scholarly journals Tau functions of (n, 1) curves and soliton solutions on nonzero constant backgrounds

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Atsushi Nakayashiki
Keyword(s):  
Soliton Solutions ◽  
Nonzero Constant ◽  
Tau Functions

scholarly journals Cusped and Smooth Solitons for the Generalized Camassa-Holm Equation on the Nonzero Constant Pedestal

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Dong Li ◽  
Yongan Xie ◽  
Shengqiang Tang
Keyword(s):  
Solitary Wave ◽  
Numerical Simulations ◽  
Mathematical Analysis ◽  
Soliton Solutions ◽  
Explicit Solutions ◽  
Solitary Wave Solutions ◽  
Nonzero Constant ◽  
Wave Solutions ◽  
Holm Equation

We investigate the traveling solitary wave solutions of the generalized Camassa-Holm equationut - uxxt + 3u2ux=2uxuxx + uuxxxon the nonzero constant pedestallimξ→±∞⁡uξ=A. Our procedure shows that the generalized Camassa-Holm equation with nonzero constant boundary has cusped and smooth soliton solutions. Mathematical analysis and numerical simulations are provided for these traveling soliton solutions of the generalized Camassa-Holm equation. Some exact explicit solutions are obtained. We show some graphs to explain our these solutions.

scholarly journals AFFINE SOLITONS: A RELATION BETWEEN TAU FUNCTIONS, DRESSING AND BÄCKLUND TRANSFORMATIONS

1993 ◽  
Vol 08 (03) ◽  
pp. 507-543 ◽  
Author(s):  
OLIVIER BABELON ◽  
DENIS BERNARD
Keyword(s):  
Lie Algebra ◽  
Vacuum Solution ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Vertex Operators ◽  
Tau Functions ◽  
Affine Lie Algebra ◽  
Backlund Transformations ◽  
Sine Gordon Model

We reconsider the construction of solitons by dressing transformations in the sine-Gordon model. We show that the N-soliton solutions are in the orbit of the vacuum, and we identify the elements in the dressing group which allow us to build the N-soliton solutions from the vacuum solution. The dressed τ functions can be computed in two different ways: either using adjoint actions in the affine Lie algebra [Formula: see text], and this gives the relation with the Bäcklund transformations, or using the level-one representations of the affine Lie algebra [Formula: see text], and this directly gives the formulae for the τ functions in terms of vertex operators.

scholarly journals Peaked and Smooth Solitons forK*(4,1)Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Yongan Xie ◽  
Hualiang Fu ◽  
Shengqiang Tang
Keyword(s):  
Explicit Form ◽  
Soliton Solution ◽  
Mathematical Analysis ◽  
Soliton Solutions ◽  
Single Peak ◽  
Nonzero Constant

This paper is contributed to explore all possible single peak solutions for theK*(4,1)equationut=uxu2+2α(uuxxx+2uxuxx). Our procedure shows that theK*(4,1)equation either has peakon, cuspon, and smooth soliton solutions when sitting on a nonzero constant pedestallimξ→±∞u=A≠0or possesses compacton solutions only whenlimξ→±∞u=A=0. We present a new smooth soliton solution in an explicit form. Mathematical analysis and numeric graphs are provided for those soliton solutions of theK*(4,1)equation.

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

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

scholarly journals Dispersive MHD waves and alfvenons in charge non-neutral plasmas

2008 ◽  
Vol 15 (4) ◽  
pp. 681-693 ◽  
Author(s):  
K. Stasiewicz ◽  
J. Ekeberg
Keyword(s):  
Soliton Solutions ◽  
Theoretical Models ◽  
Mhd Waves ◽  
Space Plasmas ◽  
Charge Effects ◽  
Novel Technique ◽  
Solitary Structures ◽  
Spatial Dimensions ◽  
Electric Potentials ◽  
Linear And Nonlinear

Abstract. Dispersive properties of linear and nonlinear MHD waves, including shear, kinetic, electron inertial Alfvén, and slow and fast magnetosonic waves are analyzed using both analytical expansions and a novel technique of dispersion diagrams. The analysis is extended to explicitly include space charge effects in non-neutral plasmas. Nonlinear soliton solutions, here called alfvenons, are found to represent either convergent or divergent electric field structures with electric potentials and spatial dimensions similar to those observed by satellites in auroral regions. Similar solitary structures are postulated to be created in the solar corona, where fast alfvenons can provide acceleration of electrons to hundreds of keV during flares. Slow alfvenons driven by chromospheric convection produce positive potentials that can account for the acceleration of solar wind ions to 300–800 km/s. New results are discussed in the context of observations and other theoretical models for nonlinear Alfvén waves in space plasmas.

scholarly journals On tau-functions for the KdV hierarchy

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Boris Dubrovin ◽  
Di Yang ◽  
Don Zagier
Keyword(s):  
Tau Functions ◽  
Kdv Hierarchy

scholarly journals Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics

2021 ◽  
Vol 20 ◽  
pp. 103762
Author(s):  
Md. Abdul Kayum ◽  
Shamim Ara ◽  
M.S. Osman ◽  
M. Ali Akbar ◽  
Khaled A. Gepreel
Keyword(s):  
Nonlinear Equations ◽  
Gas Dynamics ◽  
Soliton Solutions

scholarly journals N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Common Factors ◽  
Soliton Solutions ◽  
Higher Order ◽  
Kdv Equations ◽  
The Common ◽  
Bilinear Formulation ◽  
Generalized Kdv Equations ◽  
Hirota Bilinear

Abstract We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes.

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