We continue the study of scattering theory for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In a previous paper we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of solutions in the range of the wave operators, under a support condition on the asymptotic state required by the different propagation properties of the wave and Schr"odinger equations.Here we eliminate that condition by using an improved asymptotic form for the solutions.
We study the theory of scattering for the Maxwell-Schr"odinger system in the Coulomb gauge in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the magnetic field data in the framework of a direct method which requires smallness of the Schr"odinger data, and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.
We study the theory of scattering for the Wave-Schr"odinger system with Yukawa type coupling in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the wave data in the framework of a direct method which requires smallnes of the Schr"odinger data, and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.
We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3,in the Coulomb gauge.In the special case of vanishing asymptotic magnetic field,we prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in partially solving the Maxwell equations for the potentials,substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter…
We derive a determinant formula for the WKB exponential of singularly perturbed Zakharov-Shabat system that corresponds to the semiclassical (zero dispersion) limit of the focusing Nonlinear Schr\" odinger equation. The derivation is based on the Riemann-Hilbert Problem (RHP) representation of the WKB exponential. We also prove its independence of the branchpoints of the corresponding hyperelliptic surface assuming that the modulation equations are satisfied.
We continue the study of scattering theory for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling in space dimension 3. In previous papers, we proved the existence of modified wave operators for that system with no size restriction on the data and we determined the asymptotic behaviour in time of the solutions in the range of the wave operators, first under a support condition on the Schr"odinger asymptotic state and then without that condition, but for solutions of relatively low regularity. Here we extend the latter result to the case of more reg…
We study the theory of scattering for the Maxwell-Schr"odinger system in space dimension 3, in the Coulomb gauge. We prove the existence of modified wave operators for that system with no size restriction on the Schr"odinger and Maxwell asymptotic data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators. The method consists in partially solving the Maxwell equations for the potentials, substituting the result into the Schr"odinger equation, which then becomes both nonlinear and nonlocal in time. The Schr"odinger function is then parametrized in ter…
Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schr"odinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.
We study the theory of scattering for the system consisting of a Schr"odinger equation and a wave equation with a Yukawa type coupling,in space dimension 3.We prove in particular the existence of modified wave operators for that system with no size restriction on the data and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.The method consists in solving the wave equation, substituting the result into the Schr"odinger equation,which then becomes both nonlinear and nonlocal in time,and treating the latter by the method previously used for a family of…
We study the asymptotic behaviour in time of solutions and the theory of scattering for the modified Schr"odinger map in two space dimensions. We solve the Cauchy problem with large finite initial time, up to infinity in time, and we determine the asymptotic behaviour in time of the solutions thereby obtained. As a byproduct, we obtain global existence for small data in H^k inter FH^k with k > 1. We also solve the Cauchy problem with infinite initial time, namely we construct solutions defined in a neighborhood of infinity in time, with prescribed asymptotic behaviour of the previous type.
We study the theory of scattering for a Schr"odinger equation in an external time dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schr"odinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We a…
In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough, then such solution does not exist. For the evolution equation, we show the existence of finite time blow up of solutions with radially symmetric data with energy below the one of the ground state.
In this paper, we study the existence of traveling waves for a fourth order Schr\" odinger equations with mixed dispersion, that is, solutions to $$Δ^2 u +βΔu +i V \nabla u +αu =|u|^{p-2} u,\ in\ \R^N ,\ N\geq 2.$$ We consider this equation in the Helmholtz regime, when the Fourier symbol $P$ of our operator is strictly negative at some point. Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that $p\in (p_1 , 2N/(N-4)_+)$. The real number $p_1$ depends on the number of principal curvature of $M$ staying bounded away from $0$, whe…
We extend the notion of orbital stability to systems of nonlinear Schrodinger equations, then we prove this property under suitable assumptions of the local nonlinearity involved.
We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schr"odinger equation with repulsive delta potential (delta-NLS). We shall prove that for a given asymptotic profile, there exists a solution to (delta-NLS) which converges to the given asymptotic profile as time goes infinity. To show this result we exploit the distorted Fourier transform associated to the Schr"odinger equation with delta potential.
We review the proof of existence and uniqueness of solutions of the Maxwell-Schr"odinger system in a neighborhood of infinity in time, with prescribed asymptotic behaviour defined in terms of asymptotic data, without any restriction on the size of those data. That result is the basic step in the construction of modified wave operators for the Maxwell-Schr"odinger system.
We review some recent results on the theory of scattering and more precisely on the local Cauchy problem at infinity in time for some long range nonlinear systems including some form of the Schr"odinger equation. We consider in particular the Wave-Schr"odinger system in space dimension 3, the Maxwell-Schr"odinger system in space dimension 3, the Klein-Gordon-Schr"odinger system in space dimension 2 and the Zakharov system in space dimensions 2 and 3. By the use of a direct method which is intrinsically restricted to the case of small Schr"odinger data and to the borderline long range case, one…
In this paper, we consider the existence and the asymptotic completeness of the wave operators for Schrodinger equations with time-dependent potentials which are short-range in space.
This paper is concerned with the nonlinear Schrodinger lattice with nonlinear hopping. Via variation approach and the Nehari manifold argument, we obtain two types of solution: periodic ground state and localized ground state. Moreover, we consider the convergence of periodic solutions to the solitary wave.
We study the local behavior of solutions of the stationary Schr\" od\-inger equation with singular potentials, establishing a local decomposition into a homogeneous harmonic polynomial and a lower order term. Combining a corollary to this result with a quantitative unique continuation principle for singular potentials we obtain log-Hölder continuity for the density of states outer-measure in one, two, and three dimensions for Schr\" odinger operators with singular potentials, results that hold for the density of states measure when it exists.
We prove the uniqueness of solutions of the Maxwell-Schr"odinger system with given asymptotic behaviour at infinity in time. The assumptions include suitable restrictions on the growth of solutions for large time and on the accuracy of their asymptotics, but no restriction on their size. The result applies to the solutions with prescribed asymptotics constructed in a previous paper.
We consider the asymptotic behavior in time of solutions to the nonlinear Schr"odinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS).
We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrodinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic-quintic nonlinear Schrodinger equation with a periodic potential.
This work deals with the focusing Nonlinear Schrodinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.
In this paper we prove a subelliptic resolvent estimate for a class of semiclassical non-self-adjoint Schrödinger operators with purely imaginary potentials when the spectral parameter is in a parabolic neighborhood of the imaginary axis.
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 profile is a sum of two parts oscillating in a different way. This kind of behavior seems new. Further, several examples of systems which admit solution with several types of behavior such as modified scattering, nonlinear amplification, and nonlinear dissipation, are given. We also extend our previous classification result of nonlinear cubic systems.
This paper is a continuation of our previous study on the long time behavior of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the quadratic nonlinearity in two dimensions. More precisely, for a given asymptotic profile $u_{+}$, we construct a solution to (4NLS) which converges to $u_{+}$ as t to infinity, where $u_{+}$ is given by the leading term of the solution to the linearized equation of (4NLS) with a logarithmic phase correction.
We prove the time decay estimates $L^1({\cal R}) \rightarrow L^\infty ({\cal R}),$ where ${\cal R}$ is an infinite star-shaped network, for the Schrödinger group $e^{it(- \frac{d^2}{dx^2} + V)}$ for real-valued potentials $V$ satisfying some regularity and decay assumptions. Further we show that the solution for initial conditions with a lower cutoff frequency tends to the free solution, if the cutoff frequency tends to infinity.
We consider a derivative nonlinear Schr{ö}dinger equation with general nonlinearlity: i$\partial$tu + $\partial$ 2 x u + i|u| 2$σ$ $\partial$xu = 0, In [12], the authors prove the stability of two solitary waves in energy space for $σ$ $\in$ (1, 2). As a consequence, there exists a two-soliton trains in energy space for $σ$ $\in$ (1, 2). Our goal in this paper is proving the existence of multi-soliton trains in energy space for $σ$ 5 2. Our proofs proceed by xed point arguments around the desired prole, using Strichartz estimates.
In this paper, we study the existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} (-Δ)^su+V(x)u+ϕu=|u|^{p-1}u, & \hbox{in $\mathbb{R}^3$,} (-Δ)^sϕ=u^2,& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $2<p<2_s^{\ast}-1 = \frac{3+2s}{3-2s}$, $s\in(\frac{3}{4},1)$. Under certain assumptions on $V$, a nontrivial ground state solution $(u,ϕ)$ is established through using a monotonicity trick and global compactness Lemma. As its supplementary results, we prove some nonexistence results in the case …
We consider the following triple power nonlinear Schr{ö}dinger equation: iut + $Δ$u + a 1 |u|u + a 2 |u| 2 u + a 3 |u| 3 u = 0. We are interested in algebraic standing waves i.e standing waves with algebraic decay above equation in dimensions n (n = 1, 2, 3). We prove the instability of these solutions in the cases DDF (we use abbreviation D: defocusing (ai < 0), F:focusing (ai > 0)) and DFF when n = 2, 3 and in the case DFF with a1 = --1, a3 = 1 and a2 < 32/15$\sqrt$6 when n = 1. Under these assumptions, the standing waves are orbitally unstable in the case of small positive frequency. When t…
Consider the following system of double coupled Schrödinger equations arising from Bose-Einstein condensates etc., \begin{equation*} \left\{\begin{array}{l} -Δu + u =μ_1 u^3 + βuv^2- κv, -Δv + v =μ_2 v^3 + βu^2v- κu, u\neq0, v\neq0\ \hbox{and}\ u, v\in H^1(\R^N), \end{array} \right. \end{equation*}where $μ_1, μ_2$ are positive and fixed, $κ$ and $β$ are linear and nonlinear coupling parameters respectively. We first use critical point theory and Liouville type theorem to prove some existence and nonexistence results on the positive solutions of this system. Then using the positive and non-dege…
We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schrödinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \begin{equation} L^{\hbar}_{A,V}= -\hbar^2 Δ-\frac{2\hbar}{i} A \cdot \nabla + |A|^2- \frac{\hbar}{i}\operatorname{div}A + V(x). \end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exi…
We give necessary and sufficient conditions for the controllability of a Schr\''odinger equation involving the sub-Laplacian of a nilmanifold obtained by taking the quotient of a group of Heisenberg type by one of its discrete sub-groups.This class of nilpotent Lie groups is a major example of stratified Lie groups of step 2. The sub-Laplacian involved in these Schr\''odinger equations is subelliptic, and, contrarily to what happens for the usual elliptic Schr\''odinger equation for example on flat tori or on negatively curved manifolds, there exists a minimal time of controllability. The main…
It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Kortewegde Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves and such a system has been studied over the last few decades. However, questions about the validity of the system in the study of water waves were raised in our previous work where we presented our analysis using the fifth…
We study the following nonlinear Schrödinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-Δu +\la_1 u = μ_1 u^{2^\ast-1}+βu^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in Ω, -Δv +\la_2 v =μ_2 v^{2^\ast-1}+βv^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in $\om$},\quad u=v=0 \,\,\hbox{on $\partial\om$}.{cases}{displaymath} Here $\om\subset \R^N$ is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\la_1(\om)<\la_1,\la_2<0$, $μ_1,μ_2>0$ and $β\neq 0$, where $λ_1(\om)$ is the first eigen…
In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}.
We revisit a result by Cuccagna, Kirr and Pelinovsky about the cubic nonlinear Schr\" odinger equation (NLS) with an attractive localized potential and a time-dependent factor in the nonlinearity. We show that, under generic hypotheses on the linearization at 0 of the equation, small energy solutions are asymptotically free. This is yet a new application of the hamiltonian structure, continuing a program initiated in a paper by Bambusi and Cuccagna.
In 1956 $Å$. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunctions as the eigenvalues tend to $+\infty$. A similar question occurs naturally for Schr"\odinger operators. The first significant result has been obtained recently by the first author for the harmonic oscillator. The purpose of this paper is to consider more general potentials which are radial.…
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.
We analyse stability of observability estimates for solutions to wave and Scr\" odinger equations subjected to additive perturbations. The paper generalises the recent averaged observability/control result by allowing for systems consisting of operators of different types. The method also applies to the simultaneous observability problem by which one tries to estimate the energy of each component of a system under consideration. The analysis relies on microlocal defect tools; in particular on standard H-measures, when the main dynamic of the system is governed by the wave operator, while parab…
We consider the scattering theory for discrete Schrödinger operators on $Z^d$ with long-range potentials. We prove the existence of modified wave operators constructed in terms of solutions of a Hamilton-Jacobi equation on the torus $T^d$.
In this paper we study radial solutions of certain two-dimensional nonlinear Schrödinger equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schr ödinger equation with Lp estimates of Laguerre functions, we are able to prove an almost sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.
We study the theory of scattering for a class of Hartree type equations with long range interactions in arbitrary space dimension n > or = 1, including the case of Hartree equations with time dependent potential V(t,x) = kappa t^(mu - gamma) |x|^{- mu} with 0 < gamma < or =1 and 0 < mu < n.This includes the case of potential V(x) = kappa |x|^(-gamma) and can be extended to the limiting case of nonlinear Schr"odinger equations with cubic nonlinearity kappa t^(n- gamma) u|u|^2.Using Gevrey spaces of asymptotic states and solutions,we prove the existence of modified local wave operators at infini…
We study here the behavior of the solutions to a $2\times 2$ semi-linear cooperative system involving Schr\" odinger operators (considered in its variational form): $$LU:=(-Δ+ q(x))U = AU+μU + F(x,U) \quad{\rm in}\ \mathbb{R}^N$$ $$U(x)_{|x|\rightarrow \infty} \rightarrow 0$$ where $q$ is a continuous positive potential tending to $+\infty$ at infinity; $μ$ is a real parameter varying near the principal eigenvalue of the system; $U$ is a column vector with components $u_1$ and $u_2$ and $A$ is a square cooperative matrix with constant coefficient. $F$ is a column vector with components $f_1$ a…
We consider the cubic nonlinear Schrödinger equation posed on the spatial domain $\mathbb{R}\times \mathbb{T}^d$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leq d\leq 4)$. The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when $d\geq 2$. As a consequence, we obtain global solutions to the defocusing and focusing problems on $\mathbb{R}\times \mathbb{T}^d$ (for any $d\geq 2$) with infinitely growing high Sobol…
In this paper, we consider a Hamiltonian system combining a nonlinear Schr\" odinger equation (NLS) and an ordinary differential equation (ODE). This system is a simplified model of the NLS around soliton solutions. Following Nakanishi \cite{NakanishiJMSJ}, we show scattering of $L^2$ small $H^1$ radial solutions. The proof is based on Nakanishi's framework and Fermi Golden Rule estimates on $L^4$ in time norms.
In this paper, we study a coupled nonlinear Schrödinger system with small initial data in a product space. We establish a modified scattering of the solutions of this system and we construct a modified wave operator. The study of the resonant system, which provides the asymptotic dynamics, allows us to highlight a control of the Sobolev norms and interesting dynamics with the beating effect. The proof uses a recent work of Hani, Pausader, Tzvetkov and Visciglia for the modified scattering, and a recent work of Grébert, Paturel and Thomann for the study of the resonant system.
Using certain Ito's equation, we introduce the probability on the space of paths and show its relevance to the scattering properties of multidimensional Schrodinger operator. To relate the geometry of the support of potential to the spectral type we develop a special variant of Potential theory and prove some estimates on the modified Harmonic measure.
In this article, we prove existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg-de Vries (gKdV) equation in the scale critical ^L^r space. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to ^L^r-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schr"odinger equation.
We use modified scattering theory to demonstrate that small-data solutions to the cubic nonlinear Schrödinger equation on rescaled waveguide manifolds, $\mathbb{R} \times \mathbb{T}^d$ for $d\geq 2$, demonstrate boundedness of Sobolev norms as well as weak instability.
