Multiple solutions of discrete Schrödinger equations with growing potentials

Jia, Liqian; Chen, Guanwei

doi:10.1186/s13662-016-1003-3

Cited by 7 publications

(3 citation statements)

References 22 publications

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“…In fact, we show that indeed this is the case, see proposition 3 below, and in particular the inequality (18). Notably however, a higher dimensional version of this result does not seem to be available at the moment, see the conjecture ( 16) below.…”

Section: Discrete Szegö Inequalitymentioning

confidence: 56%

“…After combining the two and taking the limit N → ∞, we obtain (18), where f is the decreasing rearrangement of f. The equality occurs, if it occurs for both (19) and (20). Applying the criteria for equality in lemma 1 implies that one must have, say for the case f n ⩾ 0 and f 0 = max f(n), that f 0 > f 1 .…”

Section: Proposition 3 (Discrete Szegö Inequality)mentioning

confidence: 99%

“…At the same time, the DNLS has been a rich source of interesting problems in the applied mathematics literature and community. The early work of [14] (that will be central herein) set the stage for some of the important variational considerations that later extended to both periodic and decaying solutions in similar models [15], but also for both focussing and defocussing nonlinearities [16,17] and continues to be an inspiration for further variational work on the subject more recently [18,19]. On the other hand, a considerable effort has been expended to obtain information about the stability of different waveforms in one- [20,21], two [22] and even three [23] spatial dimensions.…”

Section: Introductionmentioning

confidence: 99%

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Ground states in spatially discrete non-linear Schrödinger models

2023

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In the seminal work (Weinstein 1999 Nonlinearity 12 673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on Z d . More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed power P (i.e. l 2 mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedke et al 1994 Phys. Rev. Lett. 73 1055 and Laedke et al 1996 Phys. Rev. E 54 4299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g. l 2 supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity ∂ ω ∥ φ ω ∥ l 2 2 . In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter ω varies, in contrast with the corresponding continuous NLS model.

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Section: Discrete Szegö Inequalitymentioning

confidence: 56%

Section: Proposition 3 (Discrete Szegö Inequality)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ground states in spatially discrete non-linear Schrödinger models

2023

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2‐Keto‐3‐Deoxy‐l‐Rhamnonate Aldolase (YfaU) as Catalyst in Aldol Additions of Pyruvate to Amino Aldehyde Derivatives

et al. 2017

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4-Hydroxy-2-keto acid derivatives are versatile building blocks for the synthesis of amino acids,h ydroxy carboxylic acids andc hiral aldehydes. Pyruvate aldolasesa re privileged catalysts for as traightforward access to this class of keto acid compounds.I nt his work, aC lass II pyruvate aldolase from Escherichia coli K-12, 2-keto-3-deoxy-lrhamnonate aldolase (YfaU), was evaluated for the synthesiso fa mino acid derivatives of proline,p ipecolic acid, andp yrrolizidine-3-carboxylic acid. The aldol addition of pyruvate to N-protected amino aldehydes wast he key enzymatic aldola ddition step followed by catalytic intramolecular reductivea mination. Thec orresponding N-Cbz-amino-4-hydroxy-2-keto acid (Cbz = benzyloxycarbonyl) precursors were obtained in 51-95% isolated yields and enantioselectivity ratios from 26:74 to 95:5, with chiral asubstituted N-Cbz-amino aldehydes.( S)-N-Cbzamino aldehydes gavea ldol adducts with preferentially (R)-configuration at the newly formed stereocenter, whereas the contrary is true for (R)-N-Cbzamino aldehydes.A ddition reactions to achiral amino aldehydesr enderedr acemic aldol adducts. Molecular models of the pre-reaction ternary complexesY faU-pyruvate enolate-acceptor aldehyde were constructed to explain the observeds tereochemical outcome of the reactions.C atalytic reductive amination of the aldol adducts yielded 4-hydroxy-2-pipecolic acid, and unprecedented C-5 substituted4 -hydroxyproline andp yrrolizidine-3-carboxylic acid derivatives.Keywords: aldol reaction; amino acids;a mino aldehydes; asymmetric catalysis;b iocatalysis; biotransformations;pyruvate aldolases Introduction 4-Hydroxy-2-keto acid derivatives are versatile building blocks for the synthesiso fa mino acids,h ydroxy carboxylic acids anda ldehydes among others.[1] Biocatalytic access to 4-hydroxy-2-ketoa cid moiety can be accomplished by means of pyruvate utilizing aldolases.[2]Pyruvate-dependent aldolases are ac lass of lyases that operate in sugar acid metabolism where they utilize pyruvatea st he common aldoln ucleophilic component. Practically all enzymes are type I( i.e., enamine intermediate as nucleophilic species) but there exist also some type II enzymes (i.e., metal cofactor, enolaten ucleophile) that have received less attention from the synthetic point of view.[3] Most of the Class I pyruvate-dependent aldolases,s uch as neuraminic acid aldolase (NeuAc), 2-keto-3-deoxy-6-phospho-dgluconate (GlcA) and 2-keto-3-deoxy-6-phospho-dgalactonate aldolases( GalA), usually have as tringent selectivity for the pyruvaten ucleophile and rather broad electrophiles electivity.H owever, only electrophiles equali ns ize or larger than aldopentoseso r phosphorylated short-chain aldehydes were substrates, whereasp olar, unphosphorylated short-chain aldehydesa re converted at low reaction rates and simple aliphatic or aromatica ldehydes are not substrates. [1a,4] Va riantso ft hese enzymes were identified with remarkable selectivity and catalytic efficiency for an umber of unnaturals ubstrates,a lthough st...

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Non-periodic Schrödinger lattice systems with perturbed and asymptotically linear terms: Negative energy solutions

Chen

Schechter

2019

Applied Mathematics Letters

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Multiple solutions of discrete Schrödinger equations with growing potentials

Cited by 7 publications

References 22 publications

Ground states in spatially discrete non-linear Schrödinger models

Ground states in spatially discrete non-linear Schrödinger models

2‐Keto‐3‐Deoxy‐l‐Rhamnonate Aldolase (YfaU) as Catalyst in Aldol Additions of Pyruvate to Amino Aldehyde Derivatives

Non-periodic Schrödinger lattice systems with perturbed and asymptotically linear terms: Negative energy solutions

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