Stable high dimensional solitons in nonlocal competing cubic-quintic nonlinear media

2021 ◽  
Author(s):  
Qiying Zhou ◽  
Hui-jun Li
Keyword(s):  
Propagation Constant ◽  
Nonlinear Coefficient ◽  
High Dimensional ◽  
Cubic Nonlinearity ◽  
Nonlinear Media ◽  
Nonlinear Modes ◽  
Quintic Nonlinearity ◽  
The Stability

Abstract We find and stabilize high dimensional dipole and quadrupole solitons in nonlocal competing cubic-quintic nonlinear media. By adjusting the propagation constant, cubic and quintic nonlinear coefficients, the stable intervals for dipole and quadrupole solitons which are parallel to $x$ axis and ones after rotating 45 degrees counterclockwise around the origin of coordinate are found. For the dipole solitons and ones after rotating, their stability is controlled by the propagation constant, the coefficients of cubic and quintic nonlinearity. For the quadrupole solitons, their stability is controlled by the propagation constant and the coefficient of cubic nonlinearity, rather than the coefficient of quintic nonlinearity, though there is a small effect of the quintic nonlinear coefficient on the stability. Our proposal may provide a way to generate and stabilize some novel high dimensional nonlinear modes in nonlocal system.

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

scholarly journals Quiescent Gap Solitons in Coupled Nonuniform Bragg Gratings with Cubic-Quintic Nonlinearity

2021 ◽  
Vol 11 (11) ◽  
pp. 4833
Author(s):  
Afroja Akter ◽  
Md. Jahedul Islam ◽  
Javid Atai
Keyword(s):  
Numerical Stability ◽  
Bragg Gratings ◽  
Stability Boundaries ◽  
Quintic Nonlinearity ◽  
Numerical Stability Analysis ◽  
Gap Solitons ◽  
The Stability ◽  
Dual Core

We study the stability characteristics of zero-velocity gap solitons in dual-core Bragg gratings with cubic-quintic nonlinearity and dispersive reflectivity. The model supports two disjointed families of gap solitons (Type 1 and Type 2). Additionally, asymmetric and symmetric solitons exist in both Type 1 and Type 2 families. A comprehensive numerical stability analysis is performed to analyze the stability of solitons. It is found that dispersive reflectivity improves the stability of both types of solitons. Nontrivial stability boundaries have been identified within the bandgap for each family of solitons. The effects and interplay of dispersive reflectivity and the coupling coefficient on the stability regions are also analyzed.

Compression of optical similaritons induced by cubic-quintic nonlinear media in a graded-index waveguide

2016 ◽  
Vol 25 (03) ◽  
pp. 1650033 ◽  
Author(s):  
Ritu Pal ◽  
Amit Goyal ◽  
Shally Loomba ◽  
Thokala Soloman Raju ◽  
C. N. Kumar
Keyword(s):  
Variable Coefficient ◽  
Free Parameter ◽  
Propagation Distance ◽  
Nonlinearity Coefficient ◽  
Cubic Nonlinearity ◽  
Nonlinear Media ◽  
Graded Index ◽  
Double Kink ◽  
Quintic Nonlinear Schrödinger Equation ◽  
Similarity Reductions

We employ the similarity reductions in two steps to obtain a family of bright and dark similaritons for the variable coefficient cubic–quintic nonlinear Schrödinger equation. Also, parameter domains are delineated in which kink and double-kink similaritons exist for this model. This methodology introduces a free parameter through cubic nonlinearity coefficient which gives us freedom to tune the amplitude and the propagation distance of similaritons in a tapered graded-index waveguide. Furthermore, we observe rapid beam compression of these similaritons for varying detuning parameter and the coefficient of cubic nonlinearity.

Ensemble data assimilation for systems with different degrees of nonlinearity with a hybrid nonlinear-Kalman ensemble transform filter

2021 ◽  
Author(s):  
Lars Nerger
Keyword(s):  
High Dimensional ◽  
Sampling Errors ◽  
Hybrid Filter ◽  
Nonlinear Properties ◽  
Filter Performance ◽  
Good Filter ◽  
Ensemble Transform Kalman Filter ◽  
The Stability ◽  
Lorenz 96 ◽  
Ensemble Transform

&lt;p&gt;The second-order exact particle filter NETF (nonlinear ensemble transform filter) is combined with local ensemble transform Kalman filter (LETKF) to build a hybrid filter scheme (LKNETF). The filter combines the stability of the LETKF with the nonlinear properties of the NETF to obtain improved assimilation results for smaller ensembles. Both filter components are localized in a consistent way so that the filter can be applied with high-dimensional models. The degree of filter nonlinearity is defined by a hybrid weight, which shifts the analysis between the LETKF and NETF. Since the NETF is more sensitive to sampling errors than the LETKF, the latter filter should be preferred in linear cases. It is discussed how an adaptive hybrid weight can be defined based on the nonlinearity of the system so that the adaptivity yields a good filter performance in linear and nonlinear situations. The filter behavior is exemplified based on experiments with the chaotic Lorenz-63 and Lorenz-96 models, in which the nonlinearity can be controlled by the length of the forecast phase.&lt;/p&gt;

Nonlinear Dynamics and Bifurcations of an Axially Moving Beam

1999 ◽  
Vol 122 (1) ◽  
pp. 21-30 ◽  
Author(s):  
F. Pellicano ◽  
F. Vestroni
Keyword(s):  
High Dimensional ◽  
Axially Moving Beam ◽  
Dimensional System ◽  
Global Dynamics ◽  
Dimensional Phase Space ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability ◽  
The Galerkin Method ◽  
Sample Case

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem: a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied. [S0739-3717(00)00501-8]

scholarly journals Model Reduction Methods for Complex Network Systems

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
X. Cheng ◽  
J.M.A. Scherpen
Keyword(s):  
Autonomous Systems ◽  
High Dimensional ◽  
Annual Review ◽  
Publication Date ◽  
Network Systems ◽  
Modeling And Control ◽  
Reduced Order ◽  
Reduction Methods ◽  
The Stability ◽  
And Control

Network systems consist of subsystems and their interconnections and provide a powerful framework for the analysis, modeling, and control of complex systems. However, subsystems may have high-dimensional dynamics and a large number of complex interconnections, and it is therefore relevant to study reduction methods for network systems. Here, we provide an overview of reduction methods for both the topological (interconnection) structure of a network and the dynamics of the nodes while preserving structural properties of the network. We first review topological complexity reduction methods based on graph clustering and aggregation, producing a reduced-order network model. Next, we consider reduction of the nodal dynamics using extensions of classical methods while preserving the stability and synchronization properties. Finally, we present a structure-preserving generalized balancing method for simultaneously simplifying the topological structure and the order of the nodal dynamics. Expected final online publication date for the Annual Review of Control, Robotics, and Autonomous Systems, Volume 4 is May 3, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

On the dynamics of finite-amplitude baroclinic waves as a function of supercriticality

1976 ◽  
Vol 78 (3) ◽  
pp. 621-637 ◽  
Author(s):  
Joseph Pedlosky
Keyword(s):  
Steady State ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Steady State Mode ◽  
Final State ◽  
Nonlinear Modes ◽  
Stream Structure ◽  
The Stability ◽  
Amplitude Mode ◽  
Stream Mode

A finite-amplitude model of baroclinic instability is studied in the case where the cross-stream scale is large compared with the Rossby deformation radius and the dissipative and advective time scales are of the same order. A theory is developed that describes the nature of the wave field as the shear supercriticality increases beyond the stability threshold of the most unstable cross-stream mode and penetrates regions of higher supercriticality. The set of possible steady nonlinear modes is found analytically. It is shown that the steady cross-stream structure of each finite-amplitude mode is a function of the supercriticality.Integrations of initial-value problems show, in each case, that the final state realized is the state characterized by the finite-amplitude mode with the largest equilibrium amplitude. The approach to this steady state is oscillatory (nonmonotonic). Further, each steady-state mode is a well-defined mixture of linear cross-stream modes.

Numerical analysis of the stability of optical bullets (2 + 1) in a planar waveguide with cubic–quintic nonlinearity

2009 ◽  
Vol 41 (2) ◽  
pp. 121-130 ◽  
Author(s):  
W. B. Fraga ◽  
J. W. M. Menezes ◽  
C. S. Sobrinho ◽  
A. C. Ferreira ◽  
G. F. Guimarães ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Planar Waveguide ◽  
Quintic Nonlinearity ◽  
The Stability ◽  
Optical Bullets

scholarly journals Catastrophic Interference in Reinforcement Learning: A Solution Based on Context Division and Knowledge Distillation

2021 ◽  
Author(s):  
Tiantian Zhang ◽  
Xueqian Wang ◽  
Bin Liang ◽  
Bo Yuan
Keyword(s):  
Neural Networks ◽  
Reinforcement Learning ◽  
Training Data ◽  
Learning Ability ◽  
High Dimensional ◽  
Online Clustering ◽  
Computational Overhead ◽  
Knowledge Distillation ◽  
The Stability ◽  
Catastrophic Interference

The powerful learning ability of deep neural networks enables reinforcement learning (RL) agents to learn competent control policies directly from high-dimensional and continuous environments. In theory, to achieve stable performance, neural networks assume i.i.d. inputs, which unfortunately does no hold in the general RL paradigm where the training data is temporally correlated and non-stationary. This issue may lead to the phenomenon of "catastrophic interference" (a.k.a. "catastrophic forgetting") and the collapse in performance as later training is likely to overwrite and interfer with previously learned good policies. In this paper, we introduce the concept of "context" into the single-task RL and develop a novel scheme, termed as Context Division and Knowledge Distillation (CDaKD) driven RL, to divide all states experienced during training into a series of contexts. Its motivation is to mitigate the challenge of aforementioned catastrophic interference in deep RL, thereby improving the stability and plasticity of RL models. At the heart of CDaKD is a value function, parameterized by a neural network feature extractor shared across all contexts, and a set of output heads, each specializing on an individual context. In CDaKD, we exploit online clustering to achieve context division, and interference is further alleviated by a knowledge distillation regularization term on the output layers for learned contexts. In addition, to effectively obtain the context division in high-dimensional state spaces (e.g., image inputs), we perform clustering in the lower-dimensional representation space of a randomly initialized convolutional encoder, which is fixed throughout training. Our results show that, with various replay memory capacities, CDaKD can consistently improve the performance of existing RL algorithms on classic OpenAI Gym tasks and the more complex high-dimensional Atari tasks, incurring only moderate computational overhead.

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