296 research outputs found
A Stochastic Parametrically-Forced NLS Equation
No full textIn this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse propagation through optical fibers. We establish global existence and uniqueness of mild solutions for initial data in L2(R) and H1(R).The proof is an adaptation of a fixed-point argument employed by de Bouard and Debussche [Comm. Math. Phys., 205:161-181, 1999] for the nonlinear Schrödinger equation with multiplicative noise. The fixed-point argument relies on space-time estimates on the semigroup generated by the linear parametrically-forced Schrödinger operator. We prove these so-called Strichartz estimates, originally proven for the Schrödinger operator, using Fourier methods. A key difference between the Schrödinger operator and its parametrically-forced version is that the latter is not self-adjoint. We overcome this complication by establishing fixed-time estimates on the semigroup and its adjoint, based on their Fourier representations. We also briefly discuss possible future research in the direction of stability of solitary standing wave solutions of the PFNLS equation under the influence of multiplicative noise. Using informal calculations, we demonstrate an approach to track the displacement of a soliton due to small stochastic forcing.Applied Mathematic
A COMPACT ATTRACTOR FOR ENERGY CRITICAL AND SUPER-CRITICAL NLS
No full textWe show existence of a compact attractor for NLS in energy critical and super-critical cases basing on the method of Tao [12]
Poincaré-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS
No full textWe implement an infinite iteration scheme of Poincare-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrodinger equation (NLS) in C_t L^2(T), without using any auxiliary function space. This allows us to construct weak solutions of NLS in C_t L^2(T)$ with initial data in L^2(T) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in H^s(T) for s \geq 1/6
Long Time Dynamics of the 3D Radial NLS with the Combined Terms
No full textIn this paper, we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation (NLS) with the combined terms i(ut) + Delta u = -vertical bar u vertical bar(4) u + vertical bar u vertical bar(p-1) u, 1 + 4/3 < p < 5 in energy space H-1 (R-3). The threshold energy is the energy of the ground state W of the focusing, energy critical NLS, which means that the subcritical perturbation does not affect the determination of threshold, but affects the scattering and blow-up dichotomy result with subcritical threshold energy. This extends algebraic perturbation in a previous work of Miao, Xu and Zhao [Comm. Math. Phys., 318, 767-808 (2013)] to all mass supercritical, energy subcritical perturbation.NSFC [11171033, 11231006]SCI(E)中国科技核心期刊(ISTIC)中国科学引文数据库(CSCD)[email protected]; [email protected]
The periodic Cauchy problem for PT-symmetric NLS, I: the first appearance of rogue waves, regular behavior or blow up at finite times
No full textThe recently discovered PT-symmetric nonlinear Schrödinger (PT–NLS)
equation is an integrable nonlinear and nonlocal dispersive model. In the focusing
case, monochromatic perturbations of the constant background solution with
sufficiently small wave number are modulationally unstable and one expects
the formation of rogue waves (RWs), as for the celebrated focusing nonlinear
Schrödinger (NLS) equation. In this paper we investigate the x-periodic
Cauchy problem of PT–NLS, for a generic periodic initial perturbation of the
constant background solution (what we call the periodic RW problem), in the
simplest case of one unstable mode only. We use matched asymptotic expansion
techniques to study the first appearance of RWs, well described by two recently
discovered exact solutions of PT–NLS, whose free parameters can be expressed
in terms of the initial data through elementary functions. Depending on the initial
data, the rogue waves are either regular, with arbitrarily large amplitude, or they
blow up twice at the first appearance, unlike the NLS case, in which the RWs
are always regular and with fixed amplitude. A qualitative reason for it should
be the gain-loss properties of the complex self-induced potential of PT–NLS,
that could cause extra-focusing effects with respect to the NLS case. This paper
is motivated by recent works of Grinevich and the author in which a similar
approach, as well as the finite gap method, have been used to solve the RW
periodic Cauchy problem for the focusing NLS equation
Chromatin-bound NLS proteins recruit membrane vesicles and nucleoporins for nuclear envelope assembly via importin-α/β
No full textThe mechanism for nuclear envelope (NE) assembly is not fully understood. Importin-ß and the small GTPase Ran have been implicated in the spatial regulation of NE assembly process. Here we report that chromatin-bound NLS (nuclear localization sequence) proteins provide docking sites for the NE precursor membrane vesicles and nucleoporins via importin-a and -ß during NE assembly in Xenopus egg extracts. We show that along with the fast recruitment of the abundant NLS proteins such as nucleoplasmin and histones to the demembranated sperm chromatin in the extracts, importin-a binds the chromatin NLS proteins rapidly. Meanwhile, importin-ß binds cytoplasmic NE precursor membrane vesicles and nucleoporins. Through interacting with importin-a on the chromatin NLS proteins, importin-ß targets the membrane vesicles and nucleoporins to the chromatin surface. Once encountering Ran-GTP on the chromatin generated by RCC1, importin-ß preferentially binds Ran-GTP and releases the membrane vesicles and nucleoporins for NE assembly. NE assembly is disrupted by blocking the interaction between importin-a and NLS proteins with excess soluble NLS proteins or by depletion of importin-ß from the extract. Our findings reveal a novel molecular mechanism for NE assembly in Xenopus egg extracts.Cell Research advance online publication 31 July 2012; doi:10.1038/cr.2012.113
Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations
No full textAbstractSufficient and necessary conditions for the embeddings between Besov spaces Bp,qs1 and modulation spaces Mp,qs2 are obtained. Moreover, using the frequency-uniform decomposition method, we study the Cauchy problem for the generalized BO, KdV and NLS equations, for which the global well-posedness of solutions with the small rough data in certain modulation spaces M2,1s is shown
Efficient and dynamic nuclear localization of green fluorescent protein via RNA binding
No full textClassical nuclear localization signal (NLS) sequences have been used for artificial localization of green fluorescent protein (GFP) in the nucleus as a positioning marker or for measurement of the nuclear-cytoplasmic shuttling rate in living cells. However, the detailed mechanism of nuclear retention of GFP-NLS remains unclear. Here, we show that a candidate mechanism for the strong nuclear retention of GFP-NLS is via the RNA-binding ability of the NLS sequence. GFP tagged with a classical NLS derived from Simian virus 40 (GFP-NLS(SV40)) localized not only in the nucleoplasm, but also to the nucleolus, the nuclear subdomain in which ribosome biogenesis takes place. GFP-NLS(SV40) in the nucleolus was mobile, and intriguingly, the diffusion coefficient, which indicates the speed of diffusing molecules, was 1.5-fold slower than in the nucleoplasm. Fluorescence correlation spectroscopy (FCS) analysis showed that GFP-NLS(SV40) formed oligomers via RNA binding, the estimated molecular weight of which was larger than the limit for passive nuclear export into the cytoplasm. These findings suggest that the nuclear localization of GFP-NLS(SV40) likely results from oligomerization mediated via RNA binding. The analytical technique used here can be applied for elucidating the details of other nuclear localization mechanisms, including those of several types of nuclear proteins. In addition, GFP-NLS(SV40) can be used as an excellent marker for studying both the nucleoplasm and nucleolus in living cells
Exciting extreme events in the damped and AC-driven NLS equation through plane-wave initial conditions
No full textWe investigate, by direct numerical simulations and for certain parametric regimes, the dynamics of the damped and forced nonlinear Schrödinger (NLS) equation in the presence of a time-periodic forcing. It is thus revealed that the wave number of a plane-wave initial condition dictates the number of emerged Peregrine-type rogue waves at the early stages of modulation instability. The formation of these events gives rise to the same number of transient "triangular"spatiotemporal patterns, each of which is reminiscent of the one emerging in the dynamics of the integrable NLS in its semiclassical limit, when supplemented with vanishing initial conditions. We find that the L 2-norm of the spatial derivative and the L 4-norm detect the appearance of rogue waves as local extrema in their evolution. The impact of the various parameters and noisy perturbations of the initial condition in affecting the above behavior is also discussed. The long-time behavior, in the parametric regimes where the extreme wave events are observable, is explained in terms of the global attractor possessed by the system and the asymptotic orbital stability of spatially uniform continuous wave solutions. © 2021 Author(s)
Addendum to the article: On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS Equation on the Half-Line
No full textThis is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. The publisher is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0951-7715/29/10/3206We present a short note on the extension of the results of [1] to the case of non-zero initial data. More specifically, the defocusing cubic NLS equation is considered on the half-line with decaying (in time) Dirichlet data and sufficiently smooth and decaying (in space) initial data. We prove that for this case also, and for a large class of decaying Dirichlet data, the Neumann data are sufficiently decaying so that the Fokas unified method for the solution of defocusing NLS is applicable
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