Generalized Darboux Transformation Research Articles

Generalized Darboux transformation, solitonic interactions and bound states for a coupled fourth-order nonlinear Schrödinger system in a birefringent optical fiber

The optical fiber communication system is one of the components of a supporting system in the modern Internet fields. Under investigation in this paper is a coupled fourth-order nonlinear Schrödinger system, which describes the ultrashort optical pluses in a birefringent optical fiber. By virtue of the existing Lax pair, generalized Darboux transformation, two- and three-soliton solutions are derived, with respect to the polarization components of the electric field. Based on such solutions, we graphically display (1) the elastic interactions between/among the two/three solitons on a zero-intensity background, where amplitudes of the solitons remain unchanged; (2) the inelastic interactions between/among the two/three solitons, where amplitudes of the solitons change; (3) the bound state among the three solitons; (4) the higher-order linear and nonlinear effects, represented by β, on the polarization components of the electric field: The interval between two peaks becomes smaller and the numbers of the peaks increase when the value of β increases; The three solitons move along the positive time direction when the value of β decreases; The distances between the adjacent peaks become smaller when the value of β increases.

Rational solutions and interaction solutions for (2 + 1)-dimensional nonlocal Schrödinger equation* *Project supported by LiaoNing Revitalization Talents Program, China (Grant No. XLYC1907014) and Dalian Hi-level Talents Innovation Plan (Grant No. 2017RQ101).

A chain of novel higher order rational solutions with some parameters and interaction solutions of a (2+1)-dimensional reverse space–time nonlocal Schrödinger (NLS) equation was derived by a generalized Darboux transformation (DT) which is derived by Taylor expansion and determinants. We obtained a series of higher-order rational solutions by one spectral parameter and we could get the periodic wave solution and three kinds of interaction solutions, singular breather and periodic wave interaction solution, singular breather and traveling wave interaction solution, bimodal breather and periodic wave interaction solution by two spectral parameters. We found a general formula for these solutions in the form of determinants. We also analyzed the complex wave structures of the dynamic behaviors and the effects of special parameters and presented exact solutions for the (2+1)-dimensional reverse space–time nonlocal NLS equation.

N-order rogue waves and their novel colliding dynamics for a transient stimulated Raman scattering system arising from nonlinear optics

We report the novel colliding dynamics of the rogue waves (RWs) for a three-component transient stimulated Raman scattering system arising from nonlinear optics. The N-order RW solutions are constructed by utilizing the generalized Darboux transformation scheme. Then, the dynamics of the RWs are analyzed. As $$N\ge 2$$ , the multiple RWs collide towards a central point. During the collisions and interactions, the peaks and troughs increase greatly. It is also interesting and magical that all the RWs possess exquisite symmetrical structures.

Nth-order rogue waves for the AB system via the determinants

Under investigation in this paper is the AB system, which is used to describe the baroclinic instability processes in the geophysical flows. With the generalized Darboux transformation, the Nth-order (N=1,2,3,…) rogue wave solutions of the AB system are obtained via the determinants. It is found that the rogue waves can be split into several small ones with the proper parameter selections, and moreover within the splits the maximum peak values of the rogue waves remain unchanged. Besides, we demonstrate that the configurations of the split rogue waves become adjustable.

Controllable rational solutions in nonlinear optics fibers

Based on Darboux matrix method, a generalized Darboux transformation of a generalized nonlinear Schrodinger equation possessing higher-order terms which refer to a femto second pulse propagation in a nonlinear fiber optic is constructed. This nonlinear Schrodinger equation can model the propagation of soliton in a Heisenberg spin chain. Using the above-generalized Darboux transformation, we calculate the first, two and third-order rogue wave solutions and give general formulae for generating Nth-order ones. We discuss the effect of higher-order terms contained in the above equation.

Quantum black hole seismology. I. Echoes, ergospheres, and spectra

Searches for gravitational wave echoes in the aftermath of mergers and/or formation of astrophysical black holes have recently opened a novel and surprising window into the quantum nature of their horizons. Similar to astro- and helioseismology, study of the spectrum of quantum black holes provides a promising method to understand their inner structure, what we call $\textit{quantum black hole seismology}$. We provide a detailed numerical and analytic description of this spectrum in terms of the properties of the Kerr spacetime and quantum black hole horizons, showing that it drastically differs from their classical counterparts. Our most significant findings are the following: (1) If the temperature of quantum black hole is $\lesssim 2 \times$ Hawking temperature, then it will not suffer from ergoregion instability (although the bound is looser at smaller spins). (2) We find how quantum black hole spectra pinpoint the microscopic properties of quantum structure. For example, the detailed spacing of spectral lines can distinguish whether quantum effects appear through compactness (i.e., exotic compact objects) or frequency (i.e., modified dispersion relation). (3) We find out that the overtone quasinormal modes may strongly enhance the amplitude of echo in the low-frequency region. (4) We show the invariance of the spectrum under the generalized Darboux transformation of linear perturbations, showing that it is a genuine covariant observable.

On Solutions of an Extended Nonlocal Nonlinear Schrödinger Equation in Plasmas

The parity-time symmetric nonlocal nonlinear Schrödinger equation with self-consistent sources (PTNNLSESCS) is used to describe the interaction between an high-frequency electrostatic wave and an ion-acoustic wave in plasmas. In this paper, the soliton solutions, rational soliton solutions and rogue wave solutions are derived for the PTNNLSESCS via the generalized Darboux transformation. We find that the soliton solutions can exhibit the elastic interactions of different type of solutions such as antidark-antidark, dark-antidark, and dark-dark soliton pairs on a continuous wave background. Also, we discuss the degenerate case in which only one antidark or dark soliton remains. The rogue wave solution is derived in some specially chosen situations.

Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation

Abstract An extended generalized Darboux transformation method is proposed to construct the hybrid rogue wave and breather solutions for a classical nonlinear Schrodinger equation. Three types of hybrid wave solutions are obtained: (i) the hybrid first-order rogue wave and breather; (ii) the hybrid second-order rogue wave and first-order breather; (iii) the hybrid first-order rogue wave and second-order breather. These solutions are novel and can be used to investigate the dynamical characteristic of the hybrid rogue waves and breathers. The control and interaction based on the parameters of the hybrid wave solution are graphically demonstrated. An exact link is established between the hybrid solutions and the rogue wave solutions via setting the parameter at special value.

A matrix Yajima–Oikawa long-wave-short-wave resonance equation, Darboux transformations and rogue wave solutions

Based on an introduced (2m+n)×(2m+n) matrix spectral problem, a matrix Yajima–Oikawa long-wave-short-wave resonance equation is proposed, which can be reduced to an (n+1)-component Yajima–Oikawa long-wave-short-wave resonance equation. Multi-fold generalized Darboux transformations for these two equations are constructed by using the gauge transformation between Lax pairs and their Riccati equations. Every solution to the Riccati equations can be transformed into a new solution of the matrix Yajima–Oikawa long-wave-short-wave resonance equation through the Darboux transformations. As an application of the obtained Darboux transformation, first, we derive high-order rogue wave solutions of the Yajima–Oikawa long-wave-short-wave resonance equation. Second, we obtain explicit solutions for the lower-order matrix Yajima–Oikawa long-wave-short-wave resonance equation, including soliton solutions, rogue wave solutions, wave solutions that are constant along the x-direction, and wave solutions traveling at varying speeds.

Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrödinger equation

Modulation instability, rogue waves and spectral analysis are investigated for the nonlinear Schrödinger equation with the higher-order terms. The modulation instability distribution characteristics from the sixth-order to eighth-order nonlinear Schrödinger equations are studied. Higher-order dispersion terms are closely related to the distribution of modulation stability regime, and n-order dispersion term corresponds to n−2 modulation stability curves in the modulation instability band. Based on the generalized Darboux transformation method, the higher-order rational solutions are constructed. Then the compact algebraic expression of the N-order rogue wave is given. Dynamic phenomena of the first- to third-order rogue waves are illustrated, which exhibit meaningful structures. Two arbitrary parameters play important roles in the rogue wave solution. One parameter can control the width and crest deflection of rogue wave, while the other can cause the change of width and amplitude of rogue wave. When it comes to the third-order rogue wave, three typical nonlinear wave structures, including fundamental, circular and triangular patterns, are displayed and discussed. Through spectral analysis on the first-order rogue wave, when these parameters satisfy certain conditions, it occurs a transition between W-shaped soliton and rogue wave.

Deformed breather and rogue waves for the inhomogeneous fourth-order nonlinear Schrödinger equation in alpha-helical proteins

The inhomogeneous fourth-order nonlinear Schrodinger equation is investigated, which models the transport of energy along the inhomogeneous hydrogen bonding spines in alpha-helical proteins. The deformed breather and rogue waves solutions for the equation are derived via Darboux transformation. Thereinto, the modified limit procedure in generalized Darboux transformation is proposed to construct the expressions of higher-order rogue waves. Due to the presence of inhomogeneity, despite their diversity profiles, the shape, amplitude and pulse width of each deformed breather will change with time. When the coefficients of higher-order terms are varied, the propagating trajectories of breathers are changed. Head-on collision, partially coalesced, without interaction between two deformed breathers is found under diverse selected parameters. The temporal-spatial structures of the presented rogue waves exhibit certain valleys around one or several centers, and the peak heights of those rogue waves are more than three times that of the background. Additionally, the distributed area of the rogue wave will tend to be broaden/narrow along the temporal axis, with the variation of the coefficients of inhomogeneity and higher-order terms.

Rogue wave solutions for a higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a higher-order nonlinear Schrödinger equation in an optical fiber. Based on the symbolic computation and generalized Darboux transformation, we derive the rogue wave solutions for this equation. Propagation and interaction properties of the rogue waves are discussed. We see that peaks of the higher-order rogue waves could change to different shapes, and the rogue waves will be split under certain variable coefficients.

High-order semi-rational solutions for the coherently coupled nonlinear Schrödinger equations with the positive coherent coupling

A coherently coupled nonlinear Schrödinger system with the positive coherent coupling is studied in this paper, and high-order semi-rational solutions are derived by the generalized Darboux transformation. Dynamic of the second-order semi-rational solutions is found to be of three types: collisions between the one-hump soliton and breather (double-hump rogue waves), collisions between the double-hump soliton and breather (double-hump rogue waves), and bound states of the one-hump solitons or double-hump solitons. Furthermore, for the third-order semi-rational solutions, three double-hump solitons and three one-hump solitons are found to be attracted and repulsed with each other periodically.

Rogue wave solutions for the (2+1)-dimensional complex modified Korteweg–de Vries and Maxwell–Bloch system

In this paper, we will focus on the (2+1)-dimensional complex modified Korteweg–de Vries and Maxwell–Bloch (cmKdVMB) system. According to the correlative Lax pair, the N-fold generalized Darboux Transformations (DT) will be constructed. By virtue of the DT obtained, some rogue wave solutions will be derived from the nonvanishing continuous wave (cw) backgrounds. Meanwhile, the dynamic features of those solutions are graphically analyzed.

Elastic and inelastic collisions of the semirational solutions for the coupled Hirota equations in a birefringent fiber

Under investigation in this manuscript is the coupled Hirota equations, which describe the propagation of ultrashort optical pulses in a birefringent fiber or collision of the two waves arisen by severe weather in deep ocean. Based on the generalized Darboux transformation derived before, we obtain the semirational solutions, and properties of the solitons are discussed in detail: Elastic collisions between/among the two/three solitons on the zero-intensity background are shown in figures, which means that the solitons propagate with the stable amplitudes and velocities during the collisions. Also, inelastic collisions between/among the two/three solitons are illustrated, and some of the solitons travel with amplitudes changed. Furthermore, the bound states among the three solitons are observed.

Nonlinear Dynamics of Rogue Waves in a Fifth-Order Nonlinear Schrödinger Equation

In this paper, nonlinear dynamics of higher-order rogue waves are investigated for a fifth-order nonlinear Schrodinger equation, which can depict the Heisenberg ferromagnetic spin chain. A generalized Darboux transformation is constructed based on the Lax pair. Higher-order rogue waves solutions are given in terms of a recursive formula. Using numerical simulation, the first-order to the third-order rogue waves are displayed on the basis of some free parameters, which play a crucial role in affecting the distribution of rogue waves. The results obtained should be useful in understanding the generation mechanism of rogue waves.

A generalization of the Landau-Lifschitz equation: breathers and rogue waves

A generalization of the Landau-Lifschitz equation with uniaxial anisotropy is proposed, which can also reduce to the derivative nonlinear Schrödinger equation under an infinitesimal parameter. Based on the gauge transformation between Lax pairs, an N-fold generalized Darboux transformation is constructed for the generalization of the Landau-Lifschitz equation with uniaxial anisotropy. As applications of the N-fold generalized Darboux transformations, several types of exact solutions of the generalization of the Landau-Lifschitz equation with uniaxial anisotropy are obtained, including soliton solutions, Akhmediev breather solutions and rogue-wave solutions.

A Vector General Nonlinear Schrödinger Equation with $$(m+n)$$ Components

A vector general nonlinear Schrodinger equation with $$(m+n)$$ components is proposed, which is a new integrable generalization of the vector nonlinear Schrodinger equation and the vector derivative nonlinear Schrodinger equation. Resorting to the Riccati equations associated with the Lax pair and the gauge transformations between the Lax pairs, a general N-fold Darboux transformation of the vector general nonlinear Schrodinger equation with $$(m+n)$$ components is constructed, which can be reduced directly to the classical N-fold Darboux transformation and the generalized Darboux transformation without taking limits. As an illustrative example, some exact solutions of the two-component general nonlinear Schrodinger equation are obtained by using the general Darboux transformation, including a first-order rogue-wave solution, a fourth-order rogue-wave solution, a breather solution, a breather–rogue-wave interaction, two solitons and the fission of a breather into two solitons. It is a very interesting phenomenon that, for all $$M>0$$, there exists a rogue-wave solution for the two-component general nonlinear Schrodinger equation such that the amplitude of the rogue wave is M times higher than its background wave.

Darboux transformations for a matrix long‐wave–short‐wave equation and higher‐order rational rogue‐wave solutions

A new matrix long‐wave–short‐wave equation is proposed with the of help of the zero‐curvature equation. Based on the gauge transformation between Lax pairs, both onefold and multifold classical Darboux transformations are constructed for the matrix long‐wave–short‐wave equation. Resorting to the classical Darboux transformation, a multifold generalized Darboux transformation of the matrix long‐wave–short‐wave equation is derived by utilizing the limit technique, from which rogue wave solutions, in particular, can be obtained by employing the generalized Darboux transformation. As applications, we obtain rogue‐wave solutions of the long‐wave–short‐wave equation and some explicit solutions of the three‐component long‐wave–short‐wave model, including soliton solutions, breather solutions, the first‐order and higher‐order rogue‐wave solutions, and others by using the generalized Darboux transformation.

Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

AbstractWe construct the multi‐breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and nontrivial phase‐modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multibreather solutions with different velocities in the limit , where the solution in the neighborhood of each breather tends to the simple one‐breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue wave and higher order rogue wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena, depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi‐ and higher order rogue wave solution.