A binary Darboux transformation for the Toda lattice

1986 ◽  
Vol 35 (4) ◽  
pp. 2582-2589 ◽  
Author(s):  
V. M. Babich ◽  
V. B. Matveev ◽  
M. A. Sail'
Keyword(s):  
Darboux Transformation ◽  
Toda Lattice ◽  
Binary Darboux Transformation

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

2021 ◽  
Author(s):  
Zhiguo Xu
Keyword(s):  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Integrable Hierarchy ◽  
Integrable Lattice ◽  
Pass Through ◽  
Matrix Spectral Problem ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Hierarchy

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

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.

Darboux transformation and two-dimensional Toda lattice

1983 ◽  
Vol 23 (4) ◽  
pp. 2441-2446 ◽  
Author(s):  
V. B. Matveev ◽  
M. A. Salle
Keyword(s):  
Darboux Transformation ◽  
Toda Lattice ◽  
Two Dimensional

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.

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