scholarly journals A Theoretical Study of an Extended KDV Equation

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Explicit Solution ◽  
Kdv Equation ◽  
Integrable Equation ◽  
Weakly Nonlinear ◽  
Kdv Equations ◽  
Weakly Nonlinear Waves ◽  
Extended Kdv Equation ◽  
The One ◽  
Definition Of ◽  
Fifth Order

Discovered experimentally by Russell and described theoretically by Korteweg and de Vries, KdV equation has been a nonlinear evolution equation describing the propagation of weakly dispersive and weakly nonlinear waves. This equation received a lot of attention from mathematical and physical communities as an integrable equation. The objectives of this paper are: first, providing a rigorous mathematical derivation of an extended KdV equations, one on the velocity, other on the surface elevation, next, solving explicitly the one on the velocity. In order to derive rigorously these equations, we will refer to the definition of consistency, and to find an explicit solution for this equation, we will use the sine-cosine method. As a result of this work, a rigorous justification of the extended Kdv equation of fifth order will be done, and an explicit solution of this equation will be derived.

The growth of periodic waves in gas-fluidized beds

1996 ◽  
Vol 325 ◽  
pp. 261-282 ◽  
Author(s):  
S. E. Harris
Keyword(s):  
Nonlinear Waves ◽  
Fluidized Beds ◽  
Kdv Equation ◽  
Periodic Waves ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Kdv Equation ◽  
Weakly Nonlinear Waves ◽  
Gas Fluidized Beds ◽  
The One

In this paper, we analyse the development of initially small, periodic, voidage disturbances in gas-fluidized beds. The one-dimensional model was proposed by Needham & Merkin (1983), and Crighton (1991) showed that weakly nonlinear waves satisfied a perturbed Korteweg–de Vries or KdV equation. Here, we take periodic cnoidal wave solutions of the KdV equation and follow their evolution when the perturbation terms are amplifying. Initially, all such waves grow, but at a later stage a rescaling shows that shorter wavelengths are stabilized in a weakly nonlinear state. Longer wavelengths continue to develop and eventually strongly nonlinear solutions are required. Necessary conditions for periodic waves are found and matching back onto the growing cnoidal waves is possible. It is shown further that these fully nonlinear waves also reach an equilibrium state. A comparison with numerical results from Needham & Merkin (1986) and Anderson, Sundaresan & Jackson (1995) is then carried out.

Multi-Soliton and Rational Solutions for the Extended Fifth-Order KdV Equation in Fluids

2015 ◽  
Vol 70 (7) ◽  
pp. 559-566 ◽  
Author(s):  
Gao-Qing Meng ◽  
Yi-Tian Gao ◽  
Da-Wei Zuo ◽  
Yu-Jia Shen ◽  
Yu-Hao Sun ◽  
...  
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Second Order ◽  
Wave Shape ◽  
Rational Solutions ◽  
Weakly Nonlinear ◽  
First Order ◽  
Propagation Properties ◽  
Kdv Type Equations ◽  
Fifth Order

AbstractKorteweg–de Vries (KdV)-type equations are used as approximate models governing weakly nonlinear long waves in fluids, where the first-order nonlinear and dispersive terms are retained and in balance. The retained second-order terms can result in the extended fifth-order KdV equation. Through the Darboux transformation (DT), multi-soliton solutions for the extended fifth-order KdV equation with coefficient constraints are constructed. Soliton propagation properties and interactions are studied: except for the velocity, the amplitude and width of the soliton are not influenced by the coefficient of the original equation; the amplitude, velocity, and wave shape of each soltion remain unchanged after the interaction. By virtue of the generalised DT and Taylor expansion of the solutions for the corresponding Lax pair, the first- and second-order rational solutions of the equation are obtained.

A weakly nonlinear model of long internal waves in closed basins

2002 ◽  
Vol 467 ◽  
pp. 269-287 ◽  
Author(s):  
D. A. HORN ◽  
J. IMBERGER ◽  
G. N. IVEY ◽  
L. G. REDEKOPP
Keyword(s):  
Internal Waves ◽  
Kdv Equation ◽  
Wave Interaction ◽  
Second Order ◽  
Weakly Nonlinear ◽  
Closed Basin ◽  
Kdv Equations ◽  
Main Effect ◽  
Weakly Nonlinear Model ◽  
Oscillatory Waves

A simple model is developed, based on an approximation of the Boussinesq equation, that considers the weakly nonlinear evolution of an initial interface disturbance in a closed basin. The solution consists of the sum of the solutions of two independent Korteweg–de Vries (KdV) equations (one along each characteristic) and a second-order wave–wave interaction term. It is demonstrated that the solutions of the two independent KdV equations over the basin length [0, L] can be obtained by the integration of a single KdV equation over the extended reflected domain [0, 2L]. The main effect of the second-order correction is to introduce a phase shift to the sum of the KdV solutions where they overlap. The results of model simulations are shown to compare qualitatively well with laboratory experiments. It is shown that, provided the damping timescale is slower than the steepening timescale, any initial displacement of the interface in a closed basin will generate three types of internal waves: a packet of solitary waves, a dispersive long wave and a train of dispersive oscillatory waves.

Higher-order rational soliton solutions for the fifth-order modified KdV and KdV equations

2021 ◽  
pp. 2150036
Author(s):  
Zhi-Jie Pei ◽  
Hai-Qiang Zhang
Keyword(s):  
Darboux Transformation ◽  
Fourth Order ◽  
Taylor Expansion ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Kdv Equations ◽  
Perturbation Formula ◽  
Modified Kdv Equation ◽  
Fifth Order

In this paper, we construct the generalized perturbation ([Formula: see text], [Formula: see text])-fold Darboux transformation of the fifth-order modified Korteweg-de Vries (KdV) equation by the Taylor expansion. We use this transformation to derive the higher-order rational soliton solutions of the fifth-order modified KdV equation. We find that these higher-order rational solitons admit abundant interaction structures. We graphically present the dynamics behaviors from the first- to fourth-order rational solitons. Furthermore, by the Miura transformation, we obtain the complex rational soliton solutions of the fifth-order KdV equation.

Two integrable third-order and fifth-order KdV equations with time-dependent coefficients

2019 ◽  
Vol 29 (6) ◽  
pp. 2093-2102 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Efficient Algorithm ◽  
Soliton Solutions ◽  
Integrable Equation ◽  
Time Dependent ◽  
Integrable Equations ◽  
Content Type ◽  
Kdv Equations ◽  
Propagation Of Waves ◽  
Fifth Order ◽  
Multiple Complex Soliton Solutions

Purpose The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for these two equations. Design/methodology/approach The integrability of each of the developed models has been confirmed by using the Painlev´e analysis. The author uses the complex forms of the simplified Hirota’s method to obtain two fundamentally different sets of solutions, multiple real and multiple complex soliton solutions for each model. Findings The time-dependent KdV equations feature interesting results in propagation of waves and fluid flow. Research limitations/implications The paper presents a new efficient algorithm for constructing time-dependent integrable equations. Practical implications The author develops two time-dependent integrable KdV equations of third- and fifth-order. These models represent more specific data than the constant equations. The author showed that integrable equation gives real and complex soliton solutions. Social implications The work presents useful findings in the propagation of waves. Originality/value The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

The Integrability of an Extended Fifth-Order KdV Equation in 2+1 Dimensions: Painlevé Property, Lax Pair, Conservation Laws, and Soliton Interactions

2016 ◽  
Vol 71 (6) ◽  
pp. 501-509 ◽  
Author(s):  
Gui-qiong Xu ◽  
Shu-fang Deng
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Integrable Equation ◽  
Polynomial Method ◽  
Bell Polynomial ◽  
Soliton Interactions ◽  
Painlevé Property ◽  
Bilinear Bäcklund Transformation ◽  
Fifth Order

AbstractIn this article, we apply the singularity structure analysis to test an extended 2+1-dimensional fifth-order KdV equation for integrability. It is proven that the generalized equation passes the Painlevé test for integrability only in three distinct cases. Two of those cases are in agreement with the known results, and a new integrable equation is first given. Then, for the new integrable equation, we employ the Bell polynomial method to construct its bilinear forms, bilinear Bäcklund transformation, Lax pair, and infinite conversation laws systematically. The N-soliton solutions of this new integrable equation are derived, and the propagations and collisions of multiple solitons are shown by graphs.

Understanding functional illiteracy from a policy, adult education, and cognition point of view: Towards a joint referent framework

2019 ◽  
Vol 30 (2) ◽  
pp. 109-122
Author(s):  
Aleksandar Bulajić ◽  
Miomir Despotović ◽  
Thomas Lachmann
Keyword(s):  
Adult Education ◽  
Operational Definition ◽  
Literacy Skills ◽  
Point Of View ◽  
Industrialized Countries ◽  
Functional Illiteracy ◽  
Scale Levels ◽  
Components Of Reading ◽  
The One ◽  
Definition Of

Abstract. The article discusses the emergence of a functional literacy construct and the rediscovery of illiteracy in industrialized countries during the second half of the 20th century. It offers a short explanation of how the construct evolved over time. In addition, it explores how functional (il)literacy is conceived differently by research discourses of cognitive and neural studies, on the one hand, and by prescriptive and normative international policy documents and adult education, on the other hand. Furthermore, it analyses how literacy skills surveys such as the Level One Study (leo.) or the PIAAC may help to bridge the gap between cognitive and more practical and educational approaches to literacy, the goal being to place the functional illiteracy (FI) construct within its existing scale levels. It also sheds more light on the way in which FI can be perceived in terms of different cognitive processes and underlying components of reading. By building on the previous work of other authors and previous definitions, the article brings together different views of FI and offers a perspective for a needed operational definition of the concept, which would be an appropriate reference point for future educational, political, and scientific utilization.

Democracy and Political Culture

2019 ◽  
Author(s):  
Ross McKibbin
Keyword(s):  
Political Culture ◽  
Case Studies ◽  
Social System ◽  
Political System ◽  
Democratic Politics ◽  
Social Structures ◽  
The Political ◽  
The Subject ◽  
The One ◽  
Definition Of

This book is an examination of Britain as a democratic society; what it means to describe it as such; and how we can attempt such an examination. The book does this via a number of ‘case-studies’ which approach the subject in different ways: J.M. Keynes and his analysis of British social structures; the political career of Harold Nicolson and his understanding of democratic politics; the novels of A.J. Cronin, especially The Citadel, and what they tell us about the definition of democracy in the interwar years. The book also investigates the evolution of the British party political system until the present day and attempts to suggest why it has become so apparently unstable. There are also two chapters on sport as representative of the British social system as a whole as well as the ways in which the British influenced the sporting systems of other countries. The book has a marked comparative theme, including one chapter which compares British and Australian political cultures and which shows British democracy in a somewhat different light from the one usually shone on it. The concluding chapter brings together the overall argument.

An LQ Approach to Active Control of Vibrations in Helicopters

1996 ◽  
Vol 118 (3) ◽  
pp. 482-488 ◽  
Author(s):  
Sergio Bittanti ◽  
Fabrizio Lorito ◽  
Silvia Strada
Keyword(s):  
Optimal Control ◽  
Active Control ◽  
Linear Quadratic ◽  
Control Effort ◽  
Vibration Effect ◽  
Suitable Definition ◽  
Lq Optimal Control ◽  
The One ◽  
Definition Of ◽  
And Control

In this paper, Linear Quadratic (LQ) optimal control concepts are applied for the active control of vibrations in helicopters. The study is based on an identified dynamic model of the rotor. The vibration effect is captured by suitably augmenting the state vector of the rotor model. Then, Kalman filtering concepts can be used to obtain a real-time estimate of the vibration, which is then fed back to form a suitable compensation signal. This design rationale is derived here starting from a rigorous problem position in an optimal control context. Among other things, this calls for a suitable definition of the performance index, of nonstandard type. The application of these ideas to a test helicopter, by means of computer simulations, shows good performances both in terms of disturbance rejection effectiveness and control effort limitation. The performance of the obtained controller is compared with the one achievable by the so called Higher Harmonic Control (HHC) approach, well known within the helicopter community.

scholarly journals Beyond Classical Definition: The Non-definition of Gamification

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Mario S. Staller ◽  
Swen Koerner
Keyword(s):  
Empirical Investigation ◽  
Working Life ◽  
Pedagogical Practice ◽  
Police Recruits ◽  
Design Elements ◽  
The Subject ◽  
Anchor Points ◽  
The One ◽  
Definition Of ◽  
Game Elements

AbstractGamification is regularly defined as the use of game elements in non-gaming contexts. However, discussions in the context of the pedagogical value of gamification suggest controversies on various levels. While on the one hand, the potential is seen in the design of joyful learning environments, critics point out the pedagogical dangers or the problems related to optimizing working life. It becomes apparent that the assumptions guiding action on the subject matter of gamification in educational contexts differ, which leads to different derivations for pedagogical practice—but also allows for different perspectives on initially controversial positions. Being aware of these assumptions is the claim of a reflexive pedagogy. With regard to the pedagogical use of gamifying elements and their empirical investigation, there are three main anchor points to consider from a reflexive stance: (a) the high context-specificity of the teaching undertaken and (b) the (non-)visibility of the design elements and (c) the (non-)acceptance of the gamified elements by the students. We start by providing a discussion of the definitional discourse on what is understood as gamification leading to our argument for a non-definition of gamification. We describe the potential of this non-definition of gamification and exemplify its use in a gamified concept of teaching police recruits professional reflexivity. The concept features the narrative of a potential crime that has been undertaken and that students decide for themselves if they want to engage with it.

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