Darboux Transformation Research Articles

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.

Rogue waves and breathers of the derivative Yajima-Oikawa long wave-short wave equations on theta-function backgrounds

Based on the Baker-Akhiezer functions on Riemann surfaces, a general method is proposed to construct rogue-wave solutions and breather solutions of integrable nonlinear evolution equations associated with higher-order matrix spectral problems on theta-function periodic backgrounds. The derivative Yajima-Oikawa long wave-short wave equation associated with a 3×3 matrix Lax pair is chosen as an illustrative example of our method. First, a theta-function seed solution for the derivative Yajima-Oikawa long wave-short wave equation is derived by using the Weierstrass functions. Then, the general solution of the 3×3 matrix spectral problem and its derivatives with respect to the spectral parameter are obtained with the help of the Baker-Akhiezer functions. Finally, by means of Darboux transformations, explicit exact solutions such as rogue waves and breathers on theta-function backgrounds for the derivative Yajima-Oikawa long wave-short wave equation are constructed. Taking the advantage of the Jacobi θ-functions, these new solutions are numerically calculated and illustrated. In addition, by observing the explicit solutions, it is found that two rogue waves of different shapes may appear in one solution, which is a new phenomenon different from the solutions on constant backgrounds.

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

In this article multi-soliton solutions of reverse space-time nonlocal nonlinear Schr ödinger (NLS) equation have been constructed. Darboux transformation is applied to the associated linear eigenvalue problem for the generalized NLS equation and we obtain a determinant formula for multi-soliton solutions. Under suitable reduction conditions and appropriate choice of spectral parameters, the generalized expression of first-order nontrivial solution gives some novel solutions such as double-hump and flat-top soliton solutions for reverse space-time nonlocal NLS equation. The dynamics and interaction of double-hump soliton solutions are studied in detail and it is indicated that these solutions undergo collisions without any energy redistribution. For higher-order double-hump solutions, the relative velocities of solitons play a crucial role to have humps and also induce nonlinear interference in the collision zone. The dynamics of individual decaying and growing unstable and stable double-humps as well as their interactions are explained and illustrated.

Novel optical soliton structures for a defocusing Lakshmanan–Porsezian–Daniel optical system

In this article, under investigation is a defocusing Lakshmanan–Porsezian–Daniel optical (DLPD) system holding higher-order dispersion and higher-order nonlinear effects. Via applying the Darboux transformation method, new analytical solutions are obtained for the system. Some novel intersection solitons and their characteristics are found out under the certain parameter setting. There are four basic soliton structures: (i) bright-dark soliton with reduced amplitude at the intersection point; (ii) bright-dark soliton with increased amplitude at the intersection point; (iii) doubly-bright soliton with bright peaks; (iv) doubly-dark soliton with dark troughs. Furthermore, as the order of the solutions increases, there are more abundant and more complex structures at the intersection point. This indicates that this system can carry out more optical information by exciting higher order solitons.

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.

Kuznetsov–Ma rogue wave clusters of the nonlinear Schrödinger equation

In this work, we investigate rogue wave (RW) clusters of different shapes, composed of Kuznetsov–Ma solitons (KMSs) from the nonlinear Schrödinger equation (NLSE) with Kerr nonlinearity. We present three classes of exact higher-order solutions on uniform background that are calculated using the Darboux transformation (DT) scheme with precisely chosen parameters. The first solution class is characterized by strong intensity narrow peaks that are periodic along the evolution x-axis, when the eigenvalues in DT scheme generate KMSs with commensurate frequencies. The second solution class exhibits a form of elliptical rogue wave clusters; it is derived from the first solution class when the first m evolution shifts in the nth-order DT scheme are nonzero and equal. We show that the high-intensity peaks built on KMSs of order n-2m periodically appear along the x-axis. This structure, considered as the central rogue wave, is enclosed by m ellipses consisting of a certain number of the first-order KMSs determined by the ellipse index and the solution order. The third class of KMS clusters is obtained when purely imaginary DT eigenvalues tend to some preset offset value higher than one, while keeping the x-shifts unchanged. We show that the central rogue wave at (0, 0) always retains its n-2m order. The n tails composed of the first-order KMSs are formed above and below the central maximum. When n is even, more complicated patterns are generated, with m and m-1 loops above and below the central RW, respectively. Finally, we compute an additional solution class on a wavy background, defined by the Jacobi elliptic dnoidal function, which displays specific intensity patterns that are consistent with the background wavy perturbation. This work demonstrates an incredible power of the DT scheme in creating new solutions of the NLSE and a tremendous richness in form and function of those solutions.

Hybrid structures of localized waves for the coupled extended mKdV equation

For the coupled extended modified Korteweg–de Vries (mKdV) equation, we investigate hybrid structure waves composed of localized waves. This equation can be considered as the generalization of the mKdV equation. Firstly, hybrid structures of localized waves can be derived in view of the generalized Darboux transformation. Next, we get different interactions between those hybrid waves with the parameters changing. After that, it is found that under a certain condition, the hybrid solutions will degenerate into rational solutions. In particular, we find an interesting rational rogue wave with the novel dynamical behavior. In addition, we show the striking dynamics of hybrid solutions by analyzing each figure. These new solutions can help us investigate integrable systems better.

Breathers, rogue waves and semi-rational solutions for a Heisenberg ferromagnet-type equation

The Kuralay equation, as a Heisenberg ferromagnet-type equation, is of great significance for describing nonlinear phenomena in magnets. For the Kuralay-IIA equation, based on the constructed [Formula: see text]-fold Darboux transformation, we derive its one-fold and two-fold rogue wave solutions, one-breathers and two-breathers including the Akhmediev two-breathers. By adding shifts on the time variable, we further separate the Akhmediev two-breathers into two one-breathers. In addition, the semi-rational solutions consisting of the rogue wave and one-breather solutions are also derived. The diagrams of the obtained solutions are plotted to analyze their dynamic features.

Darboux Transformation and Soliton Solutions for a Nonlocal Two-Component Complex Modified Korteweg–de Vries Equation

AbstractIn this paper, a nonlocal two-component complex modified Korteweg–de Vries (mKdV) equation is investigated and its Darboux transformation is derived. By using the Darboux transformation, soliton solutions are obtained which include dark soliton solution, quasi-W-type soliton, quasi-M-type soliton and periodic solution.

A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation

AbstractIn the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step‐type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.

Black hole greybody factors from Korteweg–de Vries integrals: Computation

It has recently been shown that the dynamics of perturbed nonrotating black holes (BHs) admits an infinite number of symmetries that are generated by the flow of the Korteweg-de Vries (KdV) equation. These symmetries lead to an infinite number of conserved quantities that can be obtained as integrals of differential polynomials in the potential appearing in the gauge-invariant master equations describing the BH perturbations, the KdV integrals. These conserved quantities are the same for all the possible potentials, which means that they are invariant under Darboux transformations, and they fully determine the BHs transmission amplitudes, or greybody factors, via a moment problem. In this paper we introduce a new semianalytical method to obtain the greybody factors associated with BH scattering processes by solving the moment problem using only the KdV integrals. The method is based on the use of Pad\'e approximants and we check it first by comparing with results from the case of a P\"oschl-Teller potential, for which we have analytical expressions for the greybody factors. Then, we apply it to the case of a Schwarzschild BH and compare with results from computations based on the Wentzel--Kramers--Brillouin (WKB) approximation. It turns out that the new method provides accurate results for the BH greybody factors for all frequencies. The method is also computationally very efficient.

A new 3-component Degasperis–Procesi hierarchy

We propose a new 3-component Degasperis–Procesi (3-DP) system. By investigating the bi-Hamiltonian structure of the corresponding hierarchy, we find that the Hamiltonian functionals of the 3-DP hierarchy are local in the negative direction and homogeneous in both directions. We show that the 3-DP system is related to the first negative flow in a modified Yajima–Oikawa hierarchy via a reciprocal transformation. With the help of this reciprocal transformation and a Darboux transformation of the associated 3-DP system, we construct a Bäcklund transformation for the 3-DP system. Furthermore, we present a short-wave model of the 3-DP system.

Generalized Darboux algorithms for Dunkl–Schrödinger equations with position-dependent mass at fixed energy

We apply generalized Darboux transformations to Schrödinger equations at fixed energy that contain Dunkl derivatives, and are equipped with a position-dependent mass. Our Darboux transformations, originating from coupled Korteweg–De Vries systems, consist of two algorithms that generate new solutions, potentials and position-dependent mass functions for the Dunkl–Schrödinger system.

The Fokas-Lenells equation on the line: Global well-posedness with solitons

In this paper, we prove the existence of global solutions in H3(R)∩H2,1(R) to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons. A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.

On the higher-order Heisenberg spin system with quintic terms in self-consistent potential

The integrable higher-order Heisenberg spin system with quintic terms in self-consistent potential (HHSQ) is an extension of Heisenberg ferromagnetic equations. In this paper, we calculate the Lax expression of the HHSQ system, then the motioning curves in [Formula: see text] dimension of Heisenberg ferromagnetic equations have been discussed. The Darboux transformation of the HHSQ system is constructed and a set of soliton solutions will be demonstrated by Darboux transformation.

Rogue wave solutions over double-periodic wave background of a fifth-order nonlinear Schrödinger equation with quintic terms

Studies on rogue waves (RWs) over constant and periodic wave background are of contemporary interest in several areas of physics. These RWs arise not only on such backgrounds but also appear on double-periodic wave background. In this paper, we derive RW solutions over a double-periodic wave background of a fifth-order nonlinear Schrödinger equation. We create the double-periodic wave background with elliptic functions (in combinations of cn, sn and dn) as seed solutions in the first iteration of Darboux transformation. We then calculate the growth rate of instability for these double-periodic solutions under different values of elliptic modulus parameter. On the top of these background waves, we generate RWs with the help of second independent solution to the eigenvalue problem. Further, we analyze the differences that occur in the appearance of RWs with reference to the lower-order and higher-order dispersions terms. In addition to the above, we examine the derived solutions in detail for certain system and elliptic modulus parameter values and highlight some interesting features that we obtain from our studies.

Bidiagonal factorization of tetradiagonal matrices and Darboux transformations

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima–Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi–Piñeiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin.

KdV breathers on a cnoidal wave background

Using the Darboux transformation for the Korteweg–de Vries equation, we construct and analyze exact solutions describing the interaction of a solitary wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like character, these wave patterns are referred to as breathers. Both elevation (bright) and depression (dark) breather solutions are obtained. The nonlinear dispersion relations demonstrate that the bright (dark) breathers propagate faster (slower) than the background cnoidal wave. Two-soliton solutions are obtained in the limit of degeneration of the cnoidal wave. In the small amplitude regime, the dark breathers are accurately approximated by dark soliton solutions of the nonlinear Schrödinger equation. These results provide insight into recent experiments on soliton-dispersive shock wave interactions and soliton gases.

Dynamic behaviors of novel nonlinear wave solutions for the Akbota equation

The Akbota equation, as a Heisenberg ferromagnet-type equation, is of great value for the study of the curve and surface geometry. Based on the J-fold Darboux transformation, some novel nonlinear wave solutions such as the one-breather, two-breather, rogue wave and the semi-rational solutions of the Akbota equation are derived. In addition, the corresponding dynamic behaviors of those solutions are discussed via graph simulations.

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.