Soliton solutions of an integrable boundary problem on the half-line for the discrete toda chain

Lump wave and line rogue wave of the (2 + 1)-dimensional Mel’nikov system are derived by taking the ansatz as the rational function. By combining a rational function and different exponential functions, mixed solutions between the lump and soliton are derived. These solutions describe the interaction phenomena of the lump-bright soliton with fission and fusion, the half-line rogue wave with a bright soliton, and a rogue wave excited from the bright soliton pair, respectively. Some special concrete interaction solutions are depicted in both analytical and graphical ways.

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Solitons: Conservation laws and dressing methods

International Journal of Modern Physics A ◽

10.1142/s0217751x19300035 ◽

2019 ◽

Vol 34 (06n07) ◽

pp. 1930003

Author(s):

Anastasia Doikou ◽

Iain Findlay

Keyword(s):

Conservation Laws ◽

Riccati Equation ◽

Integrable Systems ◽

Integrable Hierarchies ◽

Soliton Solutions ◽

Toda Chain ◽

Lax Pair ◽

Conserved Quantities ◽

Nonlinear Schrödinger ◽

Marchenko Equation

We review some of the fundamental notions associated with the theory of solitons. More precisely, we focus on the issue of conservation laws via the existence of the Lax pair and also on methods that provide solutions to partial or ordinary differential equations that are associated to discrete or continuous integrable systems. The Riccati equation associated to a given continuous integrable system is also solved and hence suitable conserved quantities are derived. The notion of the Darboux–Bäcklund transformation is introduced and employed in order to obtain soliton solutions for specific examples of integrable equations. The Zakharov–Shabat dressing scheme and the Gelfand–Levitan–Marchenko equation are also introduced. Via this method, generic solutions are produced and integrable hierarchies are explicitly derived. Various discrete and continuous integrable models are employed as examples such as the Toda chain, the discrete nonlinear Schrödinger model, the Korteweg–de Vries and nonlinear Schrödinger equations as well as the sine-Gordon and Liouville models.

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Soliton solutions of the sine-Gordon equation on the half line

Applied Mathematics Letters ◽

10.1016/j.aml.2018.06.020 ◽

2018 ◽

Vol 86 ◽

pp. 64-69 ◽

Cited By ~ 3

Author(s):

Cheng Zhang ◽

Qiang Cheng ◽

Da-jun Zhang

Keyword(s):

Gordon Equation ◽

Soliton Solutions ◽

Half Line ◽

Sine Gordon Equation

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Remarks on the asymptotic behaviour of solutions to the compressible Navier–Stokes equations in the half-line

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050200032x ◽

2002 ◽

Vol 132 (03) ◽

pp. 627

Keyword(s):

Asymptotic Behaviour ◽

Stokes Equations ◽

Half Line ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Asymptotic Behaviour Of Solutions

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On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

GUB Journal of Science and Engineering ◽

10.3329/gubjse.v5i1.47898 ◽

2018 ◽

Vol 5 (1) ◽

pp. 31-36

Author(s):

Md Monirul Islam ◽

Muztuba Ahbab ◽

Md Robiul Islam ◽

Md Humayun Kabir

Keyword(s):

Solitary Wave ◽

Acoustic Waves ◽

Water Waves ◽

Wave Equations ◽

Rectangular Channel ◽

Soliton Solutions ◽

Boussinesq Equations ◽

Travelling Wave ◽

Ion Acoustic Waves ◽

Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

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