Damped Nonlinear Schrodinger Equation Research Articles

Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato-Lefever equation (LLE), a damped nonlinear Schrodinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction-diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an inherent loss of regularity which, due to the low-order damping in the LLE, cannot be regained using standard methods. Therefore, in contrast to the case of reaction-diffusion systems, the nonlinear iteration cannot be closed for the modulated perturbation. In this work, we present a new nonlinear iteration scheme incorporating tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

We consider here a damped forced nonlinear logarithmic Schrodinger equation in \begin{document}$ \mathbb{R}^N $\end{document} . We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrodinger equations in the framework of infinite-dimensional dynamical systems.

Finite time extinction for a damped nonlinear Schrodinger equation in the whole space

We consider a nonlinear Schrodinger equation set in the whole space with a single power of interaction and an external source. We first establish existence and uniqueness of the solutions and then show, in low space dimension, that the solutions vanish at a finite time. Under a smallness hypothesis of the initial data and some suitable additional assumptions on the external source, we also show that we can choose the upper bound on which time the solutions vanish. For more information see https://ejde.math.txstate.edu/Volumes/2020/39/abstr.html

A linearly implicit structure-preserving Fourier pseudo-spectral scheme for the damped nonlinear Schrödinger equation in three dimensions

In this paper, we propose a linearly implicit Fourier pseudo-spectral scheme, which preserves the total mass and energy conservation laws for the damped nonlinear Schrodinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal L2-error estimate for the proposed method without any restriction on the grid ratio is established by analyzing the real and imaginary parts of the error function. Numerical results are addressed to confirm our theoretical analysis.

Time-dependent damping effect for the dynamics of DNA transcription

DNA is an interesting macromolecule to study because it plays a role in the body of living beings. This paper modified the Peyrard-Bishop (PB) DNA model by involving time-dependent damping. Damping on this model is considered a time-dependent perturbation. The dynamics of DNA transcription with time dependent damping effect is described by the damped nonlinear Schrodinger (DNLS) equation. Solution of the equation is obtained by applying variational method. So obtained the required parameters and done analysis using matlab. The simulation for the solution is done for nucleotide mass 5.1 × 10−25 kg, distance between base neighbor 3.4 × 10−10 m, elastic coefficient 8 N/m, depth of morse potential 0.1 eV, and width of morse potential 2 × 1010 m−1. This research showed a wave of soliton type of breather and spatial soliton. This paper also showed time-dependent damping causes changes in the amplitude and width of the DNA soliton wave.

Nonlinear coherent structures of Alfvén wave in a collisional plasma

The Alfvén wave dynamics is investigated in the framework of two-fluid approach in a compressible collisional magnetized plasma. In the finite amplitude limit, the dynamics of the nonlinear Alfvén wave is found to be governed by a modified Korteweg-de Vries Burgers equation (mKdVB). In this mKdVB equation, the electron inertia is found to act as a source of dispersion, and the electron-ion collision serves as a dissipation. The collisional dissipation is eventually responsible for the Burgers term in mKdVB equation. In the long wavelength limit, this weakly nonlinear Alfvén wave is shown to be governed by a damped nonlinear Schrödinger equation. Furthermore, these nonlinear equations are analyzed by means of analytical calculation and numerical simulation to elucidate the various aspects of the phase-space dynamics of the nonlinear wave. Results reveal that nonlinear Alfvén wave exhibits the dissipation mediated shock, envelope, and breather like structures. Numerical simulations also predict the formation of dissipative Alfvénic rogue wave, giant breathers, and rogue wave holes. These results are discussed in the context of the space plasma.

Bragg Fibers with Soliton-like Grating Profiles

Nonlinear dynamical system corresponding to the optical holography in a nonlocal nonlinear medium with dissipation contains stable localized spatio-temporal states, namely the grid dissipative solitons. These solitons display a non-uniform profile of the grating amplitude, which has the form of the dark soliton in the reflection geometry. The transformation of the grating amplitude gives rise many new atypical effects for the beams diffracted on such grating, and they are very suitable for the fiber Brass gratings. The damped nonlinear Schrodinger equation is derived that describes the properties of the grid dissipative soliton.

Existence of a finite-dimensional global attractor for a damped Parametric nonlinear Schrödinger equation

We study the existence of a global attractor for a damped parametric nonlinear Schrodinger equation. We provide sufficient conditions for this attractor to have finite dimension.

On the damped nonlinear Schödinger equation with delta functions as initial data

In this paper, we consider the Cauchy problem for the damped nonlinear Schrodinger equation with superposed delta functions as initial data. The aim of this paper is to study the influence of nonlinearity and damping (absorption) on the solution. The case where the nonlinearity makes the solution unstable is of special interest, because the dissipative nature from the damping comes into play in this case.

Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schrödinger equation in plasma

In this paper, the inhomogeneous damped nonlinear Schrodinger equation (NLS) which describes wave propagation in plasmas is investigated. Based on Husimi's transformation and lens-type transformation, we reduce the inhomogeneous NLS equation to the standard NLS equation and thus the bright soliton solutions on a continuous wave (cw) background are constructed and discussed. Besides, we also consider the inhomogeneous NLS equation with different damping coefficients. Based on the perturbation theory for bright solitons, the approximate bright soliton solutions for such an equation are obtained, which are in good agreements with direct numerical simulations.

Ergodicity for a weakly damped stochastic non-linear Schrödinger equation

We study a damped stochastic non-linear Schrodinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markov transition semi-group toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schrodinger equation in the one-dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.

Spatial Inhomogeneity of Periodic and Chaotic Attractors of a Driven Damped Nonlinear Schrödinger Equation

The driven damped nonlinear Schrodinger equation is solved numerically and dicherent attractors — periodic and chaotic ones — are obtained for dicherent ranges of the forcing freld amplitude. It is shown that these attractors are in both cases spatially nonuniform in time. Spatial inhomogeneity of the chaotic attractor is investigated by the estimation of tle correlation dimension at dicherent space points.