An integrable generalization of the Fokas–Lenells equation:Darboux transformation, reduction and explicit soliton solutions

Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly,the explicit one-and two-soliton solutions are presented and their dynamical behaviors are shown graphically.

Darboux transformation,infinite conservation laws,and exact solutions for the nonlocal Hirota equation with variable coefficients

This article presents the construction of a nonlocal Hirota equation with variable coefficients and its Darboux transformation.Using zero-seed solutions,1-soliton and 2-soliton solutions of the equation are constructed through the Darboux transformation,along with the expression for N-soliton solutions.Influence of coefficients that are taken as a function of time instead of a constant,i.e.,coefficient functionδ(t),on the solutions is investigated by choosing the coefficient functionδ(t),and the dynamics of the solutions are analyzed.This article utilizes the Lax pair to construct infinite conservation laws and extends it to nonlocal equations.The study of infinite conservation laws for nonlocal equations holds significant implications for the integrability of nonlocal equations.

DARBOUX TRANSFORMATION OF A NONLINEAR EVOLUTION EQUATION AND ITS EXPLICIT SOLUTIONS

In this paper, we study a differential-difference equation associated with discrete 3 × 3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformation of the differential-difference equation is given. In order to solve the differential-difference equation, a systematic algebraic algorithm is given. As an application, explicit soliton solutions of the differential-difference equation are given.

AUTO-DARBOUX TRANSFORMATION AND EXACT SOLUTIONS OF THE BRUSSELATOR REACTION DIFFUSION MODEL

Firstly, using the improved homogeneous balance method, an auto-Darboux transformation (ADT) for the Brusselator reaction diffusion model is found. Based on the ADT, several exact solutions are obtained which contain some authors' results known. Secondly, by using a series of transformations, the model is reduced into a nonlinear reaction diffusion equation and then through using sine-cosine method, more exact solutions are found which contain soliton solutions.

Darboux transformation and soliton solutions of a nonlocal Hirota equation

Starting from local coupled Hirota equations,we provide a reverse space-time nonlocal Hirota equation by the symmetry reduction method known as the Ablowitz–Kaup–Newell–Segur scattering problem.The Lax integrability of the nonlocal Hirota equation is also guaranteed by existence of the Lax pair.By Lax pair,an n-fold Darboux transformation is constructed for the nonlocal Hirota equation by which some types of exact solutions are found.The solutions with specific properties are distinct from those of the local Hirota equation.In order to further describe the properties and the dynamic features of the solutions explicitly,several kinds of graphs are depicted.

Hierarchy of Lax Integrable Lattice Equations, Darboux Transformation and Conservation Laws

A discrete isospectral problem and the associated hierarchy of Lax integrable lattice equations were investigated. A Darboux transformation for the discrete spectral problem was found. Finally, an infinite number of conservation laws were given for the corresponding hierarchy.

N-Fold Darboux Transformation for the Nonlocal Nonlinear Schrodinger(NNLS)Equation with the Self-Induced PT-Symmetric Potential

The Darboux transformation (DT) method is studied in a lot of local equations, but there are few of work to solve nonlocal equations by DT. In this letter, we solve the nonlocal nonlinear Schrödinger equation (NNLSE) with the self-induced PT-symmetric potential by DT. Then the N-fold DT of NNLSE is derived with the help of the gauge transformation between the Lax pairs. Then we derive some novel exact solutions including the bright soliton, breather wave soliton. In particularly, the dynamic features of one-soliton, two-soliton, three-soliton solutions and the elastic interactions between the two solitons are displayed.

N-Fold Darboux Transformation for a Nonlinear Evolution Equation

In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.

Generalized Darboux Transformation and Rational Solutions for the Nonlocal Nonlinear Schrodinger Equation with the Self-Induced Parity-Time Symmetric Potential

In this paper, I construct a generalized Darboux transformation for the nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. The N-order rational solution is derived by the iterative rule and it can be expressed by the determinant form. In particular, I calculate first-order and second-order rational solutions and obtain their figures according to different parameters.

Hamiltonian structure, Darboux transformation for a soliton hierarchy associated with Lie algebra so(4, C)

In this paper, we first introduce a Lie algebra of the special orthogonal group, g = so(4, C), whose elements are 4 × 4trace-free, skew-symmetric complex matrices. As its application, we obtain a new soliton hierarchy which is reduced to AKNS hierarchy and present its bi-Hamiltonian structure and Liouville integrability. Furthermore, for one of the equations in the resulting hierarchy, we construct a Darboux matrix T depending on the spectral parameter λ.

Darboux transformation of generalized coupled KdV soliton equation and its odd-soliton solutions

Based on the resulting Lax pairs of the generalized coupled KdV soliton equation, a new Darboux transformation with multi-parameters for the generalized coupled KdV soliton equation is derived with the help of a gauge transformation of the spectral problem. By using Darboux transformation, the generalized odd-soliton solutions of the generalized coupled KdV soliton equation are given and presented in determinant form. As an application, the first two cases are given.

Darboux Transformation and Exact Solutions for High Order Nonlocal Coupled AKNS System

In this paper, we investigate an integrable high order nonlocal coupled Ablowitz-Kaup-Newell-Segur (AKNS) system for the first time. With the aid of Lax pair of this nonlocal system, Darboux transformation (DT) and new soliton-like solutions are obtained. Different from local equations, Darboux transformation of nonlocal systems needs to meet certain conditions. In this article, under the condition of symmetry reduction, the components of Darboux transformation need to satisfy <img src="Edit_6aa5df34-2f85-4c91-a185-17195a7f82ee.bmp" alt="" />. In order to study the dynamic information of the solutions, the images of the solutions are given.

The Lax Representation and Darboux Transformation for Constrained Flows of the AKNS Hierarchy

By using a general scheme for decomposing a zero-curvature equation into two commut- ing x-and t_n-finite-dimensional integrable Hamiltonian systems (FDIHS),a systematic deduction of the Lax representation for all constrained flows of the AKNS hierarchy from the adjoint repre- sentation of the two auxiliary linear problems is presented.The Darboux transformation for these FDIHSs is derived.

Binary Darboux transformation and multi-dark solitons for a higher-order nonlinear Schrodinger equation in the inhomogeneous optical fiber

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schr?dinger equation,since dark solitons can be applied in waveguide optics as dynamic switches and junctions or optical logic devices.Based on the Lax pair,the binary Darboux transformation is constructed under certain constraints,thus the multi-dark soliton solutions are presented.Soliton propagation and collision are graphically discussed with the group-velocity dispersion,third-and fourth-order dispersions,which can affect the solitons’velocities but have no effect on the shapes.Elastic collisions between the two dark solitons and among the three dark solitons are displayed,while the elasticity cannot be influenced by the above three coefficients.

A New Negative Discrete Hierarchy and Its N-Fold Darboux Transformation

Starting from a matrix discrete spectral problem, we derive a negative discrete hierarchy. It is shown that the hierarchy is integrable in the Liouville sense and possesses a bi-Hamiltonian structure. Furthermore, its N-fold Darboux transformation is established with the help of gauge transformation of Lax pair. As an application of the Darboux transformation, some new exact solutions for a discrete equation in the negative hierarchy are obtained.

Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

In this paper,we construct Hamiltonian systems for 2 N particles whose force depends on the distances between the particles.We obtain the generalized finite nonperiodic Toda equations via a symmetric group transformation.The solutions of the generalized Toda equations are derived using the tau functions.The relationship between the generalized nonperiodic Toda lattices and Lie algebras is then be discussed and the generalized Kac-van Moerbeke hierarchy is split into generalized Toda lattices,whose integrability and Darboux transformation are studied.

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper,the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations.After this,two kinds of Darboux transforma-tions in different directions and their mixed Darboux transformations of this hierarchy are constructed.These symmetries and Darboux transformations might be useful in Gromov-Witten theory of CP1.

A FAMILY OF INTEGRABLE SYSTEMS OF LIOUVILLE AND LAX REPRESENTATION, DARBOUX TRANSFORMATIONS FOR ITS CONSTRAINED FLOWS

A family of integrable systems of Liouville are obtained by, Tu pattern. Using higher-order potential-eigenfunction constraints, the integrable systems are factorized to two x- and t(n)-integrable Hamiltonian systems whose Lax representation and three kinds of Darboux transformations are presented.

Darboux transformation and exact multisolitons for a matrix coupled dispersionless system

We present a matrix coupled dispersionless(CD)system.A Lax pair for the matrix CD system is proposed and Darboux transformation is constructed on the solutions of the matrix CD system and the associated Lax pair.We express an N soliton formula for the solutions of the matrix CD system in terms of quasideterminants.By using properties of the quasideterminants,we obtain some exact solutions,including bright and dark-type solitons,rogue wave and breather solutions of the matrix CD system.Furthermore,it has been shown that the solutions of the matrix CD system are expressed in terms of solutions to the usual CD system,sine-Gordon equation and Maxwell-Bloch system.

Darboux Transformation and Exact Solutions for Two Dimensional A_(2n-1)^((2)) Toda Equations

The Darboux transformation for the two dimensional A_(2n-1)^((2))Toda equations is constructed so that it preserves all the symmetries of the corresponding Lax pair.The expression of exact solutions of the equation is obtained by using Darboux transformation.