scholarly journals Construction of higher-order smooth positons and breather positons via Hirota’s bilinear method

2021 ◽  
Author(s):  
Zhao Zhang ◽  
Biao Li ◽  
Junchao Chen ◽  
Qi Guo
Keyword(s):  
Higher Order ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method

scholarly journals Construction of Higher-Order Smooth Positons and Breather Positons via Hirota's Bilinear Method

2021 ◽  
Author(s):  
Zhao Zhang ◽  
Biao Li ◽  
Junchao Chen ◽  
QI GUO
Keyword(s):  
Soliton Solution ◽  
Darboux Transformation ◽  
Mathematical Expression ◽  
Higher Order ◽  
Second Order ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Advantages And Disadvantages ◽  
De Vries ◽  
Generalized Darboux Transformation

Abstract Based on the Hirota's bilinear method, a more classic limit technique is perfected to obtain second-order smooth positons. Immediately afterwards, we propose an extremely ingenious limit approach in which higher-order smooth positons and breather positons can be quickly derived from N-soliton solution. Under this ingenious technique, the smooth positons and breather positons of the modified Korteweg-de Vries system are quickly and easily derived. Compared with the generalized Darboux transformation, the approach mentioned in this paper has the following advantages and disadvantages: the advantage is that it is simple and fast; the disadvantage is that this method cannot get a concise general mathematical expression of nth-order smooth positons.

Multiple Soliton Solutions for a Variety of Coupled Modified Korteweg–de Vries Equations

2011 ◽  
Vol 66 (10-11) ◽  
pp. 625-631
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Mkdv Equation ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
De Vries ◽  
Coupled Equation ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

We make use of Hirota’s bilinear method with computer symbolic computation to study a variety of coupled modified Korteweg-de Vries (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are obtained for each coupled equation. The resonance phenomenon of each coupled mKdV equation is proved not to exist.

Fission and fusion interaction phenomena of mixed lump kink solutions for a generalized (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850161 ◽  
Author(s):  
Yaqing Liu ◽  
Xiaoyong Wen
Keyword(s):  
Shallow Water ◽  
Symbolic Computation ◽  
Water Wave ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Lump Solutions ◽  
Shallow Water Wave ◽  
Petviashvili Equation ◽  
Kink Soliton ◽  
Interaction Phenomena

In this paper, a generalized (3[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili (gBKP) equation is investigated by using the Hirota’s bilinear method. With the aid of symbolic computation, some new lump, mixed lump kink and periodic lump solutions are derived. Based on the derived solutions, some novel interaction phenomena like the fission and fusion interactions between one lump soliton and one kink soliton, the fission and fusion interactions between one lump soliton and a pair of kink solitons and the interactions between two periodic lump solitons are discussed graphically. Results might be helpful for understanding the propagation of the shallow water wave.

Line Rogue Waves in the Mel’nikov Equation

2017 ◽  
Vol 72 (7) ◽  
pp. 609-615 ◽  
Author(s):  
Yongkang Shi
Keyword(s):  
Three Dimensional ◽  
Rogue Waves ◽  
Higher Order ◽  
Matrix Elements ◽  
Bilinear Method ◽  
Algebraic Expressions ◽  
General Line ◽  
Free Parameters ◽  
Hirota Bilinear ◽  
Fundamental Line

AbstractGeneral line rogue waves in the Mel’nikov equation are derived via the Hirota bilinear method, which are given in terms of determinants whose matrix elements have plain algebraic expressions. It is shown that fundamental rogue waves are line rogue waves, which arise from the constant background with a line profile and then disappear into the constant background again. By means of the regulation of free parameters, two subclass of nonfundamental rogue waves are generated, which are called as multirogue waves and higher-order rogue waves. The multirogue waves consist of several fundamental line rogue waves, which arise from the constant background and then decay back to the constant background. The higher-order rogue waves start from a localised lump and retreat back to it. The dynamical behaviours of these line rogue waves are demonstrated by the density and the three-dimensional figures.

DARK SOLITONS IN OPTICAL FIBERS WITH HIGHER ORDER EFFECTS

2002 ◽  
Vol 11 (02) ◽  
pp. 197-204 ◽  
Author(s):  
K. NAKKEERAN
Keyword(s):  
Optical Fibers ◽  
Dark Soliton ◽  
Higher Order ◽  
Order Effects ◽  
Nonlinear Wave Propagation ◽  
Bilinear Method ◽  
Fiber System ◽  
Normal Dispersion ◽  
Dispersion Regime ◽  
Higher Order Effects

We consider the higher order nonlinear Schrödinger (HNLS) equation, which governs the nonlinear wave propagation in optical fibers with higher order effects. Lax pair associated with the integrable HNLS equation for the pulse propagation in normal dispersion regime of the fiber media is constructed with the help of Ablowitz–Kaup–Newell–Segur method. Using Hirota bilinear method, dark soliton solution is explicitly derived. Similar study is also carried out for simultaneous propagation of N nonlinear pulses in the normal dispersion regime of the fiber system with higher order effects.

A COMPLETELY INTEGRABLE SYSTEM OF COUPLED MODIFIED KdV EQUATIONS

2010 ◽  
Vol 19 (01) ◽  
pp. 145-151 ◽  
Author(s):  
ABDUL-MAJID WAZWAZ
Keyword(s):  
Integrable System ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Kdv Equations ◽  
Multiple Soliton Solutions ◽  
Modified Kdv Equations ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

In this work, we study a system of coupled modified KdV (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are derived by using the Hirota's bilinear method and the Hietarinta approach. The resonance phenomenon is examined.

Integrability of coupled KdV equations

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Kdv Equation ◽  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Kdv Equations ◽  
Multiple Soliton Solutions ◽  
Coupled Kdv Equations ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

AbstractThe integrability of coupled KdV equations is examined. The simplified form of Hirota’s bilinear method is used to achieve this goal. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation. The resonance phenomenon of each model will be examined.

Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method

2014 ◽  
Vol 92 (3) ◽  
pp. 184-190 ◽  
Author(s):  
Sheng Zhang ◽  
Dong Liu
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Hirota’S Bilinear Method ◽  
Lattice Equation ◽  
Bilinear Method ◽  
Dimensional Variable ◽  
Toda Lattice Equation ◽  
Multisoliton Solutions

In this paper, Hirota’s bilinear method is extended to construct multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation. As a result, new and more general one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formula of the N-soliton solution is derived. It is shown that Hirota’s bilinear method can be used for constructing multisoliton solutions of some other nonlinear differential-difference equations with variable coefficients.

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