N-fold Darboux Transformation Research Articles

Breather–breather interactions, rational and semi-rational solutions of an integrable spin system in optical fibers

This paper investigates an integrable spin system (or Heisenberg ferromagnet-type equation or Nurshuak-IIA equation) which is an integrable generalization of the Heisenberg ferromagnet equation that raises from magneto-optical devices. The N-fold Darboux transformation of the Nurshuak-IIA equation will be constructed. With the selection of several parameters, a variety of the solutions will be produced, including breather–breather interactions, higher-order rogue waves and semi-rational solutions.

Asymptotic analysis on bright solitons and breather solutions of a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide

Abstract We study a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide. We obtain the Lax pair and N-fold Darboux transformation (DT) with N being a positive integer. Based on Lax pair obtained by us, we derive the infinitely-many conservation laws. We give the bright one-, two-, and N-soliton solutions, and the first-, second-, and Nth-order breather solutions based on the N-fold DT. We conclude that the velocities of the bright solitons are influenced by the distributed gain function, g(z), and variable coefficients in equation, h 1(z), p 1(z), r 1(z), and s 1(z) via the asymptotic analysis, where z represents the propagation variable or spatial coordinate. We also graphically observe that: the velocities of the first- and second-order breathers will be affected by h 1(z), p 1(z), r 1(z), and s 1(z), and the background wave depends on g(z).

Rogue waves in a reverse space nonlocal nonlinear Schrödinger equation

Rogue waves in a reverse space nonlocal nonlinear Schrödinger (NLS) equation with real and parity-symmetric nonlinearity-induced potential are considered. This equation has clear physical meanings since it can be derived from the Manakov system with a special reduction. The N-fold Darboux transformation and its generalized form for the nonlocal NLS equation are constructed. As an application, the multiparametric Nth-order rogue wave solution in terms of Schur polynomials for the nonlocal NLS equation with focusing case is derived by the limit technique. The significant differences of rogue wave dynamics between the nonlocal NLS equation and its usual (local) counterpart are illustrated through two types of specific rogue wave solutions. Unlike the eye-shaped (Peregrine type) rogue waves, the rogue wave doublets which involve an eye-shaped rogue wave and a dark/four-petaled rogue wave merging or separating with each other, and the rogue wave sextets that are characterized by the superpositions of three eye-shaped rogue waves and three dark/four-petaled rogue waves with fundamental, triangular and quadrilateral patterns are shown. Moreover, some wave characteristics including the difference between the light intensity and the plane-wave background, and the pulse energy of the rogue wave doublets are discussed.

Generalized [formula omitted]-fold Darboux transformation and localized waves for an integrable reduced spin Hirota-Maxwell-Bloch system in an erbium doped fiber

In this paper, under investigation is an integrable reduced spin Hirota-Maxwell-Bloch (rsHMB) system, which describes the transmission of the femtosecond pulses in an erbium doped fiber. Based on the existing N-fold Darboux transformation, we establish a generalized (n,N−n)-fold Darboux transformation for the rsHMB system, where n and N are the positive integers (n≤N). The Nth-order degenerate soliton and Nth-order rogue-wave solutions for the rsHMB system are obtained via the generalized (1,N−1)-fold Darboux transformation. By means of the generalized (2,N−2)-fold Darboux transformation, we derive the Nth-order hybrid wave solutions describing the interaction between the (N−1)th-order rogue wave and the first-order breather. The above solutions are discussed graphically, which exhibit the diverse wave structures including the triangle and pentagon, among others.

Darboux and generalized Darboux transformations for the fractional integrable derivative nonlinear Schrödinger equation

Analytical methods provide crucial mathematical insights into the stable solitary waves hidden in nonlinear phenomena. The nonlinear Schrödinger (NLS) equation is one of the most important typical integrable soliton models. From a mathematical perspective, the essence of the celebrated derivative NLS (DNLS) equation’s difference from the classical NLS equation lies in its cubic potential being differentiated once by the spatial variable and multiplied by the imaginary unit, which leads to the former having some characteristics that the latter cannot have. This paper extends the DNLS equation to the fractional integrable case with conformable derivative operators, and uses Darboux transformations (DTs) and generalized DT (GDT) to solve it exactly. Specifically, Lax pairs generating the fractional DNLS equation are first given. Based on the given Lax pairs, then the n-fold DTs and GDT for the fractional DNLS equation are derived. Some special exact solutions of the fractional DNLS equation are further obtained by employing the derived n-fold DTs and GDT. Finally, several novel space–time structures and dynamical evolutions of the obtained exact solutions are analyzed. This paper reveals through the DT and GDT methods that the double power-law fractional orders in the exact solutions of the fractional DNLS equation can be used to dominate the variable velocity propagation and anomalous diffusion in fractional dimensional media at different geometric scales.

Soliton and breather solutions of a reverse time nonlocal coupled nonlinear Schrödinger equation with four-wave mixing effect

Under investigation is a reverse time nonlocal coupled nonlinear Schrödinger equation with four-wave mixing effect. The N-fold Darboux transformation is constructed in a compact determinant representation. As an application, the multi-bright–bright soliton, multi-dark–bright soliton and multi-breather solutions are presented. Dynamical behaviors of the degenerate bright–bright solitons, the regular and periodically oscillating dark–bright solitons and the multiple breather interactions are graphically shown.

Soliton, breather and rogue wave solutions of the nonlinear Schrödinger equation via Darboux transformation on a time–space scale

Solving soliton equations on the time–space scale has always been a challenging issue. In this paper, we firstly generalize the Ablowitz–Kaup–Newel–Segur (AKNS) method to the time–space scale, concurrently obtain the nonlinear Schrödinger (NLS) equation on this scale, which unifies the continuous and the semi-discrete NLS equations. On this basis, the N-fold Darboux transformation is proposed for the NLS equation on a space scale. As applications, soliton, breather, and rogue wave solutions of NLS equation are derived from diverse seed solutions on a space scale. Specially, the rouge solution on a space scale is obtained for the first time.

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 is 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.