Determinant Representation of Binary Darboux Transformation for the AKNS Equation

2015 ◽  
Vol 70 (12) ◽  
pp. 1039-1048 ◽  
Author(s):  
Jing Yu ◽  
Jingwei Han ◽  
Jingsong He
Keyword(s):  
Darboux Transformation ◽  
Darboux Transformations ◽  
Nls Equation ◽  
Reduction Condition ◽  
Nonlinear Schrödinger ◽  
Determinant Representation ◽  
Akns Equation ◽  
New Form ◽  
Binary Darboux Transformation

AbstractIn this paper, the determinant representation of the n-fold binary Darboux transformation, which is a 2×2 matrix, for the Ablowitz–Kaup–Newell–Segur equation is constructed. In this 2×2 matrix, each element is expressed by (2n+1)-order determinants. When the reduction condition r=–q̅ is considered, we obtain one of binary Darboux transformations for the nonlinear Schrödinger (NLS) equation. As its applications, several solutions are constructed for the NLS equation. Especially, a new form of two-soliton is given explicitly.

DETERMINANT REPRESENTATION OF N-TIMES DARBOUX TRANSFORMATION FOR THE DEFOCUSING NONLINEAR SCHRÖDINGER EQUATION

2013 ◽  
Vol 27 (29) ◽  
pp. 1350216 ◽  
Author(s):  
JINGWEI HAN ◽  
JING YU ◽  
JINGSONG HE
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Determinant Representation

The determinant expression T[N] of a new Darboux transformation (DT) for the Ablowitz–Kaup–Newell–Segur equation are given in this paper. By making use of this DT under the reduction r = q*, we construct determinant expressions of dark N-soliton solution for the defocusing NLS equation. Except known one-soliton, we provide smooth two-soliton and smooth N-soliton on a certain domain of parameter for the defocusing NLS equation.

Darboux transformation of the second-type nonlocal derivative nonlinear Schrödinger equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950123 ◽  
Author(s):  
De-Xin Meng ◽  
Kuang-Zhong Li
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Darboux Transformations ◽  
First Order ◽  
Nonlinear Schrödinger ◽  
Derivative Nonlinear Schrödinger Equation

The second-type nonlocal derivative nonlinear Schrödinger (NDNLSII) equation is studied in this paper. By constructing its [Formula: see text]-order Darboux transformations (DT) from the first-order DT, Vandermonde-type determinant solutions of the NDNLSII equation are obtained from zero seed solutions, which would be singular unless the square of eigenvalues are purely imaginary.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

2016 ◽  
Vol 30 (10) ◽  
pp. 1650106 ◽  
Author(s):  
Hai-Qiang Zhang ◽  
Jian Chen
Keyword(s):  
Darboux Transformation ◽  
Variable Coefficient ◽  
Rogue Wave ◽  
Variable Coefficients ◽  
Higher Order ◽  
Optical Systems ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Order Variable ◽  
Generalized Darboux Transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

scholarly journals A hidden life of Peregrine's soliton: Rouge waves in the oceanic depths

2014 ◽  
Vol 11 (06) ◽  
pp. 1450057 ◽  
Author(s):  
Alla Yurova
Keyword(s):  
Mathematical Method ◽  
Darboux Transformation ◽  
Frontal Zone ◽  
Nls Equation ◽  
Nonlinear Schrödinger

Although the Peregrine-type solutions of the nonlinear Schrödinger (NLS) equation have long been associated mainly with the infamous "rouge waves" on the surface of the ocean, they might have a much more interesting role in the oceanic depths; in this paper we show that these solutions play an important role in the evolution of the intrathermocline eddies, also known as the "oceanic lenses". In particular, we show that the collapse of a lens is determined by the particular generalization of the Peregrine soliton — the so-called exultons — of the NLS equation. In addition, we introduce a new mathematical method of construction of a vortical filament (a frontal zone of a lens) from a known one by the Darboux transformation.

scholarly journals N-Fold Darboux Transformation for the Classical Three-Component Nonlinear Schrödinger Equations and Its Exact Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 733
Author(s):  
Yu-Shan Bai ◽  
Peng-Xiang Su ◽  
Wen-Xiu Ma
Keyword(s):  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Lax Pairs ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
The Given

In this paper, by using the gauge transformation and the Lax pairs, the N-fold Darboux transformation (DT) of the classical three-component nonlinear Schrödinger (NLS) equations is given. In addition, by taking seed solutions and using the DT, exact solutions for the given NLS equations are constructed.

New Rational Homoclinic Solution and Rogue Wave Solution for the Coupled Nonlinear Schrödinger Equation

2014 ◽  
Vol 69 (8-9) ◽  
pp. 441-445 ◽  
Author(s):  
Long-Xing Li ◽  
Jun Liu ◽  
Zheng-De Dai ◽  
Ren-Lang Liu
Keyword(s):  
Schrödinger Equation ◽  
Wave Solution ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Homoclinic Solution ◽  
Nonlinear Schrodinger Equation ◽  
Nls Equation ◽  
Nonlinear Schrödinger ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Rogue Wave Solution

In this work, the rational homoclinic solution (rogue wave solution) can be obtained via the classical homoclinic solution for the nonlinear Schrödinger (NLS) equation and the coupled nonlinear Schrödinger (CNLS) equation, respectively. This is a new way for generating rogue wave comparing with direct constructing method and Darboux dressing technique

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