scholarly journals Inverse scattering transform and soliton solutions of an integrable nonlocal Hirota equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuan Li ◽  
Shou-Fu Tian
Keyword(s):  
Inverse Scattering ◽  
Hilbert Problem ◽  
Scattering Problem ◽  
Soliton Solutions ◽  
Inverse Scattering Transform ◽  
Scattering Data ◽  
Symmetric Relation ◽  
Hirota Equation ◽  
Direct Scattering ◽  
Scattering Transform

<p style='text-indent:20px;'>In this work, we study the inverse scattering transform of a nonlocal Hirota equation in detail, and obtain the corresponding soliton solutions formula. Starting from the Lax pair of this equation, we obtain the corresponding infinite number of conservation laws and some properties of scattering data. By analyzing the direct scattering problem, we get a critical symmetric relation which is different from the local equations. A novel left-right Riemann-Hilbert problem is proposed to develop the inverse scattering theory. The potentials are recovered and the pure soliton solutions formula is obtained when the reflection coefficients are zero. Based on the zero types of scattering data, nine types of soliton solutions are obtained and three typical types are described in detail. In addition, some dynamic behaviors are given to illustrate the soliton characteristics of the space symmetric nonlocal Hirota equation.</p>

scholarly journals Inverse scattering transform for new mixed spectral Ablowitz-Kaup-Newell-Segur equations

2020 ◽  
Vol 24 (4) ◽  
pp. 2437-2444
Author(s):  
Sheng Zhang ◽  
Caihong You
Keyword(s):  
Inverse Scattering ◽  
Spectral Parameter ◽  
Analytical Methods ◽  
Linear Problem ◽  
Soliton Solutions ◽  
Inverse Scattering Transform ◽  
Time Varying ◽  
Spatial Structures ◽  
New Exact Solutions ◽  
Scattering Transform

The inverse scattering transform plays a very important role in promoting the development of analytical methods to solve non-linear PDE exactly. In this paper, new and more general mixed spectral Ablowitz-Kaup-Newell-Segur equations are derived and solved by embedding a novel time-varying spectral parameter in-to an associated linear problem and the inverse scattering transform. As a result, new exact solutions and n-soliton solutions are obtained. To gain more insights into the embedded spectral parameter and the obtained solutions, some dynamical evolutions, and spatial structures are simulated. It is shown that the derived Ablowitz-Kaup-Newell-Segur equations are Lax integrable and the obtained soliton solutions possess time-varying amplitudes.

Toda lattice with corrections via inverse scattering transform

2020 ◽  
pp. 2150084
Author(s):  
Yanpei Zhen ◽  
Xiaodan Wang ◽  
Junyi Zhu
Keyword(s):  
Perturbation Theory ◽  
Approximate Solution ◽  
Conservation Laws ◽  
Time Evolution ◽  
Inverse Scattering ◽  
Toda Lattice ◽  
Inverse Scattering Transform ◽  
Scattering Data ◽  
Scattering Transform

The perturbation theory based on the inverse scattering transform is extended to discuss the Toda lattice with corrections. The time evolution of the associated scattering data is given by some summation representations for corrections and eigenfunctions. The perturbation correction of the conservation laws is investigated. The adiabatic approximate solution and its correction are considered.

The Eigenfunctions and Exact Solutions of Discrete mKdV Hierarchy with Self-Consistent Sources via the Inverse Scattering Transform

2015 ◽  
Vol 7 (5) ◽  
pp. 663-674 ◽  
Author(s):  
Q. Li ◽  
J. B. Zhang ◽  
D. Y. Chen
Keyword(s):  
Exact Solutions ◽  
Spectral Problem ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Inverse Scattering Transform ◽  
The Self ◽  
Scattering Transform ◽  
Self Consistent ◽  
Exact Soliton Solutions ◽  
Mkdv Hierarchy

AbstractAnother form of the discrete mKdV hierarchy with self-consistent sources is given in the paper. The self-consistent sources is presented only by the eigenfunctions corresponding to the reduction of the Ablowitz-Ladik spectral problem. The exact soliton solutions are also derived by the inverse scattering transform.

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