This paper is concerned with the following fractional Schrödinger equations involving critical exponents: \begin{eqnarray*} (-Δ)^αu+V(x)u=k(x)f(u)+λ|u|^{2_α^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N}, \end{eqnarray*} where $(-Δ)^α$ is the fractional Laplacian operator with $α\in(0,1)$, $N\geq2$, $λ$ is a positive real parameter and $2_α^{*}=2N/(N-2α)$ is the critical Sobolev exponent, $V(x)$ and $k(x)$ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the subcritical nonlinearity.
We develop an elementary and self-contained differential approach, in an $L^\infty$ setting, for well-posedness (existence, uniqueness and smooth dependence with respect to the data) for the multi-marginal Schr{ö}dinger system which arises in the entropic regularization of optimal transport problems.
In this note we prove global well-posedness for the defocusing, cubic nonlinear Schr{ö}dinger equation with initial data lying in a critical Sobolev space.
We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{ö}dinger equation to the magnetic Schr{ö}dinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schr{ö}dinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resolvent estimate. Although the approach is classical, these results on loga…
In [19] and [26], the authors proved the stability of multi-solitons for derivative nonlinear Schr{ö}dinger equations. Roughly speaking, sum of finite stable solitons is stable. We predict that if there is one unstable solition then multi-soliton is unstable. This prediction is proved in [7] for classical nonlinear Schr{ö}dinger equations. In this paper, we proved this prediction for derivative nonlinear Schr{ö}dinger equations by using the method of C{ô}te-Le Coz [7] with the help of Gauge transformation.
We give an example of a linear, time-dependent, Schr{ö}dinger operator with optimal growth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of the linear Lowest Landau Level equation with a time-dependent potential.
In this paper we prove a Liouville theorem for the Chern--Simons--Schr{ö}dinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [KKO22] for the soliton resolution result in a weighted space.
In this note we prove scattering for a defocusing nonlinear Schr{ö}dinger equation with initial data lying in a critical Besov space. In addition, we obtain polynomial bounds on the scattering size as a function of the critical Besov norm.
The Schr{ö}dinger problem is an entropy minimisation problem on the space of probability measures. Its optimal value is a cost between two probability measures. In this article we investigate some regularity properties of this cost: continuity with respect to the marginals and time derivative of the cost along probability measures valued curves.
We prove existence results for a stationary Schr{ö}dinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.
We consider the logarithmic Schr{ö}dinger equations with damping, also called Schr{ö}dinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are asymptotically stable in Sobolev regularity. In the case without damping, we prove that for almost all value of the nonlinear parameter, these solutions are stable in high Sobolev regularity for arbitrary long times when the solution is close to a plane wave. We also show and discuss numerical experiments illustrating our results.
Global well-posedness and scattering for the defocusing, energy -critical, nonlinear Schr{ö}dinger equation in the exterior of a convex obstacle when $d = 4$
We consider the nonlinear Schr{ö}dinger-Langevin equation for both signs of the logarithmic nonlinearity. We explicitly compute the dynamics of Gaussian solutions for large times, which is obtained through the study of a particular nonlinear differential equation of order 2. We then give the asymptotic behavior of general energy weak solutions under some regularity assumptions. Some numerical simulations are performed in order to corroborate the theoretical results.
We pursue our work on the asymptotic regimes of the Landau-Lifshitz equation for bi-axial ferromagnets. We put the focus on the cubic Schr{ö}dinger equation, which is known to describe the dynamics in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of the Landau-Lifshitz equation and quantify their convergence towards the solitons of the one-dimensional cubic Schr{ö}dinger equation.
The problem we consider is given a mixed model $F(γ) \rightarrow γ\rightarrow \tildeγ$ and a potential $\widehat{V}$ such that $\log \widehat{V} \in L^{1}$, to determine a movement $U$ such that $F(U γ)=F(γ) + \widehat{V}$.
The nonlinear Schr{ö}dinger equation with derivative cubic nonlinearity admits a family of solitons, which are orbitally stable in the energy space. In this work, we prove the orbital stability of multi-solitons configurations in the energy space, under suitable assumptions on the speeds and frequencies of the composing solitons. The main ingredients of the proof are modulation theory, energy coercivity and monotonicity properties.
We study the Littlewood-Paley-Stein functions associated with Hodge-de Rham and Schr{ö}dinger operators on Riemannian manifolds. Under conditions on the Ricci curvature we prove their boundedness on L p for p in some interval (p 1 , 2] and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For p > 2 we give a criterion to obtain the boundedness of the vertical Littlewood-Paley-Stein function associated with Schr{ö}dinger operators on L p .
We consider the linear and nonlinear Schr{ö}dinger equation with a spatial white noise as a potential in dimension 2. We prove existence and uniqueness of solutions thanks to a change of unknown originally used in [8] and conserved quantities.
We consider the Schrödinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves.
We show how the solutions to a $2\times2$ linear system involving Schr{ö}dinger operators blow up as the parameter $μ$ tends to some critical value which is the principal eigenvalue of the system; here the potential is continuous positive with superquadratic growth and the square matrix of the system is with constant coefficients and may have a double eigenvalue.
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{ö}dinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a kinetic theory similar to the Vlasov equation which is used for describing a classical collisionless plasma. In this paper we mainly show Landau damping in the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson system. At the same time, we also prove the existence and the stability of the nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability assumption.
In this paper we consider the Schr{ö}dinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces X k (R) and in $ϕ$ + H k (R). Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.
In this paper we continue our study [DSS20] of the nonlinear Schrödinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove global well-posedness for the 1D NLS with initial data lying in $L^{p}$ for any $2 < p < \infty$, provided the initial data is sufficiently smooth. We do not use the complete integrability of the cubic nonlinear Schr{ö}dinger equation.
We study the stability issue in the inverse problem of determining the magnetic field and the time-dependent electric potential appearing in the Schrödinger equation, from boundary observations. We prove in dimension 3 or greater, that the knowledge of the Dicrichlet-to-Neumann map stably determines the magnetic field and the electric potential.
In this paper we prove that the defocusing, cubic nonlinear Schr{ö}dinger initial value problem is globally well-posed and scattering for $u_{0} \in L^{2}(\mathbf{R}^{2})$. To do this, we will prove a frequency localized interaction Morawetz estimate similar to the estimate made in \cite{CKSTT4}. Since we are considering an $L^{2}$ - critical initial value problem we will localize to low frequencies.
It was proved by H. Bahouri, P. G{é}rard and C.-J. Xu in [9] that the Schr{ö}dinger equation on the Heisenberg group $\mathbb{H}^d$, involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schr{ö}dinger equation, by a refined study of the Schr{ö}dinger kernel $S_t$ on $\mathbb{H}^d$. The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kern…
We prove exponential decay for a system of two Schr{ö}dinger equations in a wave guide, with coupling and damping at the boundary. This relies on the spectral analysis of the corresponding coupled Schr{ö}dinger operator on the one-dimensional cross section. We show in particular that we have a spectral gap and that the corresponding generalized eigenfunctions form a Riesz basis.
In this paper, we consider the Cauchy problem in $\mathbb{R}^N$, $N\geq1$, for semi-linear Schrödinger equations with space-time fractional derivatives. We discuss the nonexistence of global $L^1$ or $L^2$ weak solutions in the subcritical and critical cases under some conditions on the initial data and the nonlinear term. Furthermore, the nonexistence of local $L^1$ or $L^2$ weak solutions in the supercritical case are studied.
We look for solutions to derivative nonlinear Schrodinger equations built upon solitons. We prove the existence of multi-solitons i.e. solutions behaving at large time as the sum of finite solitons. We also show that one can attach a kink at the begin of the sum of solitons i.e multi kink-solitons. Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates.
We consider a nonlinear Schr{ö}dinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the solutions vanish at a finite time. Under a smallness hypothesis of the initial data and some suitable additional assumptions on the external source, we also show that we can choose the upper bound on which time the solutions vanish.
In this paper, we study some properties of multi-solitons for the non-linear Schr{ö}dinger equations in R^d with general non-linearities. Multi-solitons have already been constructed in H^1, successively by Merle, by Martel and Merle, and by C{ô}te, Martel and Merle. We show here that multi-solitons are smooth, depending on the regularity of the non-linearity. We obtain also a result of uniqueness in some class, either when the ground states are all stable, or in the mass-critical case.
In this paper we deal with the so-called "spectral inequalities", which yield a sharp quantification of the unique continuation for the spectral family associated with the Schrödinger operator in $ \mathbb{R}^d$ \begin{equation*} H_{g,V} = Δ_g + V(x), \end{equation*} where $Δ_g$ is the Laplace-Beltrami operator with respect to an analytic metric $g$, which is a perturbation of the Euclidean metric, and $V(x)$ a real valued analytic potential vanishing at infinity.
I review an information-theoretic variant of the quantum de Finetti theorem due to Brand{ã}o and Harrow and discuss its applications to the topic of bosonic mean-field limits. This leads to slightly improved methods for the derivation of the local non-linear Schr{ö}dinger energy functional from many-body quantum mechanics.
For the double power one dimensional nonlinear Schr{ö}dinger equation, we establish a complete classification of the stability or instability of standing waves with positive frequencies. In particular, we fill out the gaps left open by previous studies. Stability or instability follows from the analysis of the slope criterion of Grillakis, Shatah and Strauss. The main new ingredients in our approach are a reformulation of the slope and the explicit calculation of the slope value in the zero-frequency case. Our theoretical results are complemented with numerical experiments.
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.
We study pointwise spatial decay of multi-solitons of the generalized Korteweg-de Vries equations. We obtain that, uniformly in time, these solutions and their derivatives decay exponentially in space on the left of and in the solitons region, and prove rapid decay on the right of the solitons. We also prove the corresponding result for multi-solitons of the nonlinear Schr{ö}dinger equations, that is, exponential decay in the solitons region and rapid decay outside.
We study a stochastic Schr{ö}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schr{ö}dinger group, which allows us to co…
We prove a Limiting Absorption Principle for Schr{ö}dinger operators in tubes about infinite curves embedded in the Euclidian space with different types of boundary conditions. The argument is based on the Mourre theory with conjugate operators different from the generator of dilations which is usually used in this case, and permits to prove a Limiting Absorption Principle for Schr{ö}dinger operators in singular waveguides.
We consider an anisotropic model case for a strictly convex domain of dimension $d\geq 2$ with smoothboundary and we describe dispersion forthe semi-classical Schr{ö}dinger equation with Dirichlet boundary condition. More specifically, we obtain the following fixed time decay rate for the linear semi-classical flow : a loss of $(\frac ht)^{1/4}$ occurs with respect to the boundary less case due to repeated swallowtail type singularities, and is proven optimal. Corresponding Strichartz estimates allow to solve the cubic nonlinear Schödinger equation on such a 3D model convex domain, hence match…
