scholarly journals The stability spectrum for elliptic solutions to the focusing NLS equation

2017 ◽  
Vol 346 ◽  
pp. 1-19 ◽  
Author(s):  
Bernard Deconinck ◽  
Benjamin L. Segal
Keyword(s):  
Nls Equation ◽  
Stability Spectrum ◽  
Elliptic Solutions ◽  
The Stability

scholarly journals The stability spectrum for elliptic solutions to the sine-Gordon equation

2017 ◽  
Vol 360 ◽  
pp. 17-35 ◽  
Author(s):  
Bernard Deconinck ◽  
Peter McGill ◽  
Benjamin L. Segal
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation ◽  
Stability Spectrum ◽  
Elliptic Solutions ◽  
The Stability

scholarly journals Evolution of a narrow-band spectrum of random surface gravity waves

2003 ◽  
Vol 478 ◽  
pp. 1-10 ◽  
Author(s):  
KRISTIAN B. DYSTHE ◽  
KARSTEN TRULSEN ◽  
HARALD E. KROGSTAD ◽  
HERVÉ SOCQUET-JUGLARD
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Band Spectrum ◽  
Nls Equation ◽  
Random Surface ◽  
Narrow Bandwidth ◽  
Quasi Stationary State ◽  
The Stability ◽  
Three Dimensional Simulations

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

Analytical simulations of the Fokas system; extension (2 + 1)-dimensional nonlinear Schrödinger equation

2021 ◽  
Author(s):  
Mostafa M. A. Khater
Keyword(s):  
Optical Fibers ◽  
Pulse Propagation ◽  
Dynamical Behavior ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Contour Plots ◽  
System Extension ◽  
Inverse Spectral Method ◽  
Nonlinear Pulse Propagation ◽  
The Stability

This paper studies novel analytical solutions of the extended [Formula: see text]-dimensional nonlinear Schrödinger (NLS) equation which is also known with [Formula: see text]-dimensional complex Fokas ([Formula: see text]D–CF) system. Fokas derived this system in 1994 by using the inverse spectral method. This model is considered as an icon model for nonlinear pulse propagation in monomode optical fibers. Many novel computational solutions are constructed through two recent analytical schemes (Ansatz and Projective Riccati expansion (PRE) methods). These solutions are represented through sketches in 2D, 3D, and contour plots to demonstrate the dynamical behavior of pulse propagation in breather, rogue, periodic, lump, and solitary characteristics. The stability property of the obtained solutions is examined based on the Hamiltonian system’s properties. The obtained solutions are checked by putting them back into the original equation through Mathematica 12 software.

Phase-field model: Boundary layer, velocity of propagation, and the stability spectrum

1992 ◽  
Vol 46 (24) ◽  
pp. 16045-16057 ◽  
Author(s):  
Raz Kupferman ◽  
Ofer Shochet ◽  
Eshel Ben-Jacob ◽  
Zeev Schuss
Keyword(s):  
Boundary Layer ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Stability Spectrum ◽  
Layer Velocity ◽  
Velocity Of Propagation ◽  
The Stability

Experiments on nonlinear gravity–capillary waves

1999 ◽  
Vol 380 ◽  
pp. 205-232 ◽  
Author(s):  
LEV SHEMER ◽  
MELAD CHAMESSE
Keyword(s):  
Narrow Band ◽  
Numerical Study ◽  
Three Dimensional ◽  
Capillary Waves ◽  
Carrier Wave ◽  
Nls Equation ◽  
Zakharov Equation ◽  
The Stability ◽  
Nonlinear Gravity

Benjamin–Feir instability of nonlinear gravity–capillary waves is studied experimentally. The experimental results are compared with computations performed for values of wavelength and steepness identical to those employed in the experiments. The theoretical approach is based on the Zakharov nonlinear equation which is modified here to incorporate weak viscous dissipation. Experiments are performed in a wave ume which has an accurately controlled wavemaker for generation of the carrier wave, as well as an additional independent conical wavemaker for generation of controlled three-dimensional disturbances. The approach adopted in the present experimental investigation allows therefore the determination of the actual boundaries of the instability domain, and not just the most unstable disturbances. Instantaneous surface elevation measurements are performed with capacitance-type wave gauges. Multipoint measurements make it possible to determine the angular dependence of the amplitude of the forced and unforced disturbances, as well as their variation along the tank. The limits of the instability domains obtained experimentally for each set of carrier wave parameters agree favourably with those computed numerically using the model equation. The numerical study shows that application of the Zakharov equation, which is free of the narrow-band approximation adopted in the derivation of the nonlinear Schrödinger (NLS) equation, may lead to qualitatively different results regarding the stability of nonlinear gravity–capillary waves. The present experiments support the results of the numerical investigation.

Multi-peak breather stability in a dissipative discrete Nonlinear Schrödinger (NLS) equation

2014 ◽  
Vol 23 (04) ◽  
pp. 1450044 ◽  
Author(s):  
Panayotis Panayotaros ◽  
Felipe Rivero
Keyword(s):  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Dissipation Parameter ◽  
Stabilizing Effect ◽  
Localized Forcing ◽  
The Stability

We study the stability of breather solutions of a dissipative cubic discrete NLS with localized forcing. The breathers are similar to the ones found for the Hamiltonian limit of the system. In the case of linearly stable multi-peak breathers the combination of dissipation and localized forcing also leads to stability, and the apparent damping of internal modes that make the energy around multi-peak breathers nondefinite. This stabilizing effect is however accompanied by overdamping for relatively small values of the dissipation parameter, and the appearance of near-zero stable eigenvalues.

scholarly journals Stability of multi-solitons in the cubic NLS equation

2014 ◽  
Vol 11 (02) ◽  
pp. 329-353 ◽  
Author(s):  
Andres Contreras ◽  
Dmitry Pelinovsky
Keyword(s):  
Initial Data ◽  
Inverse Scattering ◽  
Orbital Stability ◽  
Inverse Scattering Transform ◽  
Nls Equation ◽  
Transform Methods ◽  
Nonlinear Schrödinger ◽  
Scattering Transform ◽  
The Stability

We address the stability of multi-solitons for the cubic nonlinear Schrödinger (NLS) equation on the line. By using the dressing transformation and the inverse scattering transform methods, we establish the orbital stability of multi-solitons in the L2(ℝ) space when the initial data is in a weighted L2(ℝ) space.

scholarly journals Numerical investigation of stability of breather-type solutions of the nonlinear Schrödinger equation

2013 ◽  
Vol 1 (5) ◽  
pp. 5087-5115
Author(s):  
A. Calini ◽  
C. M. Schober
Keyword(s):  
Numerical Investigation ◽  
Rogue Waves ◽  
Maximal Dimension ◽  
Noisy Environments ◽  
Nls Equation ◽  
Stability Behavior ◽  
Nonlinear Schrödinger ◽  
Large Ensembles ◽  
The Stability ◽  
The One

Abstract. In this article we present the results of a broad numerical investigation on the stability of breather-type solutions of the nonlinear Schrödinger (NLS) equation, specifically the one- and two-mode breathers for an unstable plane wave, which are frequently used to model rogue waves. The numerical experiments involve large ensembles of perturbed initial data for six typical random perturbations. Ensemble estimates of the "closeness", A(t), of the perturbed solution to an element of the respective unperturbed family indicate that the only neutrally stable breathers are the ones of maximal dimension, that is: given an unstable background with N unstable modes, the only neutrally stable breathers are the N-dimensional ones (obtained as a superimposition of N simple breathers via iterated Backlund transformations). Conversely, breathers which are not fully saturated are sensitive to noisy environments and are unstable. Interestingly, A(t) is smallest for the coalesced two-mode breather indicating the coalesced case may be the most robust two-mode breather in a laboratory setting. The numerical simulations confirm and provide a realistic realization of the stability behavior established analytically by the authors.

Elliptic solutions of the defocusing NLS equation are stable

2011 ◽  
Vol 44 (28) ◽  
pp. 285201 ◽  
Author(s):  
Nathaniel Bottman ◽  
Bernard Deconinck ◽  
Michael Nivala
Keyword(s):  
Nls Equation ◽  
Elliptic Solutions

On the number of independent partitions

1985 ◽  
Vol 50 (3) ◽  
pp. 809-814 ◽  
Author(s):  
Akito Tsuboi
Keyword(s):  
The Other ◽  
Stable Theory ◽  
Independence Property ◽  
Basic Properties ◽  
Stability Spectrum ◽  
Other Hand ◽  
The Stability ◽  
Independent Partitions

In [3], Shelah defined the cardinals κn(T) and , for each theory T and n < ω. κn(T) is the least cardinal κ without a sequence (pi)i<κ of complete n-types such that pi is a forking extension of pj for all i < j < κ. It is essential in computing the stability spectrum of a stable theory. On the other hand is called the number of independent partitions of T. (See Definition 1.2 below.) Unfortunately this invariant has not been investigated deeply. In the author's opinion, this unfortunate situation of is partially due to the fact that its definition is complicated in expression. In this paper, we shall give equivalents of which can be easily handled.In §1 we shall state the definitions of κn(T) and . Some basic properties of forking will be stated in this section. We shall also show that if = ∞ then T has the independence property.In §2 we shall give some conditions on κ, n, and T which are equivalent to the statement . (See Theorem 2.1 below.) We shall show that does not depend on n. We introduce the cardinal ı(T), which is essential in computing the number of types over a set which is independent over some set, and show that ı(T) is closely related to . (See Theorems 2.5 and 2.6 below.) The author expects the reader will discover the importance of via these theorems.Some of our results are motivated by exercises and questions in [3, Chapter III, §7]. The author wishes to express his heartfelt thanks to the referee for a number of helpful suggestions.

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