Multi-dark Soliton Research Articles

Optical solitons, dynamics of bifurcation, and chaos in the generalized integrable (2+1)-dimensional nonlinear conformable Schrödinger equations using a new Kudryashov technique

In the present paper, the new Kudryashov approach is utilized to construct several novel optical soliton solutions for the generalized integrable (2 + 1)-dimensional nonlinear Schrödinger system with conformable derivative. Additionally, the dynamics of bifurcation behavior and chaos analysis in this system are investigated. We applied bifurcation and chaos theories to enhance our understanding of the planar dynamical system derived from the current model while we obtained and illustrated the chaotic solutions for the perturbed dynamical system using graphs. The study yields a class of new optical soliton solutions, including bell-shaped, wave, dark, dark-bright, dark, multi-dark, and singular soliton solutions. Three-dimensional, two-dimensional, and contour plots are presented to visually demonstrate the physical implications and dynamic characteristics of the current conformable equation system. Further, an analysis is discussed on how the conformable derivative parameter and the parameter of time impact the present optical solutions, demonstrating the system’s importance. It is believed that the solutions analyzed in this study are entirely new and have not been previously reported. These discoveries have the potential to significantly enhance our understanding of nonlinear physical phenomena, especially in nonlinear optics and traffic signaling effects with optical dromion transmission.

KP reductions and various soliton solutions to the Fokas–Lenells equation under nonzero boundary condition

AbstractIn this paper, we clarify the connection of the Fokas–Lenells (FL) equation to the Kadomtsev–Petviashvili (KP)–Toda hierarchy by using a set of bilinear equations as a bridge and confirm multidark soliton solution to the FL equation previously given by Matsuno (J. Phys. A 2012 45 (475202). We also show that the set of bilinear equations in the KP–Toda hierarchy can be generated from a single discrete KP equation via Miwa transformation. Based on this finding, we further deduce the multibreather and general rogue wave solutions to the FL equation. The dynamical behaviors and patterns for both the breather and rogue wave solutions are illustrated and analyzed.

Soliton molecules for combined mKdV-type bilinear equation

Starting from the multi-soliton solutions obtained by the Hirota bilinear method, the soliton molecule structures for the combined mKdV-type bilinear equation are investigated using the velocity resonance mechanism. The two-soliton molecules of the mKdV-35 equation and the three-soliton molecules of the mKdV-357 equation are specifically demonstrated in this paper. With particular selections of the involved arbitrary parameters, especially the wave numbers, it is confirmed that, besides the usual multi-bright soliton molecules, the multi-dark soliton molecules and the mixed multi-bright-dark soliton molecules can also be obtained. In addition, we discuss the existence of the multi-soliton molecules for the combined mKdV-type bilinear equation with more higher order nonlinear terms and dispersions. The results demonstrate that when N ≥ 4, the combined mKdV-type bilinear equation no longer admits soliton molecules comprising more than four solitons.

Multi-solitons in the spin-orbit-coupled spin-1 Bose-Einstein condensates

We find a restrictive integrable reduction for the one-dimensional spin-orbit-coupled spin-1 Bose-Einstein condensate system, which allows us to reduce the three-component Gross-Pitaevskii system into the two-component Manakov system or the single cubic Schrödinger equation. Based on the reduction, we obtain the multi-breathers, higher-order rouge waves and multi-dark soliton solutions for ferromagnetic and anti-ferromagnetic system, respectively. Furthermore, the impacts of the spin-orbit coupling to the multi-soliton dynamics, and the stability of these soliton solutions have also been discussed.

Darboux transformation and general soliton solutions for the reverse space–time nonlocal short pulse equation

We construct the Darboux transformation for the reverse space–time (RST) nonlocal short pulse equation by considering a nonlocal symmetry reduction of the Ablowitz–Kaup–Newell–Segur spectral problem and a covariant hodograph transformation. Some essential differences between the RST nonlocal models and the usual (local) ones are demonstrated by means of a series of explicit solutions and the solving process. The multi-bright soliton solution on vanishing background, as well as the multi-dark soliton, multi-breather and higher-order rogue wave solutions corresponding to nonvanishing background of the RST nonlocal defocusing short pulse equation are derived through the Darboux transformation. The classification and dynamics of these explicit solutions with loop-, cuspon- and smooth-type are presented. The asymptotic analysis is rigorously performed for the two-bright soliton, two-dark soliton and two-breather solutions. Some novel wave patterns such as the double-loop and mixed loop-cuspon rogue waves which are different from their regular counterparts are shown. Other types of singular solutions along certain space–time lines are also obtained.

Optical solitons in Fiber Bragg Gratings for the coupled form of the nonlinear [formula omitted]-dimensional Kundu-Mukherjee-Naskar equation via four powerful techniques

In this article, four powerful techniques namely, the three wave method, double exponential, homoclinic breather, M-Shaped rational, M-Shaped with one kink, and M-Shaped with two kink are putting into practice to bolster the new optical solutions of the coupled form of the nonlinear (2+1)-dimensional Kundu-Mukherjee-Naskar equation in Fiber Bragg Gratings (FBGs). Implementation of the appropriate functions of the solutions results in a different form of multi-wave solutions with their interaction phenomena such as dark soliton solutions, singular soliton solutions, bright soliton solutions, kink soliton solutions, Multi-Peak soliton solutions, multi dark-bright soliton solutions, multi bright soliton solutions, multi dark soliton solutions with different structure, multi M-Shaped soliton solutions, multi M-Shaped soliton solutions, M-Shaped soliton solutions with different structure, kink soliton solutions with different structure, and periodical M-Shaped soliton solutions. Three-dimensional graphics demonstrating the dependability and productivity of the suggested methodologies are used to describe the dynamics of the produced solutions.

Tau‐function formulation for bright, dark soliton and breather solutions to the massive Thirring model

AbstractIn the present paper, we are concerned with the link between the Kadomtsev–Petviashvili–Toda (KP–Toda) hierarchy and the massive Thirring (MT) model. First, we bilinearize the MT model under both the vanishing and nonvanishing boundary conditions. Starting from a set of bilinear equations of two‐component KP–Toda hierarchy, we derive multibright solution to the MT model. Then, considering a set of bilinear equations of the single‐component KP–Toda hierarchy, multidark soliton and multibreather solutions to the MT model are constructed by imposing constraints on the parameters in two types of tau function, respectively. The dynamics and properties of one‐ and two‐soliton for bright, dark soliton and breather solutions are analyzed in details.

A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al . 2016 Physica D 327 , 13–29 ( doi:10.1016/j.physd.2016.03.012 ); Feng et al . 2016 Phys. Rev. E 93 , 052227 ( doi:10.1103/PhysRevE.93.052227 )).

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

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schrödinger 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.

Dynamics of interaction between lumps and solitons in the Mel’nikov equation

Two families of semi-rational solutions to the Mel’nikov equation (ME), which is a known model of the interaction between long and short waves in two dimensions, are reported. These semi-rational solutions describe interaction between lumps and solitons. The first family of semi-rational solutions is derived by employing the KP-hierarchy reduction method. The fundamental (first-order) semi-rational solutions, consisting of a lump and a dark soliton, feature three different interactions, depending on speeds of the corresponding lump, Vlump, and soliton, Vsoliton: fusion of the lump into the dark soliton (at Vlump < Vsoliton); splitting of the lump from the soliton (at Vlump > Vsoliton); and lump-soliton bound states (at Vlump=Vsoliton). Three subclasses of non-fundamental semi-rational solutions, namely, higher-order, multi-, and mixed semi-rational solutions, are produced. These non-fundamental semi-rational solutions also represent different interaction: fusion of mutli-lumps into mutli-dark solitons, fission of multi-lumps from multi-dark solitons, multi-lump-soliton bound states, partial merger or partial splitting of lumps into/from lump-soliton bound states, etc. In particular, by selecting specific parameter constraints, the first family of semi-rational solutions reduces to solutions of a newly proposed partially spatial-reversed nonlocal ME. The second family of semi-rational solutions is constructed by using the Hirota method combined with a perturbative expansion and a long-wave limit, which describes a lump permanently propagating on the background of a dark soliton. The second family of the solutions indicates not all interactions between lumps and solitons in the ME give rise to fission or fusion. Besides that, a semi-rational solution to the coupled Schrödinger-Boussinesq equation, consisting of a rogue wave and a dark soliton, is obtained as a reduction of a semi-rational solution belonging to the second family.

Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

Based on reduction of the KP hierarchy, the general multi-dark soliton solutions in Gram type determinant forms for the (2[Formula: see text]+[Formula: see text]1)-dimensional multi-component Maccari system are constructed. Especially, the two component coupled Maccari system comprising of two component short waves and single-component long waves are discussed in detail. Besides, the dynamics of one and two dark-dark solitons are analyzed. It is shown that the collisions of two dark-dark solitons are elastic by asymptotic analysis. Additionally, the two dark-dark solitons bound states are studied through two different cases (stationary and moving cases). The bound states can exist up to arbitrary order in the stationary case, however, only two-soliton bound state exists in the moving case. Besides, the oblique stationary bound state can be generated for all possible combinations of nonlinearity coefficients consisting of positive, negative and mixed cases. Nevertheless, the parallel stationary and the moving bound states are only possible when nonlinearity coefficients take opposite signs.

Dark and multi-dark solitons in the three-component nonlinear Schrödinger equations on the general nonzero background**Project supported by the National Natural Science Foundation of China (Grant No. 11771151), the Guangdong Natural Science Foundation of China (Grant No. 2017A030313008), the Guangzhou Science and Technology Program of China (Grant No. 201904010362), and the Fundamental Research Funds for the Central Universities of China (Grant

We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrödinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrödinger systems, and require more in-depth studies in the future.

The general coupled Hirota equations: modulational instability and higher-order vector rogue wave and multi-dark soliton structures.

The general coupled Hirota equations are investigated, which describe the wave propagations of two ultrashort optical fields in a fibre. Firstly, we study the modulational instability for the focusing, defocusing and mixed cases. Secondly, we present a unified formula of high-order rational rogue waves (RWs) for the focusing, defocusing and mixed cases, and find that the distribution patterns for novel vector rational RWs of focusing case are more abundant than ones in the scalar model. Thirdly, the Nth-order vector semirational RWs can demonstrate the coexistence of Nth-order vector rational RWs and N breathers. Fourthly, we derive the multi-dark-dark solitons for the defocsuing and mixed cases. Finally, we derive a formula for the coexistence of dark solitons and RWs. These results further enrich and deepen the understanding of localized wave excitations and applications in vector nonlinear wave systems.

Multi-dark soliton solutions for a coupled AB system in the geophysical flows

In this manuscript, we pay our attention to a coupled AB system, which describes the baroclinic instability processes in the geophysical flows. Based on the Lax pair for the system, the multi-fold Darboux transformation in terms of simple determinants is constructed, and multi-dark soliton solutions are derived, simultaneously. With the dark solutions derived, the properties of dark solitons are illustrated in the figures for several sample solutions. Our results will be meaningful for the investigation of the future development of vector multi-dark solitons in many physical systems.

Bright–dark soliton solutions of the multi-component AB system

In this paper, we investigate a multi-component AB system that appears in geophysical fluid dynamics. We construct bright–dark soliton solutions of the system using the Hirota’s bilinear method. We obtain asymptotic behaviors of two-soliton solution for the two-component AB system and prove that the interactions between two bright and two dark solitons are elastic. Under various parameter conditions, the oblique interactions and bound states of solitons are analyzed in detail. By use of the Pfaffian technique, we present N-bright and N−dark soliton solutions of the two- and multi-component AB system. The results obtained in this paper can be helpful for the study of vector multi-dark solitons in many physical systems such as nonlinear optics and fluid dynamics.

Dynamics and elastic interactions of the discrete multi-dark soliton solutions for the Kaup–Newell lattice equation

Under consideration in this paper is the Kaup–Newell (KN) lattice equation which is an integrable discretization of the KN equation. Infinitely, many conservation laws and discrete N-fold Darboux transformation (DT) for this system are constructed and established based on its Lax representation. Via the resulting N-fold DT, the discrete multi-dark soliton solutions in terms of determinants are derived from non-vanishing background. Propagation and elastic interaction structures of such solitons are shown graphically. Overtaking interaction phenomena between/among the two, three and four solitons are discussed. Numerical simulations are used to explore their dynamical behaviors of such multi-dark solitons. Numerical results show that their evolutions are stable against a small noise. Results in this paper might be helpful for understanding the propagation of nonlinear Alfvén waves in plasmas.

Multi-dark soliton solutions for the higher-order nonlinear Schrödinger equation in optical fibers

In optical fibers with the higher-order dispersive and nonlinear effects, the propagation of the ultrashort pulse can be governed by the higher-order dispersive nonlinear Schrodinger equation. In this paper, the binary Darboux transformation is applied to this higher-order model, and the n-times iterative transformation is expressed in terms of the determinants. By the limit technique, the multi-dark soliton solutions are derived from the nonzero background. Based on obtained solution, the elastic collisions between two and among three-dark solitons with different velocities are presented. The bound-state dark solitons can be also formed when two-dark solitons with the same velocity propagate in parallel.

General soliton solutions to a coupled Fokas–Lenells equation

In this paper, we firstly establish the multi-Hamiltonian structure and infinitely many conservation laws for the vector Kaup–Newell hierarchy of the positive and negative orders. The first nontrivial negative flow corresponds to a coupled Fokas–Lenells equation. By constructing a generalized Darboux transformation and using a limiting process, all kinds of one-soliton solutions are constructed including the bright–dark soliton, the dark–anti-dark soliton and the breather-like solutions. Furthermore, multi-bright and multi-dark soliton solutions are derived and their asymptotic behaviors are investigated.

Multi-dark-dark solitons of the integrable repulsive AB system via the determinants.

We investigate the integrable repulsive AB system and construct its Darboux transformation using the loop group method. The associated N-fold Darboux transformation is found in terms of simple determinants. Moreover, we derive multi-dark-dark solitons of the repulsive AB system with a non-vanishing background through the Darboux transformation with a limit procedure. Particularly, we exhibit the one-, two-, and three-dark-dark solitons. The results will be meaningful for the study of vector multi-dark solitons in many physical systems such as geophysical fluid dynamics and nonlinear optics.

General N-dark vector soliton solution for multi-component defocusing Hirota system in optical fiber media

In the normal dispersion regime of optical fibers with third-order dispersion and self-steepening higher-order effects, the propagation of ultrashort pulses is govern by the multi-component defocusing Hirota system. The general N-dark vector soliton solution is firstly constructed by the binary Darboux transformation. From obtained multi-dark soliton solutions, it is found that the collisions between or among vector dark solitons in this multi-component system are elastic, and vector dark solitons retain their shape and intensity only with a slight change in their phase. The results might be useful for applications about vector dark soliton in fiber directional couplers, optical switching and quantum information processors.