Symmetries of the nonlinear Schrödinger equation

Benoît Grébert; Thomas Kappeler

Displaying similar documents to “Symmetries of the nonlinear Schrödinger equation”

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

Similarity:

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $Δ u - u + f (u) = 0$ in $ℝ^{N}$ , $u \in H^{1} (ℝ^{N})$ , where $N \geq 2$ . Under natural conditions on the nonlinearity $f$ , we prove the existence of $𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠$ in any dimension $N \geq 2$ . Our result complements earlier works of Bartsch and Willem $(N = 4 𝚘𝚛 N \geq 6)$ and Lorca-Ubilla $(N = 5)$ where solutions invariant under the action of $O (2) \times O (N - 2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_{k} \times O (N - 2)$ where $D_{k} \subset O (2)$ denotes the dihedral...

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

Similarity:

Let $ϕ$ be a substitution of Pisot type on the alphabet $𝒜 = {1, 2, ..., d}$ ; $ϕ$ satisfies theif for every $i, j \in 𝒜$ , there are integers $k, n$ such that $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same $k$ -th letter, and the prefixes of length $k - 1$ of $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d = 2$ and provide a partial result for $d \geq 2$ .

Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space

Andrea R. Nahmod, Gigliola Staffilani (2015)

Journal of the European Mathematical Society

Similarity:

We also prove a long time existence result; more precisely we prove that for fixed $T > 0$ there exists a set $Σ_{T}$ , $ℙ (Σ_{T}) > 0$ such that any data $φ^{ω} (x) \in H^{γ} (𝕋^{3}), γ < 1, ω \in Σ_{T}$ , evolves up to time $T$ into a solution $u (t)$ with $u (t) - e^{i t Δ} φ^{ω} \in C ([0, T]; H^{s} (𝕋^{3}))$ , $s = s (γ) > 1$ . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space $H^{1} (𝕋^{3})$ , that is in the supercritical scaling regime.

Some results on spaces with $ℵ_{1}$ -calibre

Wei-Feng Xuan, Wei-Xue Shi (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We prove that, assuming , if $X$ is a space with $ℵ_{1}$ -calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_{δ}$ -diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with $ℵ_{1}$ -calibre.

$H^{p}$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

Similarity:

Let A = -Δ + V be a Schrödinger operator on $ℝ^{d}$ , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of $H_{A}^{p}$ if the maximal function $s u p_{t > 0} | T_{t} f (x) |$ belongs to $L^{p} (ℝ^{d})$ , where ${T_{t}}_{t > 0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space $H_{A}^{p}$ admits a special atomic decomposition.

On multiset colorings of generalized corona graphs

Yun Feng, Wensong Lin (2016)

Mathematica Bohemica

Similarity:

A vertex $k$ -coloring of a graph $G$ is a if $M (u) \neq M (v)$ for every edge $u v \in E (G)$ , where $M (u)$ and $M (v)$ denote the multisets of colors of the neighbors of $u$ and $v$ , respectively. The minimum $k$ for which $G$ has a multiset $k$ -coloring is the $χ_{m} (G)$ of $G$ . For an integer $ℓ \geq 0$ , the $ℓ$ - of a graph $G$ , ${cor}^{ℓ} (G)$ , is the graph obtained from $G$ by adding, for each vertex $v$ in $G$ , $ℓ$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for $ℓ$ - of all complete graphs, the regular complete...

Weakly nonlinear stochastic CGL equations

Sergei B. Kuksin (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

We consider the linear Schrödinger equation under periodic boundary conditions, driven by a random force and damped by a quasilinear damping: $\frac{d}{d t} u + i (- Δ + V (x)) u = ν (Δ u - γ_{R} {| u |}^{2 p} u - i γ_{I} {| u |}^{2 q} u) + \sqrt{ν} η (t, x) . (*)$ The force $η$ is white in time and smooth in $x$ ; the potential $V (x)$ is typical. We are concerned with the limiting, as $ν \to 0$ , behaviour of solutions on long time-intervals $0 \leq t \leq ν^{- 1} T$ , and with behaviour of these solutions under the double limit $t \to \infty$ and $ν \to 0$ . We show that these two limiting behaviours may be described in terms of solutions for the( $*$ ) which is a well...

On standard norm varieties

Nikita A. Karpenko, Alexander S. Merkurjev (2013)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let $p$ be a prime integer and $F$ a field of characteristic $0$ . Let $X$ be theof a symbol in the Galois cohomology group $H^{n + 1} (F, μ_{p}^{\otimes n})$ (for some $n \geq 1$ ), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F (X)$ has the following property: for any equidimensional variety $Y$ , the change of field homomorphism $CH (Y) \to CH (Y_{F (X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (dim X) / (p - 1)$ . One of the main ingredients of the proof is a computation...

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

D. Erhard, F. den Hollander, G. Maillard (2014)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

In this paper we study the parabolic Anderson equation $\partial u (x, t) / \partial t = κ 𝛥 u (x, t) + ξ (x, t) u (x, t)$ , $x \in ℤ^{d}$ , $t \geq 0$ , where the $u$ -field and the $ξ$ -field are $ℝ$ -valued, $κ \in [0, \infty)$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $ξ$ -field plays the role of athat drives the equation. The initial condition $u (x, 0) = u_{0} (x)$ , $x \in ℤ^{d}$ , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2 d κ$ ,...

Cutting the loss of derivatives for solvability under condition $(Ψ)$

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

Similarity:

For a principal type pseudodifferential operator, we prove that condition $(ψ)$ implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from $ϵ + 3 / 2$ for any $ϵ > 0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $(ψ)$ doesimply local solvability with a loss of 1...

Cobham's theorem for substitutions

Fabien Durand (2011)

Journal of the European Mathematical Society

Similarity:

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $α$ and $β$ be two multiplicatively independent Perron numbers. Then a sequence $x \in A^{ℕ}$ , where $A$ is a finite alphabet, is both $α$ -substitutive and $β$ -substitutive if and only if $x$ is ultimately...

Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

Similarity:

Let $𝒜$ be a commutative algebraic group defined over a number field $k$ . We consider the following question:A complete answer for the case of the multiplicative group $𝔾_{m}$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $𝒜$ is either an elliptic curve or a torus of small dimension with respect to $r$ . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime. ...

Simultaneous solutions of operator Sylvester equations

Sang-Gu Lee, Quoc-Phong Vu (2014)

Studia Mathematica

Similarity:

We consider simultaneous solutions of operator Sylvester equations $A_{i} X - X B_{i} = C_{i}$ (1 ≤ i ≤ k), where $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ are commuting k-tuples of bounded linear operators on Banach spaces and ℱ, respectively, and $(C ₁, . . ., C_{k})$ is a (compatible) k-tuple of bounded linear operators from ℱ to , and prove that if the joint Taylor spectra of $(A ₁, . . ., A_{k})$ and $(B ₁, . . ., B_{k})$ do not intersect, then this system of Sylvester equations has a unique simultaneous solution.

The equation $- Δ 𝑢 - λ \frac{𝑢}{{| 𝑥 |}^{2}} = {| \nabla 𝑢 |}^{𝑝} + 𝑐 𝑓 (𝑥)$ : The optimal power

Boumediene Abdellaoui, Ireneo Peral (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

We will consider the following problem $- Δ u - λ \frac{u}{{| x |}^{2}} = {| \nabla u |}^{p} + c f, u > 0 in Ω,$ where $Ω \subset ℝ^{N}$ is a domain such that $0 \in Ω$ , $N \geq 3$ , $c > 0$ and $λ > 0$ . The main objective of this note is to study the precise threshold $p_{+} = p_{+} (λ)$ for which there is novery weak supersolutionif $p \geq p_{+} (λ)$ . The optimality of $p_{+} (λ)$ is also proved by showing the solvability of the Dirichlet problem when $1 \leq p < p_{+} (λ)$ , for $c > 0$ small enough and $f \geq 0$ under some hypotheses that we will prescribe.

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti, Veronica Felli, Andrea Malchiodi (2005)

Journal of the European Mathematical Society

Similarity:

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials $V (x) \sim {| x |}^{- α}$ , $0 < α < 2$ , and $K (x) \sim {| x |}^{- β}$ , $β > 0$ . Working in weighted Sobolev spaces, the existence of ground states $v_{ε}$ belonging to $W^{1, 2} (ℝ^{N})$ is proved under the assumption that $σ < p < (N + 2) / (N - 2)$ for some $σ = σ_{N, α, β}$ . Furthermore, it is shown that $v_{ε}$ are spikes concentrating at a minimum point of $𝒜 = V^{θ} K^{- 2 / (p - 1)}$ , where $θ = (p + 1) / (p - 1) - 1 / 2$ .

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

Similarity:

We show that the dimer model on a bipartite graph $Γ$ on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space $Ł_{Γ}$ of line bundles with connections on the graph $Γ$ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs $Γ_{1}$ and $Γ_{2}$ areif the Newton polygons of the corresponding partition functions coincide up to translation....

The $σ$ -property in $C (X)$

Anthony W. Hager (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

The $σ$ -property of a Riesz space (real vector lattice) $B$ is: For each sequence ${b_{n}}$ of positive elements of $B$ , there is a sequence ${λ_{n}}$ of positive reals, and $b \in B$ , with $λ_{n} b_{n} \leq b$ for each $n$ . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “ $σ$ ” obtains for a Riesz space of continuous real-valued functions $C (X)$ . A basic result is: For discrete $X$ , $C (X)$ has $σ$ iff the cardinal $| X | < 𝔟$ , Rothberger’s bounding number. Consequences...

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

Similarity:

Let $k$ be a $p$ -adic field. Let $G$ be the group of $k$ -rational points of a connected reductive group $𝖦$ defined over $k$ , and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$ , wethe tempered dual $\hat{G}$ of $G$ via the Plancherel formula on $𝔤$ , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$ . As a consequence, we prove that any tempered representation contains a good minimal $𝖪$ -type; we extend this result to irreducible...

Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $ℝ^{3}$

M. Burak Erdoğan, Michael Goldberg, Wilhelm Schlag (2008)

Journal of the European Mathematical Society

Similarity:

We present a novel approach for bounding the resolvent of $H = - Δ + i (A \cdot \nabla + \nabla \cdot A) + V = : - Δ + L 1$ for large energies. It is shown here that there exist a large integer $m$ and a large number $λ_{0}$ so that relative to the usual weighted $L^{2}$ -norm, $∥ {(L {(- Δ + (λ + i 0))}^{- 1})}^{m} ∥ < \frac{1}{2} 2$ for all $λ > λ_{0}$ . This requires suitable decay and smoothness conditions on $A, V$ . The estimate (2) is trivial when $A = 0$ , but difficult for large $A$ since the gradient term exactly cancels the natural decay of the free resolvent. To obtain (2), we introduce a conical decomposition of the resolvent and...