scholarly journals Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 23
Author(s):  
Hai Jing Xu ◽  
Song Lin Zhao
Keyword(s):  
Gordon Equation ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Jordan Block ◽  
Sine Gordon Equation ◽  
Third Order ◽  
The Third ◽  
Negative Order ◽  
Double Wronskian ◽  
Akns Equation

In this paper, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, the third order nonisospectral AKNS equation and the negative order nonisospectral AKNS equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral AKNS equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.

Generalized double Wronskian solutions of the third-order isospectral AKNS equation

2009 ◽  
Vol 39 (2) ◽  
pp. 926-935 ◽  
Author(s):  
Fu-mei Yin ◽  
Peng Chen ◽  
Guang-sheng Wang ◽  
Deng-yuan Chen
Keyword(s):  
Third Order ◽  
The Third ◽  
Double Wronskian ◽  
Akns Equation

Corrigenda

1978 ◽  
Vol 362 (1711) ◽  
pp. 572-572
Keyword(s):  
Gordon Equation ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Multiple Soliton Solutions

Proc. R. Soc. Lond . A 359, 479-495 (1978) Exact, multiple soliton solutions of the double sine Gordon equation By P. G. Burt Equation (A 10), exponent of P should read - 1/(2 p ). Equation (A 15), exponents of P should read - 1/(2 p ) and - (1/(2 p ) + 1) respectively. Proc. R. Soc. Lond . A 362, 281-303 (1978) The Bakerian Lecture, 1977: in vitro models for photosynthesis By Sir George Porter, F. R. S. Page 300, line —8: for λ < 455 nm read λ > 455 nm

Noncommutative negative order AKNS equation and its soliton solutions

2018 ◽  
Vol 33 (35) ◽  
pp. 1850209 ◽  
Author(s):  
H. Wajahat A. Riaz ◽  
Mahmood ul Hassan
Keyword(s):  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Curvature Condition ◽  
Integrable Structure ◽  
Zero Curvature ◽  
Negative Order ◽  
Akns Equation

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.

scholarly journals Darboux transformation and multi-soliton solutions of the discrete sine-Gordon equation

2020 ◽  
Vol 2020 (6) ◽  
Author(s):  
Y Hanif ◽  
U Saleem
Keyword(s):  
Darboux Transformation ◽  
Gordon Equation ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Dynamical Features ◽  
Discrete Darboux Transformation ◽  
Explicit Expressions

Abstract We study the discrete Darboux transformation and construct multi-soliton solutions in terms of the ratio of determinants for the integrable discrete sine-Gordon equation. We also calculate explicit expressions of single-, double-, triple-, and quadruple-soliton solutions as well as single- and double-breather solutions of the discrete sine-Gordon equation. The dynamical features of discrete kinks and breathers are also illustrated.

Representations of affine Lie algebras and soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 391-391
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Gordon Equation ◽  
Vries Equation ◽  
Affine Lie Algebras ◽  
Loop Groups ◽  
Sine Gordon Equation ◽  
Kyoto School ◽  
Hidden Symmetries ◽  
Soliton Equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.

Integrable and solvable systems, and relations among them

1985 ◽  
Vol 315 (1533) ◽  
pp. 451-457 ◽  
Keyword(s):  
Gordon Equation ◽  
Soliton Solutions ◽  
The Self ◽  
Sine Gordon Equation ◽  
Field Equations ◽  
Four Dimensions ◽  
Painlevé Property ◽  
All Solutions ◽  
Special Cases ◽  
Integrable Dynamical Systems

There are several different classes of differential equations that may be described as ‘integrable’ or ‘solvable’. For example, there are completely integrable dynamical systems; equations such as the sine—Gordon equation, which admit soliton solutions; and the self-dual gauge-field equations in four dimensions (with generalizations in arbitrarily large dimension). This lecture discusses two ideas that link all of these together: one is the Painlevé property, which says (roughly speaking) that all solutions to the equations are meromorphic; the other is that many of the equations are special cases (i.e. reductions) of others.

Soliton Solutions for a Negative Order AKNS Equation

2008 ◽  
Vol 50 (5) ◽  
pp. 1033-1035 ◽  
Author(s):  
Ji Jie ◽  
Zhao Song-Lin ◽  
Zhang Da-Jun
Keyword(s):  
Soliton Solutions ◽  
Negative Order ◽  
Akns Equation

scholarly journals Solitons, Rogues and Interaction Behaviors of Third-Order Nonlinear Schrodinger Equation

2022 ◽  
Author(s):  
Ren Bo ◽  
Shi Kai-Zhong ◽  
Shou-Feng Shen ◽  
Wang Guo-Fang ◽  
Peng Jun-Da ◽  
...  
Keyword(s):  
Bilinear Form ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Ultrashort Pulses ◽  
Rogue Waves ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
Femtosecond Regime ◽  
Wave Limit

Abstract In this paper, we investigate the third-order nonlinear Schr\"{o}dinger equation which is used to describe the propagation of ultrashort pulses in the subpicosecond or femtosecond regime. Based on the independent transformation, the bilinear form of the third-order NLSE is constructed. The multiple soliton solutions are constructed by solving the bilinear form. The multi-order rogue waves and interaction between one-soliton and first-order rogue wave are obtained by the long wave limit in multi-solitons. The dynamics of the first-order rogue wave, second-order rogue wave and interaction between one-soliton and first-order rogue wave are presented by selecting the appropriate parameters. In particular parameters, the positions and the maximum of amplitude of rogue wave can be confirmed by the detail calculations.PACS numbers: 02.30.Ik, 05.45.Yv.

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