We consider the cubic nonlinear Schrödinger equation in two space dimensions with an attractive potential. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small, localized in space initial data, converge to the set of bound states. Therefore, the center manifold in this problem is a global attractor. The proof hinges on dispersive estimates that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics around the bound states.
Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t ψ+ Δψ+ |ψ|^2 ψ= 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(Γ+ vx - t|v|^2 + α^2 t)} ϕ(x-2tv-D, α), where ϕ(x, α) is a positive ground state solution of the semilinear elliptic equation -Δϕ+ α^2ϕ= ϕ^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a sol…
We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + Δu + |u|^2 u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + Δu + |u|^4 u=0, that blow-up on a circle.
The Fourier transforms of the products of two respectively three solutions of the free Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schroedinger (NLS) equation with data u_0 in the function space ^L^r:=^H^r_0, where for s \in R the spaces ^H^r_s are defined by the norms ||u_0||_{^H^r_s}:=||^u_0||_{L^r'_ξ}, 1/r + 1/r'=1. Similar arguments, combined with a…
Sharp spectral asymptotics for multidimensional Schroedinger operators with the strong magnetic field are derived under rather weak smoothness conditions. I assume that magnetic intensity matrix has constant defect r>0 at each point. In comparison with version 1 of 5.5 year ago this version contains more results (we also study some degenerations), improvements and some minor corrections.
A system of two coupled nonlinear Schroedinger equations is treated. In addition, a linear coupling which models an external driven field described by the Rabi frequency is considered. Asymptotics for large Rabi frequency are carried out. Convergence in the appropriate Strichartz space is proven. As a consequence, the global existence of the limiting system gives a criterion for the long time behavior of the original system.
We consider a cubic nonlinear Schroedinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.
We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we treat the energy-critical case of a quintic dissipation in three space dimensions.
We consider 2- and 3-dimensional Schrödinger or generalized Schrödinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schrödinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} < \|Q\|_{L^2} \|\nabla Q\|_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle (2006), we control via momentum conservation…
We prove the local-in-time well-posedness and the mass and energy conservation laws for a 3d cubic nonlinear Schroedinger equation with a real-valued potential.
We study the focusing 3d cubic NLS equation with H^1 data at the mass-energy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^-; here Q is the ground state, and Q^+, Q^- exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^- scattering in the negative time direction. S…
We consider a class of nonlinear Schroedinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time d…
For the 3d cubic nonlinear Schrödinger (NLS) equation, which has critical (scaling) norms $L^3$ and $\dot H^{1/2}$, we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time blow-up. For the rest of the paper, we focus on the study of finite-time radial blow-up solutions, and prove a result on the concentration of the $L^3$ norm at the origin. Two disparate possibilities emerge, one which coincides with solutions typically observed in numerical experiments that consist of a specific bump profile with maximum at the origin and focu…
In this article we obtain eigenvalue asymptotics for 2D and 3D-Schroedinger, Schroedinger-Pauli and Dirac operators in the situations in which the role of the magnetic field is important. These operators are essentially different and there is a significant difference between 2D and 3D-operators.
We consider a singular Schrödinger operator in $L^2(\mathbb{R}^2)$ written formally as $-Δ- βδ(x-γ)$ where $γ$ is a $C^4$ smooth open arc in $\mathbb{R}^2$ of length $L$ with regular ends. It is shown that the $j$th negative eigenvalue of this operator behaves in the strong-coupling limit, $β\to +\infty$, asymptotically as \[ E_j(β)=-\frac{β^2}{4} +μ_j +\mathcal{O}\Big(\dfrac{\logβ}β\Big), \] where $μ_j$ is the $j$th Dirichlet eigenvalue of the operator \[ -\frac{d^2}{ds^2} -\frac{κ(s)^2}{4}\, \] on $L^2(0,L)$ with $κ(s)$ being the signed curvature of $γ$ at the point $s\in(0,L)$.
Sharp spectral asymptotics for 2- and 3-dimensional Schroedinger operators with strong magnetic field are derived under rather weak smoothness conditions. In comparison with version 1 of 2005 new results are added and minor errors corrected.
We give a short description of the proof of asymptotic-completeness for NLS-type equations, including time dependent potential terms, with radial data in three dimensions. We also show how the method applies for the two-body Quantum Scattering case.
I derive sharp semiclassical asymptotics of \int |e_h(x,y,0)|^2ω(x,y)dxdy where e_h(x,y,τ) is the Schwartz kernel of the spectral projector of Magnetic Schroedinger operator and ω(x,y) is singular as x=y. I also consider asymptotics of more general expressions.
The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques are proposed. Using these, we exhaustively describe admissible point transformations in classes of nonlinear (1+1)-dimensional Schroedinger equations, in particular, in the class of nonlinear (1+1)-dimensional Schroedinger equations with modular nonlinearities and potentials and some subclasses thereof. We then carry out a complete group classification in t…
Singularly perturbed vector nonlinear Schroedinger equations (PVNLS) are investigated. Emphasis is placed upon the relation with their restriction: The singularly perturbed scalar nonlinear Schroedinger equation (PNLS) studied earlier by Li. It turns out that the persistent homoclinic orbit for the PNLS is the only one for the PVNLS, asymptotic to the same saddle.
We consider the problem of identifying sharp criteria under which radial $H^1$ (finite energy) solutions to the focusing 3d cubic nonlinear Schrödinger equation (NLS) $i\partial_t u + Δu + |u|^2u=0$ scatter, i.e. approach the solution to a linear Schrödinger equation as $t\to \pm \infty$. The criteria is expressed in terms of the scale-invariant quantities $\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}$ and $M[u]E[u]$, where $u_0$ denotes the initial data, and $M[u]$ and $E[u]$ denote the (conserved in time) mass and energy of the corresponding solution $u(t)$. The focusing NLS possesses a soliton solutio…
In this paper, we study the discrete cubic nonlinear Schroedinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Backlund-Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent invariant manifolds and Fenichel fibers for the perturbed lattice. Finally based upon the above machinery, existence of codimension 2 transversal homoclinic tubes is established through a Melnikov type ca…
We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schroedinger and Gross-Pitaevskii equations on the exterior of a non-trapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate with a smoothing effect on exterior domains.
A slightly modified variant of the cubic periodic one-dimensional nonlinear Schroedinger equation is shown to admit weak solutions for all initial data in certain function spaces wider than L^2. These solutions depend uniformly continuously on the initial data, in the norms considered. The solutions are constructed as sums of infinite series of multilinear operators applied to initial data; no fixed point argument or energy inequality are used. In a companion paper we have shown that weak solutions in these same function spaces are however not unique.
We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy solutions converge to a harmonic map as t goes to infinity (asymptotic stability), extending previous work down to degree m = 3. Due to slow spatial decay of the harmonic map components, a new approach is needed for m=3, involving (among other tools) a "normal form" for the parameter dynamics, and the 2D radial double-endpoint Strichartz estimate for Schroedinger op…
We consider 2-dimensional Schroedinger operator with the non-degenerating magnetic field in the domain with the boundary and under certain non-degeneracy assumptions we derive spectral asymptotics with the remainder estimate better than $O(h^{-1})$, up to $O(μ^{-1}h^{-1})$ and the principal part $\asymp h^{-2}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field; $μh \le 1$. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to O(1) and with the prin…
I consider 4-dimensional Schrödinger operator with the generic non-degenerating magnetic field and for a generic potential I derive spectral asymptotics with the remainder estimate $O(μ^{-1}h^{-3})$ and the principal part $\asymp h^{-4}$ where $h\ll 1$ is Planck constant and $μ\gg 1$ is the intensity of the magnetic field. For general potentials remainder estimate $O(μ^{-1}h^{-3}+μ^2h^{-2})$ is achieved.
With derive sharp spectral asymptotics (with the remainder estimate $O(μ^{-1}h^{1-d}+μ^{\frac{d} {2}-1}h^{1-\frac{d}{2}})$ for $d$-dimensional Schrödinger operator with a strong magnetic field; here $h$ and $μ$ are Plank and binding constants respectively and magnetic intensity matrix has full rank at each point. In comparison with version 1 of 4.5 year ago this version contains more results (we also study some degenerations), improvements and some minor corrections.
Consider the focusing cubic semilinear Schroedinger equation in R^3 i \partial_t ψ+ Δψ+ | ψ|^2 ψ= 0. It admits an eight-dimensional manifold of special solutions called ground state solitons. We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant. Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a soliton and radiation. Conve…
Existence of homoclinic orbits in the cubic nonlinear Schrödinger equation under singular perturbations is proved. Emphasis is placed upon the regularity of the semigroup $e^{\e t \pa_x^2}$ at $\e = 0$. This article is a substantial generalization of \cite{LMSW96}, and motivated by the effort of Dr. Zeng \cite{Zen00a} \cite{Zen00b}. The mistake of Zeng in \cite{Zen00b} is corrected with a normal form transform approach. Both one and two unstable modes cases are investigated.
We consider 2-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider 3-dimensional Schrödinger operator with the non-degenerating magnetic field and we discuss spectral asymptotics with the remainder estimate $o(μ^{-1}h^{-1})$ or better. We also consider generalized Schrödinger-Pauli operator in the same framework albeit with $μh\ge 1$ and derive spectral asymptotics with the remainder estimate up to $O(h^{-1}|\log h|)$ and with the principal part $\asymp μh^{-2}$.
We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - Δu + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.
Resonances which result from perturbation of embedded eigenvalues are studied by time dependent methods. A general theory is developed, with new and weaker conditions, allowing for perturbations of threshold eigenvalues and relaxed Fermi Golden rule. The exponential decay rate of resonances is addressed; its uniqueness in the time dependent picture is shown is certain cases. The relation to the existence of meromorphic continuation of the properly weighted Green's function to time dependent resonance is further elucidated, by giving an equivalent time dependent asymptotic expansion of the solu…
We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate in the argument is a Morawetz-type inequality for the solution $ϕ$. This estimate bounds $\|ϕ(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(ϕ(x,t))^4}{|x|} dx dt
We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein-Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic nonlinearities. Vector-field techniques to deal with the long-range nature of the cubic nonlinearity become problematic in the presence of variable coefficients. We introduce a novel approach based on pointwise-in-time local decay estimates for the Klein-Gordon propagator to overcome this impasse.
We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$ of the…
We consider the cubic Nonlinear Schroedinger Equation (NLS) in one space dimension, either focusing or defocusing. We prove that the solutions satisfy a-priori local in time Hs bounds in terms of the Hs size of the initial data for s >=-1/4 . This improves earlier results of Christ-Colliander-Tao [2] and of the authors [12]. The new ingredients are a localization in space and local energy decay, which we hope to be of independent interest.
We prove two new results about the Cauchy problem for nonlinear Schroedinger equations on four-dimensional compact manifolds. The first one concerns global wellposedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere. The second one provides local wellposedness for quadratic nonlinearities in the case of zonal data on the sphere. Both results are based on new multilinear Strichartz-type estimates for the Schroedinger group.
We prove for a class of nonlinear Schrödinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as {\it ground state selection}. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dyn…
The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space $H\sp 1$.
We use the inverse scattering transform, the auto-Backlund transformation and the steepest descent method of Deift and Zhou to obtain the asymptotic stability of the solitons in the cubic NLS (nonlinear Schrodinger) equation.
We study the cubic non linear Schrödinger equation (NLS) on compact surfaces. On the sphere $\mathbb{S}^2$ and more generally on Zoll surfaces, we prove that, for $s>1/4$, NLS is uniformly well-posed in $H^s$, which is sharp on the sphere. The main ingredient in our proof is a sharp bilinear estimate for Laplace spectral projectors on compact surfaces. On étudie l'équation de Schrödinger non linéaire (NLS) sur une surface compacte.Sur la sphère $\mathbb{S}^2$ et plus généralement sur toute surface de Zoll, on démontre que pour $s>1/4$, NLS est uniformément bien posée dans $H^s$, ce qui est opt…
This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable spectra, Schroedinger - computable spectra, Discrete spectral theorem via sesquilinear forms, Laplace eigenfunctions, Natural boundary conditions, Magnetic Laplacian, Schroedinger in confining well, Variational characterizations, Monotonicity of eigenvalues, Weyl's asymptotic, Polya's conjecture, Reaction-diffusion stability, Thin fluid film stability) Part …
We consider examples of discrete nonlinear Schroedinger equations in Z admitting ground states which are orbitally but not asymptotically stable in l ^2(Z). The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schroedinger equations in eucliedean spaces. We do not investigate connections with work on quasi periodic solutions as in M.Johansson & S.Aubry and of Bamb…
In this paper, we study a coupled nonlinear Schr{ö}dinger system with small initial data in the one dimension Euclidean space. Such a system appears in the context of the coupling between two different optical waveguides. We establish an asymptotic nonlinear behavior and a decay estimate for solutions of this system. The proof uses a recent work of Kato and Pusateri.
We prove local in time well-posedness for the Zakharov system in two space dimensions with large initial data in L^2 x H^{-1/2} x H^{-3/2}. This is the space of optimal regularity in the sense that the data-to-solution map fails to be smooth at the origin for any rougher pair of spaces in the L^2-based Sobolev scale. Moreover, it is a natural space for the Cauchy problem in view of the subsonic limit equation, namely the focusing cubic nonlinear Schroedinger equation. The existence time we obtain depends only upon the corresponding norms of the initial data - a result which is false for the cu…
We consider a semiclassical asymptotics of local trace for the 3D-Schroedinger operator with self-generated magnetic field; it is given by Weyl expression with O(h^{-1}) error and under standard condition to Hamiltonian trajectories even o(h^{-1}). In comparison to v1,2 errors are corrected, new results are added and more details are provided. Misprint correction in comparison to v3
Consider a regular $d$-dimensional metric tree $Γ$ with root $o$. Define the Schroedinger operator $-Δ- V$, where $V$ is a non-negative, symmetric potential, on $Γ$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-γ}$ at infinity, where $1 < γ\leq d \leq 2, γ\neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-Δ- V$. In other words, we will describe the asymptotical behavior of $\inf σ(-Δ- αV)$ as $α\to 0+$
We consider in the whole plane the Hamiltonian coupling of semilinear Schroedinger equations which have critical growth in the sense of Moser. We prove that the (nonempty) set S of ground state solutions is compact up to translations. Moreover, ground states are uniformly bounded and uniformly decaying at infinity. Then we prove that actually the ground state is positive and radially symmetric. We apply those results to prove the existence of semiclassical ground states solutions to the singularly perturbed system when the external potential stays bounded away from zero. Namely, as the adimens…
Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives related to the physics of Bose-Einstein condensates and Anti-de Sitter spacetime) that admit special analytic solutions and an extra conserved quantity. Here, we develop and explore a complex plane representation for these systems modelled on the related cubic Szego and LLL equations. To demonstrate the power of this representation, we use it t…
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 ope…
For the Schrödinger equation with a cubic-quintic, focusing-defocusing nonlinearity in one space dimension, we prove the asymptotic stability of solitary waves for a large range of admissible frequencies. For this model, the linearized problem around the solitary waves does not have internal mode nor resonance.
In this article we consider the Cauchy problem for the cubic focusing nonlinear Schrö\-dinger (NLS) equation on the line with initial datum close to a particular $N$-soliton. Using inverse scattering and the $\bar{\partial}$ method we establish the decay of the $L^{\infty}$ norm of the residual term in time.
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.
