition for the one-dimensional fourth order Schrödinger equation

     

Variational iteration method for solving cubic nonlinear Schrödinger equation

The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.     

A KAM algorithm for the resonant non-linear Schrödinger equation

We prove, through a KAM algorithm, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in  and  and a generalization of the quasi-Töplitz functions defined in [31].     

Biharmonic nonlinear Schrödinger equation and the profile decomposition

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L²-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H²(R^N), the best constant of a Gagliardo-Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the solution to the biharmonic nonlinear Schrödinger equation is given.     

Positive solutions for a weakly coupled nonlinear Schrödinger system

Existence of a nontrivial solution is established, via variational methods, for a system of weakly coupled nonlinear Schrödinger equations. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Generalizations for nonautonomous systems are considered.     

A quadratic nonlinear Schrödinger equation in one space dimension

In this paper, we study the Cauchy problem for the quadratic derivative nonlinear Schrödinger equation

(∗)     

Scattering theory for nonlinear Schrödinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x|⁻²$a{|x|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with _Pa=−Δ+a|x|⁻²$P_a=-\mathrm{\Delta }+a{|x|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space H¹(Rⁿ)$H^1({\mathbb{R}}^n)$.     

Profile decompositions of fractional Schrödinger equations with angularly regular data

We study the fractional Schrödinger equations in R^1+d${\mathbb{R}}^{1+d}$, d?3$d?3$, of order d/(d−1)<α<2$d/(d-1)<\alpha <2$. Under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results [9] to data without radial assumption. As applications we show blowup phenomena of solutions to mass-critical fractional Hartree equations.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.     

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.     

On the role of quadratic oscillations in nonlinear Schrödinger equations

We consider a nonlinear semi-classical Schrödinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrödinger equation with harmonic potential.