N-fold Darboux Transformation Research Articles

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

Abstract 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 are given. Particularly, the explicit one- and two-soliton solutions are presented and their dynamical behaviors are shown graphically.

Magnetic soliton and breather interactions for the higher-order Heisenberg ferromagnetic equation via the iterative N-fold Darboux transformation

This paper focuses on a higher-order Heisenberg ferromagnetic equation, which may describe the motion of the magnetic vector of isotropic ferromagnetism. The iterative N-fold Darboux transformation is first constructed to generate the dark and anti-dark magnetic solitons on the non-zero constant backgrounds, bright and dark breathers on the trigonometric function and non-zero constant backgrounds as well as breathers on the trigonometric function and vanishing backgrounds. We discover that the soliton structures of three different components can generate rotation with different constant seed solutions. Meanwhile, the trajectory curve and the direction of the magnetic vector are also discussed from the perspective of magnetism, we find that for constant seed solutions, the motion of the magnetic vector is limited to the hemisphere, while for trigonometric seed solutions, the motion of the magnetic vector can be distributed throughout the whole sphere. These novel phenomena may be helpful to understand the dynamics of the magnetic vector in the magnetic materials.

Soliton solutions of a novel nonlocal Hirota system and a nonlocal complex modified Korteweg–de Vries equation

This paper proposes a novel nonlocal Hirota system by adding constraints to the coupled Hirota equation, which can be transformed into a nonlocal complex modified Korteweg–de Vries (mKdV) equation with a Galilean transformation. Since the nonlocal Hirota equation and nonlocal complex mKdV equation can be converted into each other, we only need to solve one of them. Starting from the eigenfunctions of the Lax pair and the adjoint Lax pair of the nonlocal complex mKdV equation, the N-fold Darboux transformation (DT) of this equation is constructed. Subsequently, bright soliton solutions, double-hump soliton solutions, breather-like solutions under zero background, and multi-breather solutions under nonzero constant background are obtained for the nonlocal complex mKdV equation. Besides that, the interaction behaviors of the multi-soliton waves are also studied. There are some interesting phenomenons. The breather-II soliton wave, that can be seen as a combination of bright wave and dark wave, will appear. It does not appear alone but in the multi-breather solution.

On limit fractional Volterra hierarchies

For the limit fractional Volterra (LFV) hierarchy, we construct the n-fold Darboux transformation and the soliton solutions. Furthermore, we extend the LFV hierarchy to the noncommutative LFV (NCLFV) hierarchy, and construct the Darboux transformation expressed by quasi determinant of the noncommutative version. Meanwhile, we establish the relationship between new and old solutions of the NCLFV hierarchy. Finally, the quasi determinant solutions of the NCLFV hierarchy are obtained.

The new soliton solution types to the Myrzakulov-Lakshmanan-XXXII-equation

<abstract> <p>Our attention concenters on deriving diverse forms of the soliton arising from the Myrzakulov-Lakshmanan XXXII (M-XXXII) that describes the generalized Heisenberg ferromagnetic equation. This model has been solved numerically only using the N-fold Darboux Transformation method, not solved analytically before. We will derive new types of the analytical soliton solutions that will be constructed for the first time in the framework of three impressive schemas that are prepared for this target. These three techniques are the Generalized Kudryashov scheme (GKS), the (G'/G)-expansion scheme and the extended direct algebraic scheme (EDAS). Moreover, we will establish the 2D, 3D graphical simulations that clear the new dynamic properties of our achieved solutions.</p> </abstract>

Quasi-Grammian solutions of the coupled Gerdjikov–Ivanov equation

We construct the N-fold standard binary Darboux transformation (bDT) for the coupled Gerdjikov–Ivanov (cGI) equation. Then, using the bDT presented in this article, we explicitly construct the exact solutions of the cGI equation in terms of quasi-Grammians. As applications of the bDT, we present various particular solutions for the cGI equation, including soliton, breather, and rogue wave solutions.

Multiple nonlinear wave solutions of a generalized Heisenberg ferromagnet model and their interactions

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers.

Darboux transformation and explicit solutions for the Kaup-Kupershmidt equation

In this paper, the N-fold Darboux transformation of the Kaup-Kupershmidt equation associated with 3×3 order spectral problem is constructed by means of the gauge transformation. By applying the Darboux transformation, some soliton solutions of this equation are derived.

From breather solutions to lump solutions: A construction method for the Zakharov equation

Periodic solutions of the Zakharov equation are investigated. By performing the limit operation λ 2l–1 → λ 1 on the eigenvalues of the Lax pair obtained from the n-fold Darboux transformation, an order-n breather-positon solution is first obtained from a plane wave seed. It is then proven that an order-n lump solution can be further constructed by taking the limit λ 1 → λ 0 on the breather-positon solution, because the unique eigenvalue λ 0 associated with the Lax pair eigenfunction Ψ(λ 0) = 0 corresponds to the limit of the infinite-periodic solutions. A convenient procedure of generating higher-order lump solutions of the Zakharov equation is also investigated based on the idea of the degeneration of double eigenvalues in multi-breather solutions.

Hybrid relativistic and modified Toda lattice-type system: equivalent form, N-fold Darboux transformation and analytic solutions

Hybrid relativistic and modified Toda lattice-type system: equivalent form, N-fold Darboux transformation and analytic solutions

Darboux transformation for nonlinear Schrödinger type hierarchies

The Darboux transformation for the nonlinear Schrödinger type hierarchies associated to the Hermitian symmetric spaces can be gotten by using the factorization theory of corresponding group. Based on this, we give the n-fold Darboux transformations in determinant representation, and some explicit soliton solutions are constructed.

Semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries equations

The ubiquitous Korteweg–de Vries equation is widely used in oceanic, atmospheric dynamics, and fluid mechanics and can explain almost all kinds of physical phenomena of waves, while explicit solutions are studied in nonlinear mechanics and optics. We investigate explicit solutions for the semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries (cmKdV) equations, which describe sound waves in an inharmonic lattice. A semi-discrete four-component Frobenius-coupled cmKdV equation is presented by utilizing a zero-curvature equation. We use two different reduction methods to get a semi-discrete local Frobenius-coupled cmKdV equation and a semi-discrete nonlocal Frobenius-coupled cmKdV equation. In the paper, we study the above two new types of semi-discrete Frobenius-coupled cmKdV equations. Then the bi-Hamiltonian structures of the above two types of equations are constructed, which show that the two types of equations possess Liouville integrability. The covariant properties of the both new equations are shown by constructing their corresponding Darboux transformation (DT), respectively. Soliton solutions, semi-rational soliton solutions, and rogue wave solutions of the above two types of equations are constructed by N-fold DT and generalized (n,N−n)-fold DT. Three-dimensional plots and density profiles of these explicit solutions are presented. The characteristics of these figures demonstrate the relationship between the above two types of equations.

Soliton solutions for a two-component generalized Sasa-Satsuma equation

As is well known, the Sasa-Satsuma equation is an important integrable high order nonlinear Schrödinger equation. In this paper, a two-component generalized Sasa-Satsuma (gSS) equation is investigated. We construct the n-fold Darboux transformation for the two-component gSS equation. Based on the Darboux transformation, we obtain some interesting solutions, such as a breather soliton solution, kink solution, anti-soliton solution and a periodic-like solution.

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

Inhomogeneous reduced Maxwell-Bloch system in nonlinear optics: Darboux-transformation and solitonic issues

Studies on nonlinear optics are active, providing the applications in modulators, fiber lasers, optical sensors, etc. In this paper, we investigate an inhomogeneous reduced Maxwell-Bloch system, which could describe the ultra-short optical pulses through an inhomogeneous medium. N-fold Darboux transformation is constructed via an existing Lax pair, where N is a positive integer. Generalized Darboux transformation is obtained via a limit procedure. One- and two-soliton solutions are determined with the aid of the N-fold Darboux transformation. Double-pole soliton solutions are derived via the generalized Darboux transformation. With respect to the electric field, microscopic polarization, phase information, and atomic inversion related to the inhomogeneous medium, one soliton, interaction between the two solitons, and double-pole soliton are graphically depicted. Effects of the coefficients in that system on those solitons are shown via the trigonometric and linear functions.

Multi-pole solitons in an inhomogeneous multi-component nonlinear optical medium

Investigations on nonlinear optics are active, with the applications in the modulators, fiber lasers, optical sensors, etc. In this paper, we focus our attention on a system for the ultra-short pulses in an inhomogeneous multi-component nonlinear optical medium. Starting from the existing Lax pair and one-fold Darboux transformation (DT), we construct the N-fold DT of that system, which involves N distinct spectral parameters, where N is a positive integer. The N-fold generalized DT with one spectral parameter is obtained through resorting to the Taylor-series-expansion coefficients of a special solution to that Lax pair. Double-pole soliton solutions of that system are derived via that N-fold generalized DT with N=2. With the aid of the N-fold DT, an N-fold Darboux matrix is constructed, based on which the multi-pole soliton solutions in the determinant form with respect to the electromagnetic field E are determined. Graphically, we find that those double-pole soliton solutions are a kind of the bound-state soliton solutions which represent the elastic interactions between the two solitons. Effects of the coefficients in that system on the double-pole soliton are shown via choosing the trigonometric, linear and quadratic functions. Furthermore, we present the triple-pole soliton and quadruple-pole soliton with respect to E. Our results might be useful in understanding the ultra-short pulses in the nonlinear optical media.

Hodograph transformation, various exact solutions and dynamical analysis for the complex Wadati–Konno–Ichikawa-II equation

This paper is devoted to studying the second type of complete integrable complex Wadati–Konno–Ichikawa-II (cWKI-II) equation. We first study the conservation laws and construct the hodograph transformation for this equation. Based on the resulting transformation, we construct a new two-component associated cWKI-II equation, which is completely equivalent to the cWKI-II equation. Based on the corresponding Lax pair for this new associated equation, we construct its N-fold Darboux transformation and generalized (m,N−m)-fold Darboux transformation. Then, the associated Bäcklund transformation is used to give various exact solutions, such as smooth soliton, bursting soliton, singular-loop soliton, position controllable semi-rational soliton, breather and rogue wave solutions. The soliton structures along with dynamical evolution behaviors are discussed and shown graphically. Moreover, we investigate some important physical quantities of some soliton solutions such as amplitude, wave number, wave width, velocity, energy and phase by using the asymptotic analysis technique. Finally, we also summarize the formation reasons for some singular structures under some parameter constraints.

Exact solutions and Darboux transformation for the reverse space–time nonlocal Lakshmanan–Porsezian–Daniel equation

The integrable reverse space–time nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation is proposed by nonlocal reduction of the AKNS scattering problem. The 1-fold and N-fold Darboux transformations of the nonlocal LPD equation are constructed, from which some interesting nonlinear wave solutions including soliton and rogue wave solutions are obtained, whose dynamical properties are discussed by evolution plots. It is worth mentioning that the real and imaginary parts of the soliton solutions show the characteristics of the breathing solution. These results may be useful in nonlinear optics and related fields.

Bright-dark soliton solutions and their elastic interaction analysis for a reduced integrable spin Hirota–Maxwell–Bloch equation

Under consideration is a reduced integrable spin Hirota–Maxwell–Bloch (rsHMB) equation, which may have potential applications in describing femtosecond pulse propagation through an erbium doped fiber. With the aid of symbolic computation, we establish the N-fold Darboux transformation (DT) of rsHMB equation based on its known 2 × 2 matrix spectral problem. As an application of the resulting DT, we can get bright-dark soliton and breather solutions of rsHMB equation. Through graphical and asymptotic analysis, the soliton elastic collisions are shown and discussed graphically, and some important soliton physical features including wave velocity, amplitude and energy are analyzed. The consequences and properties presented in this paper may be helpful to comprehend the transmission of optical soliton in erbium doped fiber.

Conservation laws, [formula omitted]-fold Darboux transformation, [formula omitted]-dark-bright solitons and the [formula omitted]th-order breathers of a variable-coefficient fourth-order nonlinear Schrödinger system in an inhomogeneous optical fiber

Optical fiber communication system is one of the core supporting systems of the modern internet age. For the simultaneous propagation of nonlinear waves in an inhomogeneous optical fiber, in this work, a coupled variable-coefficient fourth-order nonlinear Schrödinger system is studied. Infinitely-many conservation laws and N-fold Darboux transformation (DT) based on the existing Lax pair under certain constraints are derived, where N is a positive integer. Via the N-fold DT, the N-dark-bright soliton and Nth-order breather solutions under certain constraints are given. Interactions between the two dark-bright solitons are depicted graphically when the dispersion coefficient in that system, γ1(t), and the group velocity dispersion coefficient in that system, σ(t), are the constants and periodic functions, where t means the normalized retarded time. It is found that the velocities of the two dark-bright solitons increase as γ1(t) and σ(t) increase when γ1(t) and σ(t) are the constants. Interactions between the two breathers are presented graphically.