Solitons and breather waves for a (2+1)-dimensional Sawada–Kotera equation

2017 ◽  
Vol 31 (22) ◽  
pp. 1750129 ◽  
Author(s):  
Shu-Liang Jia ◽  
Yi-Tian Gao ◽  
Wen-Qiang Hu ◽  
Jing-Jing Su ◽  
Gao-Fu Deng
Keyword(s):  
Soliton Solutions ◽  
Analytic Solutions ◽  
The Other ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Ansatz Method ◽  
Velocity Amplitude ◽  
Straight Line ◽  
Time Periodic

Under investigation in this letter is a (2[Formula: see text]+[Formula: see text]1)-dimensional Sawada-Kotera equation. With the aid of the bilinear forms derived from the Bell polynomials, the Nth-order soliton solutions are obtained via the Pffafian method, and breather solutions are derived with the ansätz method. Analytic solutions obtained via the Pffafian method are the bell-type solitons. Two different kinds of the homoclinic breathers are seen, one of which is real and the other of which is complex, with two breathers interacting with each other. Homoclinic breather wave can evolve periodically along a straight line with a certain angle with the x axis and y axis, and its velocity, amplitude and width remain unchanged during the propagation. Homoclinic breather wave is not only space-periodic but also time-periodic. Interaction between the two breathers is elastic, which is similar to that of the solitons.

Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

2018 ◽  
Vol 32 (08) ◽  
pp. 1750268 ◽  
Author(s):  
Xue-Hui Zhao ◽  
Bo Tian ◽  
Yong-Jiang Guo ◽  
Hui-Min Li
Keyword(s):  
Water Waves ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Linear Superposition ◽  
Elastic Interactions ◽  
The One ◽  
Dimensional Variable

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

Solitons, Bäcklund transformations, Lax pair and conservation laws for the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients

2016 ◽  
Vol 30 (03) ◽  
pp. 1650008 ◽  
Author(s):  
Lei Liu ◽  
Bo Tian ◽  
Wen-Rong Sun ◽  
Yu-Feng Wang ◽  
Yun-Po Wang
Keyword(s):  
Conservation Laws ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Time Dependent ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Bell Polynomial ◽  
Backlund Transformations

The transition phenomenon of few-cycle-pulse optical solitons from a pure modified Korteweg–de Vries (mKdV) to a pure sine-Gordon regime can be described by the nonautonomous mKdV–sinh-Gordon equation with time-dependent coefficients. Based on the Bell polynomials, Hirota method and symbolic computation, bilinear forms and soliton solutions for this equation are obtained. Bäcklund transformations (BTs) in both the binary Bell polynomial and bilinear forms are obtained. By virtue of the BTs and Ablowitz–Kaup–Newell–Segur system, Lax pair and infinitely many conservation laws for this equation are derived as well.

On a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli Equation

2015 ◽  
Vol 70 (5) ◽  
pp. 309-316 ◽  
Author(s):  
Da-Wei Zuo ◽  
Yi-Tian Gao ◽  
Xin Yu ◽  
Yu-Hao Sun ◽  
Long Xue
Keyword(s):  
Incompressible Fluid ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Dispersion Relations ◽  
Bell Polynomials ◽  
Nonlinear Dispersion ◽  
Lax Pairs ◽  
Soliton Propagation ◽  
Periodic Property

AbstractThe Boiti–Leon–Manna–Pempinelli (BLMP) equation is seen as a model for the incompressible fluid. In this article, a (3+1)-dimensional BLMP equation is investigated. With the aid of the Bell polynomials, bilinear form of such an equation is obtained. By virtue of the bilinear form, two kinds of soliton solutions with different nonlinear dispersion relations and another kind of analytic solutions are derived. Lax pairs and Bäcklund transformations are also constructed. Soliton propagation and interaction are analysed: (i) solitions with different nonlinear dispersion relations have different velocities and backgrounds; (ii) for another kind of analytic solutions with different nonlinear dispersion relations, the periodic property is displayed.

Multi-Soliton Solutions and Interaction for a Generalized Variable- Coefficient Calogero–Bogoyavlenskii–Schiff Equation

2014 ◽  
Vol 69 (5-6) ◽  
pp. 239-248 ◽  
Author(s):  
Long Xue ◽  
Yi-Tian Gao ◽  
Da-Wei Zuo ◽  
Yu-Hao Sun ◽  
Xin Yu
Keyword(s):  
Symbolic Computation ◽  
Variable Coefficient ◽  
Soliton Solutions ◽  
Auxiliary Variable ◽  
Hirota Method ◽  
Bell Polynomials ◽  
Bilinear Forms ◽  
Soliton Interaction ◽  
Wave Numbers

In this paper, a generalized variable-coefficient Calogero-Bogoyavlenskii-Schiff equation is investigated. Based on the Bell polynomials and an auxiliary variable, bilinear forms of such an equation are obtained. One-, two-, and three-soliton solutions are given through the Hirota method and symbolic computation. N-soliton solutions are also constructed. Multi-soliton interaction and propagation are investigated and illustrated: (i) properties of the multi-soliton interaction on different planes in space depend on the forms of the only variable coefficient; (ii) positions of the solitons change when the wave numbers have the reverse signs.

Observations Regarding Numerical Results Obtained by the Floquet-Method

2013 ◽  
Author(s):  
Till J. Kniffka ◽  
Horst Ecker
Keyword(s):  
Numerical Methods ◽  
Monodromy Matrix ◽  
Mathieu Equation ◽  
Numerical Results ◽  
The Other ◽  
Benchmark Problem ◽  
Stability Studies ◽  
Floquet Method ◽  
System Equations ◽  
Time Periodic

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

scholarly journals XXIII.—On the Disruptive Discharge of Electricity: An Experimental Thesis for the Degree of Doctor of Science, Department A

1878 ◽  
Vol 28 (2) ◽  
pp. 633-671 ◽  
Author(s):  
Alexander Macfarlane
Keyword(s):  
Copper Wire ◽  
Late Summer ◽  
The Other ◽  
Straight Line ◽  
Summer Session ◽  
The Difference ◽  
A Chain ◽  
The One ◽  
Metallic Parts ◽  
The University

The experiments to which I shall refer were carried out in the physical laboratory of the University during the late summer session. I was ably assisted in conducting the experiments by three students of the laboratory,—Messrs H. A. Salvesen, G. M. Connor, and D. E. Stewart. The method which was used of measuring the difference of potential required to produce a disruptive discharge of electricity under given conditions, is that described in a paper communicated to the Royal Society of Edinburgh in 1876 in the names of Mr J. A. Paton, M. A., and myself, and was suggested to me by Professor Tait as a means of attacking the experimental problems mentioned below.The above sketch which I took of the apparatus in situ may facilitate tha description of the method. The receiver of an air-pump, having a rod capable of being moved air-tight up and down through the neck, was attached to one of the conductors of a Holtz machine in such a manner that the conductor of the machine and the rod formed one conducting system. Projecting from the bottom of the receiver was a short metallic rod, forming one conductor with the metallic parts of the air-pump, and by means of a chain with the uninsulated conductor of the Holtz machine. Brass balls and discs of various sizes were made to order, capable of being screwed on to the ends of the rods. On the table, and at a distance of about six feet from the receiver, was a stand supporting two insulated brass balls, the one fixed, the other having one degree of freedom, viz., of moving in a straight line in the plane of the table. The fixed insulated ball A was made one conductor with the insulated conductor of the Holtz and the rod of the receiver, by means of a copper wire insulated with gutta percha, having one end stuck firmly into a hole in the collar of the receiver, and having the other fitted in between the glass stem and the hollow in the ball, by which it fitted on to the stem tightly. A thin wire similarly fitted in between the ball B and its insulating stem connected the ball with the insulated half ring of a divided ring reflecting electrometer.

The Mechanical Performance Evaluation of a Mechanism with Curved Edge Driving Component

2013 ◽  
Vol 834-836 ◽  
pp. 1290-1294
Author(s):  
Xin Qin Liu
Keyword(s):  
Performance Evaluation ◽  
Fast Moving ◽  
Mechanical Performance ◽  
Gain Coefficient ◽  
The Other ◽  
Force Transmission ◽  
Transmission Performance ◽  
Connecting Rod ◽  
Numerical Examples ◽  
Straight Line

Mechanicalmethods were employed to study the motion and force transmission performance ofa kind of connecting rod slider mechanism with a curved edge driving component.The deduction methods and the computation formulae of the slider displacement,velocity, acceleration and the executive force gain coefficient were given.Considering two cases of the driving components with straight line edge andexponential function edge, the numerical examples was computed respectively,the results show that the former one is suitable for the force transmission andcan be used in the grip design and the other one is suitable for the motiontransmission which can be used in the fast moving mechanism

On indirect Methods of acquiring Knowledge - The Method of History. - The First Table of Mortality

1851 ◽  
Vol 1 (1) ◽  
pp. 40-46
Author(s):  
Edwin James Farren
Keyword(s):  
Real Line ◽  
English Language ◽  
The Other ◽  
Adult Student ◽  
Indirect Methods ◽  
The Real ◽  
Straight Line ◽  
Point To Point ◽  
The One

The term scholar, as current in the English language, has two extreme acceptations, tyro and proficient; or what the later Greeks fancifully termed the alpha and omega of acquirement. If we attempt to trace the steps by which even the adult student of any especial branch of professional or literary knowledge has fairly passed the boundary defined by the one meaning in passing on to that position denoted by the other, it will commonly be found, that in place of that lucid order, that straight line from point to point, which theory and resolve generally premise, the real order of acquirement has been desultory—the real line of progression, circuitous and uncertain.

scholarly journals Compactons and topological solitons of the Drinfel'd–Sokolov system

2014 ◽  
Vol 19 (2) ◽  
pp. 209-224
Author(s):  
Mustafa Inc ◽  
Eda Fendoglu ◽  
Houria Triki ◽  
Anjan Biswas
Keyword(s):  
Elliptic Function ◽  
Function Method ◽  
Soliton Solutions ◽  
Ansatz Method ◽  
Topological Soliton ◽  
Topological Solitons ◽  
Wave Solutions

This paper presents the Drinfel’d–Sokolov system (shortly D(m, n)) in a detailed fashion. The Jacobi’s elliptic function method is employed to extract the cnoidal and snoidal wave solutions. The compacton and solitary pattern solutions are also retrieved. The ansatz method is applied to extract the topological 1-soliton solutions of the D(m, n) with generalized evolution. There are a couple of constraint conditions that will fall out in order to exist the topological soliton solutions.

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