Journal ArticleDOI
Integrable evolution systems based on Gerdjikov-Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation
TLDR
In this article, a spectral problem and the associated Gerdjikov-Ivanov (GI) hierarchy of nonlinear evolution equations is presented, and an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique.Abstract:
A spectral problem and the associated Gerdjikov–Ivanov (GI) hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known GI equation of derivative nonlinear Schrodinger equations is obtained. It is shown that the GI hierarchy is integrable in a Liouville sense and possesses bi-Hamiltonian structure. Moreover, the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the Bargmann constraint between the potentials and the eigenfunctions. In particular, an explicit N-fold Darboux transformation for the GI equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique. Some explicit solitonlike solutions of the GI equation are given by applying its Darboux transformation.read more
Citations
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Journal ArticleDOI
The rogue wave and breather solution of the Gerdjikov-Ivanov equation
Shuwei Xu,Jingsong He +1 more
TL;DR: The Gerdjikov-Ivanov (GI) system of q and r is defined by a quadratic polynomial spectral problem with 2 × 2 matrix coefficients and each element of the matrix of n-fold Darboux transformation (DT) for this system is expressed by a ratio of (n + 1) × (n+ 1) determinant and n × n determinant of eigenfunctions.
Journal ArticleDOI
Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model
TL;DR: In this paper, the Gerdjikov-Ivanov envelope solitons were derived and discussed under suitable hypothesis for the current velocity, and the fluid density satisfies a generalized stationary Gardner equation, which possesses bright-and dark-type (including gray and black) solitary waves due to associated parametric constraints.
Journal ArticleDOI
Optical soliton perturbation for Gerdjikov-Ivanov equation via two analytical techniques
Saima Arshed,Anjan Biswas,Anjan Biswas,Mahmoud Abdel-Aty,Qin Zhou,Seithuti P. Moshokoa,Milivoj R. Belic +6 more
TL;DR: In this paper, two integration procedures are employed to obtain soliton solutions to the perturbed Gerdjikov-Ivanov equation, i.e., G′/G2expansion method and sine-cosine method.
Journal ArticleDOI
Madelung fluid description on a generalized mixed nonlinear Schrödinger equation
TL;DR: In this paper, the authors derived bright and dark envelope solutions for a generalized mixed nonlinear Schrodinger model, including gray-and black-soliton envelope solutions, under suitable assumptions for the current velocity associated with different boundary conditions of the fluid density.
Journal ArticleDOI
A generalized multi-component Glachette–Johnson (GJ) hierarchy and its integrable coupling system
TL;DR: In this paper, a new loop algebra G M is constructed, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A 1, and a general scheme for generating multi-component integrable hierarchies is proposed.
References
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TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
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Solitons, Nonlinear Evolution Equations and Inverse Scattering
M. A. Ablowitz,Peter A. Clarkson +1 more
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Darboux transformations and solitons
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TL;DR: In this paper, the authors developed a systematic algebraic approach to solve linear and non-linear partial differential equations arising in soliton theory, such as the non-stationary linear Schrodinger equation, Korteweg-de Vries and Kadomtsev-Petviashvili equations, the Davey Stewartson system, Sine-Gordon and nonlinearSchrodinger equations 1+1 and 2+1 Toda lattice equations, and many others.
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L. D. Faddeev,Leon A. Takhtajan +1 more
TL;DR: The Nonlinear Schrodinger Equation (NS Model) and Zero Curvature Representation (ZCR) as discussed by the authors have been used for the classification and analysis of Integrable Evolution Equations.
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A Simple model of the integrable Hamiltonian equation
TL;DR: In this paper, a method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested, based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equation with their symmetries by using symplectic operators.
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