Numerical Simulations of X-Ray Free Electron Lasers (XFEL)

Antonelli, Paolo; Athanassoulis, Agissilaos; Huang, Zhongyi; Markowich, Peter A.

doi:10.1137/130927838

Cited by 11 publications

(18 citation statements)

References 15 publications

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“…If u 2 L 2 < c, then (5.19) contradicts with (5.14) in Lemma 5.4. Hence u n → u in L 2 , and by the interpolation, u n → u in L q (R 3 ), for all q ∈ [2,6). Then by (4.25) and (4.24), we have…”

Section: Solutions As Global Minimizersmentioning

confidence: 87%

“…Proof of Theorem 1.6. (1) and (2) have been proved in [19]. Here we only prove (3) and (4), in both cases, we always assume that λ 1 , λ 2 ∈ R and λ 3 > 0.…”

Section: Normalized Solutions With Partial Confinementioning

confidence: 94%

“…(2) If Q(u) > 0 and u 2 L 2 = c, then v n → u in L 2 as n → ∞. This implies that v n → u in L q as n → ∞, for any q ∈ [2,6) . On the other hand, we deduce from Q(u) > 0 that Q(v n − u) ≤ 0 for sufficiently large n. Thus, we can obtain v n → u in H 1 r as n → ∞.…”

Section: Normalized Solutions Without Partial Confinementioning

confidence: 99%

“…Recently, when the magnetic potential A depends only on time t, Antonelli et al in [1] investigated the asymptotic behavior of solution of equation (1.1) with 0 < p < 4 3 in the highly oscillating regime. Antonelli et al in [2] considered similar problems by numerical simulations. The first author has extended the study of Antonelli et al in [1] to power nonlinearity with exponent 0 < p < 4 and proved the stronger form of convergence, see [16].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

Feng¹,

Zhao²

2019

SIAM J. Control Optim.

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In this paper, we are concerned with the standing waves for the following nonlinear Schrödinger equationwhere 0 < p < 4. This equation arises as an effective single particle model in X-ray Free ElectronLasers. We mainly study the existence and stability/instability properties of standing waves for this equation, in two cases: the first one is that no magnetic potential is involved, (i.e. b = 0 in the equation) and the second one is that b = 0. To be precise, in the first case, by considering a minimization problem on a suitable Pohozaev manifold, we prove the existence of radial solutions, and show further that the corresponding standing waves are strongly unstable by blow-up in finite time. Moreover, by making use of the ideas of these proofs, we are able to prove the existence and instability of normalized solutions, whose proofs seem to be new, compared with the studies of normalized solutions in the existing literature. This study also indicates that there is a close connection between the study of the strong instability and the one of the existence of normalized solutions. In the second case, the situation is more difficult to be treated, due to the additional term of the partial harmonic potential. We manage to prove the existence of stable standing waves for p ∈ (0, 4) and with some assumptions on the coefficients, solutions are obtained as global minimizers if p ∈ (0, 4 3 ], and as local minimizers if p ∈ [ 4 3 , 4). In the mass-critical and supercritical cases p ∈ [ 4 3 , 4), we also establish the variational characterization of the ground states on a new manifold which is different from the one neither of the Nehari type nor of the Pohozaev type, and then prove the existence of ground states. Finally under some assumptions on the coefficients, we prove that the ground state standing waves are strongly unstable.2010 Mathematics Subject Classification. 35Q55, 35J50, 37K45.for all t ∈ [0, T * ). This completes the proof.Proof of Theorem 1.11. When 4 3 ≤ p < 2, let u be the ground state related to (1.4), under the assumptionfor all λ > 1. By the same argument as Theorem 1.4, we can obtain the initial data χ M 0 u λ 0 ∈ Σ∩K ω such that the corresponding solution ψ λ 0 (t) of (1.3) blows up in finite time. Hence, we can prove this theorem.When 2 ≤ p < 4, we see from Lemma 6.2 that S b,ω (λu) < S b,ω (u), K b,ω (λu) < 0 and Q b (λu) < 0, for all λ > 1. Therefore, we can prove the strong instability along the above lines by replacing u λ 0 by λ 0 u. This completes the proof.

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Section: Solutions As Global Minimizersmentioning

confidence: 87%

“…Proof of Theorem 1.6. (1) and (2) have been proved in [19]. Here we only prove (3) and (4), in both cases, we always assume that λ 1 , λ 2 ∈ R and λ 3 > 0.…”

Section: Normalized Solutions With Partial Confinementioning

confidence: 94%

Section: Normalized Solutions Without Partial Confinementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

Feng¹,

Zhao²

2019

SIAM J. Control Optim.

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“…In this paper, we investigate a first-principles model for beammatter interaction in X-ray free electron lasers (XFEL) [7,13]. The fundamental model for XFEL is the following nonlinear Schrödinger equation with a timedependent electromagnetic field and a Coulomb potential i∂ t u = (i∇ − A(α(t), x)) 2…”

mentioning

confidence: 99%

On the Cauchy problem for the XFEL Schrödinger equation

Feng¹,

Zhao²

2018

Discrete &Amp; Continuous Dynamical Systems - B

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In this paper, we consider the Cauchy problem for the nonlinear Schrödinger equation with a time-dependent electromagnetic field and a Coulomb potential, which arises as an effective single particle model in X-ray free electron lasers(XFEL). We firstly show the local and global well-posedness for the Cauchy problem under the assumption that the magnetic potential is unbounded and time-dependent, and then obtain the regularity by a fixed point argument.

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On the ground states for the X‐ray free electron lasers Schrödinger equation

Zhao

2022

Math Methods in App Sciences

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We consider the following X‐ray free electron lasers Schrödinger equation false(i∇−Afalse)2u+Vfalse(xfalse)u−μfalse|xfalse|u=()1false|xfalse|∗false|ufalse|2u−Kfalse(xfalse)false|ufalse|q−2u,0.1em0.1emx∈ℝ3,$$ {\left(i\nabla -A\right)}^2u+V(x)u-\frac{\mu }{\mid x\mid }u=\left(\frac{1}{\mid x\mid}\ast {\left|u\right|}^2\right)u-K(x){\left|u\right|}^{q-2}u,x\in {\mathbb{R}}^3, $$ where A∈Lloc2false(ℝ3,ℝ3false)$$ A\in {L}_{loc}^2\left({\mathbb{R}}^3,{\mathbb{R}}^3\right) $$ denotes the magnetic potential such that the magnetic field B=curl0.4emA$$ B=\operatorname{curl}\kern0.4em A $$ is ℤ3$$ {\mathbb{Z}}^3 $$‐periodic, μ∈ℝ,0.1emK∈L∞()ℝ3$$ \mu \in \mathbb{R},K\in {L}^{\infty}\left({\mathbb{R}}^3\right) $$ is ℤ3$$ {\mathbb{Z}}^3 $$ periodic and non‐negative, q∈false(2,4false)$$ q\in \left(2,4\right) $$. Using the variational method, based on a profile decomposition of the Cerami sequence in HA1()ℝ3$$ {H}_A^1\left({\mathbb{R}}^3\right) $$, we obtain the existence of the ground state solution for suitable μ≥0$$ \mu \ge 0 $$. When μ<0$$ \mu <0 $$ is small, we also obtain the non‐existence. Furthermore, we give a description of the asymptotic behavior of the ground states as μ→0+$$ \mu \to {0}^{+} $$.

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Numerical Simulations of X-Ray Free Electron Lasers (XFEL)

Cited by 11 publications

References 15 publications

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

Optimal Bilinear Control for the X-Ray Free Electron Laser Schrödinger Equation

On the Cauchy problem for the XFEL Schrödinger equation

On the ground states for the X‐ray free electron lasers Schrödinger equation

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