Discrete nonlocal N-fold Darboux transformation and soliton solutions in a reverse space-time nonlocal nonlinear self-dual network equation

2021 ◽  
pp. 2150314
Author(s):  
Cui-Lian Yuan ◽  
Xiao-Yong Wen
Keyword(s):  
Darboux Transformation ◽  
Soliton Solutions ◽  
Space Time ◽  
Local Equation ◽  
Electrical Circuits ◽  
Electrical Signals ◽  
Integrable Lattice ◽  
Graphic Analysis ◽  
Potential Applications ◽  
Nonlocal Nonlinear

In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.

scholarly journals Darboux Transformation, Exact Solutions and Conservation Laws for the Reverse Space-Time Fokas-Lenells Equation

2021 ◽  
Author(s):  
Jiang-Yan Song ◽  
Yu Xiao ◽  
Chi-Ping Zhang
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Space Time ◽  
Local Equation ◽  
Seed Solution ◽  
Dynamical Behaviors ◽  
Local Case ◽  
Physical Phenomena

Abstract In this paper, we firstly deduce a reverse space-time Fokas-Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas-Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics, and related fields.

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.

Decomposition, Darboux transformation and soliton solutions for the (2 + 1)-dimensional integrable nonlocal “breaking soliton” equation

2020 ◽  
Vol 34 (24) ◽  
pp. 2050251
Author(s):  
Xiaoming Zhu ◽  
Kelei Tian
Keyword(s):  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Interaction Mechanisms ◽  
Dark Solitons ◽  
De Vries ◽  
Nonlocal Nonlinear Schrödinger Equation ◽  
Nonlocal Nonlinear

In this paper, we investigate an integrable nonlocal “breaking soliton” equation, which can be decomposed into the nonlocal nonlinear Schrödinger equation and the nonlocal complex modified Korteweg–de Vries equation. As an application, with the use of this decomposition and Darboux transformation, the dark solitons, antidark solitons, rational dark solitons and rational antidark solitons of the considered equation are given explicitly. In particular, the interaction mechanisms of these solutions are discussed and illustrated through some figures.

N-fold Darboux transformation of a six-field integrable lattice system

2021 ◽  
pp. 2150248
Author(s):  
Yanan Qin
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Lattice System ◽  
Local Conservation ◽  
Integrable Lattice ◽  
Coupled Equation ◽  
Math Phys

In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct [Formula: see text]-fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when [Formula: see text].

scholarly journals Infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system

2019 ◽  
Vol 33 (14) ◽  
pp. 1950147 ◽  
Author(s):  
Fangcheng Fan ◽  
Shaoyun Shi ◽  
Zhiguo Xu
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Lattice System ◽  
Darboux Transformations ◽  
Lattice Equation ◽  
Seed Solution ◽  
Integrable Lattice ◽  
Special Cases

In this paper, we study a 6-field integrable lattice system, which, in some special cases, can be reduced to the self-dual network equation, the discrete second-order nonlinear Schrödinger equation and the relativistic Volterra lattice equation. With the help of the Lax pair, we construct infinitely many conservation laws and a new Darboux transformation for system. Exact solutions resulting from the obtained Darboux transformation are presented by using a given seed solution. Further, we generate the soliton solutions and plot the figures of one-soliton solutions with properly parameters.

Conservation laws and Darboux transformations for a 3-coupled integrable lattice equations

2020 ◽  
Vol 34 (21) ◽  
pp. 2050218
Author(s):  
Fangcheng Fan ◽  
Shaoyun Shi ◽  
Zhiguo Xu
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Lax Pair ◽  
Polynomial Form ◽  
Darboux Transformations ◽  
Logarithmic Form ◽  
Integrable Lattice

In this paper, we firstly establish infinitely many conservation laws of the 3-coupled integrable lattice equations by using the Riccati method. Comparing with the results obtained by Sahadevan and Balakrishnan, we not only get infinite conserved densities of the polynomial form, but also some conserved densities of logarithmic form. Secondly, Darboux transformation for the system is derived with the help of the Lax pair and gauge transformation. Finally, we obtain the exact solutions of the system with the obtained Darboux transformation, and present the soliton solutions and their figures with properly parameters.

Translocations In Space-Time and Simultaneous States of the Universe: Convergent Quantifications and Alternative Solutions from the Principles of Physics for the Challenges of “Time Travel” and “Parallel Universes”

2016 ◽  
pp. 4058-4069
Author(s):  
Michael A Persinger
Keyword(s):  
Single Photon ◽  
Rest Mass ◽  
Space Time ◽  
Hydrogen Line ◽  
Magnetic Forces ◽  
Universal Space ◽  
Potential Applications ◽  
Order Of Magnitude ◽  
Alternative Solutions ◽  
The Universe

                                Translation of four dimensional axes anywhere within the spatial and temporal boundaries of the universe would require quantitative values from convergence between parameters that reflect these limits. The presence of entanglement and volumetric velocities indicates that the initiating energy for displacement and transposition of axes would be within the upper limit of the rest mass of a single photon which is the same order of magnitude as a macroscopic Hamiltonian of the modified Schrödinger wave function. The representative metaphor is that any local 4-D geometry, rather than displaying restricted movement through Minkowskian space, would instead expand to the total universal space-time volume before re-converging into another location where it would be subject to cause-effect. Within this transient context the contributions from the anisotropic features of entropy and the laws of thermodynamics would be minimal.  The central operation of a fundamental unit of 10-20 J, the hydrogen line frequency, and the Bohr orbital time for ground state electrons would be required for the relocalized manifestation. Similar quantified convergence occurs for the ~1012 parallel states within space per Planck’s time which solve for phase-shift increments where Casimir and magnetic forces intersect.  Experimental support for these interpretations and potential applications is considered. The multiple, convergent solutions of basic universal quantities suggest that translations of spatial axes into adjacent spatial states and the transposition of four dimensional configurations any where and any time within the universe may be accessed but would require alternative perspectives and technologies.

scholarly journals Darboux transformation for classical acoustic spectral problem

2003 ◽  
Vol 2003 (49) ◽  
pp. 3123-3142 ◽  
Author(s):  
A. A. Yurova ◽  
A. V. Yurov ◽  
M. Rudnev
Keyword(s):  
Spectral Problem ◽  
Inhomogeneous Medium ◽  
Darboux Transformation ◽  
Maxwell Equations ◽  
Soliton Solutions ◽  
Moutard Transformation ◽  
Acoustic Problem ◽  
Spatial Dimensions ◽  
Dielectric Permitivity ◽  
Harry Dym Equation

We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows steps similar to those for the Schrödinger operator. However, there is no one-to-one correspondence between the two problems. The technique developed enables one to construct new families of integrable potentials for the acoustic problem, in addition to those already known. The acoustic problem produces a nonlinear Harry Dym PDE. Using the technique, we reproduce a pair of simple soliton solutions of this equation. These solutions are further used to construct a new positon solution for this PDE. Furthermore, using the dressing-chain approach, we build a modified Harry Dym equation together with its LA pair. As an application, we construct some singular and nonsingular integrable potentials (dielectric permitivity) for the Maxwell equations in a 2D inhomogeneous medium.

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