Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

2021 ◽  
Author(s):  
Tian-Yu Zhou ◽  
Bo Tian ◽  
Su-Su Chen ◽  
Cheng-Cheng Wei ◽  
Yu-Qi Chen
Keyword(s):  
Burgers Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Ocean Dynamics ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Dispersive Waves ◽  
Backlund Transformations ◽  
Hybrid Solutions ◽  
Kink Waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

Bäcklund transformation according to Gerdzikov and Kullish for the Korteweg–de Vries equation

1982 ◽  
Vol 60 (11) ◽  
pp. 1599-1606 ◽  
Author(s):  
Henri-François Gautrin
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Scattering Parameters ◽  
Backlund Transformation ◽  
Specific Change ◽  
Backlund Transformations ◽  
De Vries

A study of solutions of the Gel'fand–Levitan equation permits one to establish new Bäcklund transformations for the Korteweg–de Vries equation. To a specific change in the scattering parameters, there corresponds a family of Bäcklund transformations. A means to construct these transformations is presented.

scholarly journals Bäcklund Transformation and New Exact Solutions of the Sharma-Tasso-Olver Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Lin Jianming ◽  
Ding Jie ◽  
Yuan Wenjun
Keyword(s):  
Exact Solutions ◽  
Bäcklund Transformation ◽  
Analytic Study ◽  
Bäcklund Transformations ◽  
Painlevé Analysis ◽  
Backlund Transformation ◽  
New Exact Solutions ◽  
Backlund Transformations ◽  
Painleve Analysis

The Sharma-Tasso-Olver (STO) equation is investigated. The Painlevé analysis is efficiently used for analytic study of this equation. The Bäcklund transformations and some new exact solutions are formally derived.

scholarly journals Bäcklund transformations for several cases of a type of generalized KdV equation

2004 ◽  
Vol 2004 (63) ◽  
pp. 3369-3377
Author(s):  
Paul Bracken
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Generalized Kdv Equation ◽  
Backlund Transformations ◽  
De Vries ◽  
Derivative Term

An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are generated and studied.

scholarly journals Bäcklund transformations of the first kind associated with Monge-Ampère’s equations

1973 ◽  
Vol 51 ◽  
pp. 161-184
Author(s):  
Michihiko Matsuda
Keyword(s):  
Integrable Systems ◽  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Contact Transformation ◽  
Backlund Transformation ◽  
Backlund Transformations ◽  
The Given

Due to Clairin and Goursat, a Bäcklund transformation of the first kind can be associated with Monge-Ampère’s equation. We shall consider Monge-Ampère’s equation of the form s + f(x, y, z, p, q) + g(x, y, z, p, q) t = 0, where p = ∂z/∂x, q = ∂z/∂y, s = ∂2z/∂x∂y, t = ∂2z/∂y2. The following theorems will be obtained:1. The transformed equation takes on the same form s′ + f′ + g′t′ = 0 if and only if the given equation can be transformed to a Teixeira equation s + L(x, y, z, q)t + M(x, y, z, q)p + N(x, y, z, q) = 0 by a contact transformation.2. Teixeira equation s + tL + pM + N = 0 is solved by integrable systems of order n if and only if the transformed equation is solved by integrable systems of order n — 1.

On Type-II Bäcklund Transformation for the MKdV Hierarchy

2017 ◽  
Vol 72 (4) ◽  
pp. 291-293
Author(s):  
Hui Mao ◽  
Shuqiang Lv
Keyword(s):  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Type I ◽  
Type Ii ◽  
Backlund Transformation ◽  
Backlund Transformations ◽  
Compound Type ◽  
New Type ◽  
Mkdv Hierarchy

AbstractThe study of new integrable defects leads to new type of Bäcklund transformations named as the type-II Bäcklund transformations. In this article, we show, for the MKdV hierarchy, that the type-II Bäcklund transformation is the compound type-I Bäcklund transformation.

scholarly journals Geometry of Bäcklund transformations II: Monge–Ampère invariants

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Yuhao Hu
Keyword(s):  
Bäcklund Transformation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Backlund Transformations ◽  
Local Invariants ◽  
Monge Ampere

Abstract This article is concerned with the question: For which pairs of hyperbolic Euler–Lagrange systems in the plane does there exist a rank-$1$ Bäcklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler–Lagrange systems. In addition, we discover a class of Bäcklund transformations relating two hyperbolic Euler–Lagrange systems of distinct types.

Bäcklund transformations for the sine–Gordon equations

1976 ◽  
Vol 351 (1667) ◽  
pp. 499-523 ◽  
Keyword(s):  
Pulse Propagation ◽  
Numerical Solutions ◽  
Bäcklund Transformation ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Backlund Transformation ◽  
Optical Pulse Propagation ◽  
Backlund Transformations

The generalized sine–Gordon equations z ,xt = F ( z ) in two independent variables x , t include the sine–Gordon z ,xt = sin z and the multiple sine–Gordon’s like z ,xt = sin z + ½ sin ½ z . Among other physical applications all these sine–Gordon’s are significant to the theory of intense ultra-short optical pulse propagation. The sine–Gordon itself has analytical multi-soliton solutions. It also has an infinity of polynomial conserved densities and has auto-Bäcklund transformations which generate a second solution of the sine–Gordon from a first solution – particularly from the solution z ≡ 0. We prove first that the generalized multi-dimensional sine–Gordon in two or more space variables x 1 , x 2 , . . . has no auto-Bäcklund transformations. Next we prove that the generalized sine–Gordon’s z ,xt = F ( z ) and z ' ,xt = G ( z ') have an invertible Bäcklund transformation between solutions z and z ' if and only if F and G are solutions of F ¨ = α 2 F , G ¨ = β 2 G where, in general, β = αh -1 , α is a complex number and h 2 (≠ 0) is real. In case h = 1 and F and G are the same function z ,xt = F ( z ) has an auto-Bäcklund transformation if and only if F ¨ = α 2 F . We exhibit the B. ts and a. B. ts in these cases as well as the other B. ts for the generalized sine–Gordon. We conclude that the multiple sine–Gordon’s do not have a. B. ts and infer that, despite the soliton character of the numerical solutions, the multiple sine–Gordon’s are not soluble by present simplest formulations of the two by two inverse scattering method.

scholarly journals BÄCKLUND TRANSFORMATIONS IN 2-D DILATON GRAVITY

1998 ◽  
Vol 13 (25) ◽  
pp. 2049-2055 ◽  
Author(s):  
D. J. NAVARRO ◽  
J. NAVARRO-SALAS
Keyword(s):  
Field Theory ◽  
Canonical Transformation ◽  
Bäcklund Transformation ◽  
Free Field ◽  
Gravity Theory ◽  
Bäcklund Transformations ◽  
Dilaton Gravity ◽  
Backlund Transformation ◽  
Backlund Transformations ◽  
Generally Covariant

We give a Bäcklund transformation connecting a generic 2-D dilaton gravity theory to a generally covariant free field theory. This transformation provides an explicit canonical transformation relating both theories.

scholarly journals On Backlund transformations of surfaces by extended Harry-Dym flow

2019 ◽  
Vol 23 (Suppl. 6) ◽  
pp. 1823-1831
Author(s):  
Muhammed Sariaydin
Keyword(s):  
Mathematical Model ◽  
Bäcklund Transformation ◽  
Mapping Method ◽  
Bäcklund Transformations ◽  
Extended Version ◽  
Backlund Transformation ◽  
Backlund Transformations ◽  
Bonnet Surface ◽  
Schrodinger Flow ◽  
The Mathematical Model

The present paper deals with the introduction of Backlund transformations by extended Harry-Dym flow and with the aid of the extended version of the Riccati mapping method is obtained new solutions. Then, we give the Backlund transformation of the Schrodinger flow and obtain its the Bonnet surface. In finally, results obtained with the mathematical model are evaluated by applying to mathematica.

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