(2+1) Dimensional Soliton Equations Covariant with respect to the Binary Darboux Transformation

1996 ◽  
Vol 65 (1) ◽  
pp. 53-57 ◽  
Author(s):  
Kenji Imai ◽  
Kazuhiro Nozaki
Keyword(s):  
Darboux Transformation ◽  
Soliton Equations ◽  
Binary Darboux Transformation ◽  
Dimensional Soliton

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.

Bound-state solitons for a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber

2020 ◽  
Vol 34 (36) ◽  
pp. 2050423
Author(s):  
Jie Zhang ◽  
Bo Tian ◽  
Qi-Xing Qu ◽  
Chen-Rong Zhang ◽  
Xia-Xia Du ◽  
...  
Keyword(s):  
Darboux Transformation ◽  
Bound State ◽  
Optical Pulses ◽  
Coherent Coupling ◽  
Non Linear ◽  
Optical Modes ◽  
Binary Darboux Transformation ◽  
Birefringent Fiber ◽  
Schrödinger System

In this paper, we study a non-linear Schrödinger system with the negatively coherent coupling in a weakly birefringent fiber for two orthogonally polarized optical pulses. With respect to the slowly-varying envelopes of two interacting optical modes and based on the existing binary Darboux transformation, we obtain four types of the bound-state solitons: degenerate-I, degenerate-II, degenerate–non-degenerate, and non-degenerate–non-degenerate bound-state solitons. We graphically analyze the interactions between the degenerate or non-degenerate solitons and four types of the bound-state solitons. When the degenerate solitons interact with the bound-state solitons, amplitudes and widths of the degenerate solitons remain unchanged. When the non-degenerate solitons interact with the bound-state solitons, amplitudes and widths of the bound-state solitons remain unchanged.

SOLITON EIGENFUNCTION EQUATIONS: THE IST INTEGRABILITY AND SOME PROPERTIES

1990 ◽  
Vol 02 (04) ◽  
pp. 399-440 ◽  
Author(s):  
B.G. KONOPELCHENKO
Keyword(s):  
Differential Equations ◽  
Linear Systems ◽  
Wide Class ◽  
Eigenvalue Problems ◽  
Nonlinear Differential Equations ◽  
Soliton Equations ◽  
Spectral Transform ◽  
Dimensional Soliton ◽  
Operator Representations ◽  
General Method

Eigenfunctions of the linear eigenvalue problems for the soliton equations obey nonlinear differential equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform (IST) method. They have triad operator representations. Eigenfunction equations are the generating equations and possess other interesting properties. Eigenfunction equations form a new wide class of nonlinear integrable equations. Eigenfunction equations for several typical, well-known (1+1)-, (2+1)- and multi-dimensional soliton equations are considered. A general method for constructing the auxiliary linear systems for the eigenfunction equations is proposed. It is shown that the vertical hierarchies of the eigenfunction equations contain only finite numbers of different members in the cases considered. The properties of such hierarchies for soliton equations are closely connected with their Painleve properties. Some “linear” properties of the eigenfunction equations are also discussed.

Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions

2017 ◽  
Vol 72 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Lili Feng ◽  
Fajun Yu ◽  
Li Li
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Non Linear ◽  
And Mathematics

AbstractStarting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.

THE GENERALIZED MULTI-COMPONENT AKNS HIERARCHY AND N-FOLD DARBOUX TRANSFORMATION

2006 ◽  
Vol 20 (25) ◽  
pp. 1575-1589 ◽  
Author(s):  
HONG-XIANG YANG ◽  
DAO-LIN WANG ◽  
CHANG-SHENG LI
Keyword(s):  
Spectral Problem ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Lax Pairs ◽  
Akns Hierarchy ◽  
Soliton Equations ◽  
Gauge Transformations ◽  
Set Up ◽  
Liouville Integrable

Starting from a 3×3 spectral problem, by using the Tu scheme, a hierarchy of generalized multi-component AKNS soliton equations are derived. It is shown that each equation in the resulting hierarchy is Liouville integrable. With the help of gauge transformations of the Lax pairs, an N-fold Darboux transformation (DT) with multi-parameters for the spectral problem is set up. For application, the soliton solutions of the first nonlinear soliton equation are explicitly given.

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