Double-hump Solitons Research Articles

The dynamic behaviors between double-hump solitons in birefringent fibers

In this paper, we research the fractional coupled Hirota equations with variable coefficients describing the collisions of two waves in deep oceans and the propagation of ultrashort light pulses in birefringent fibers and successfully acquire the double-hump one-soliton, two-solitons and N-solitons solutions via the Hirota bilinear method. At the same time, the Bäcklund transformation and the corresponding soliton solutions are also obtained. Based on the precise forms of the solitons solutions, we gain double-hump solitons images with different shapes including U-shape, V-shape and wave-type by assigning proper functions to the group velocity dispersion and the third-order dispersion and analyze the interaction dynamics of double-hump solitons. It is worth noting that the Hirota bilinear operators involved here are fractional rather than integer, which has never appeared in previous literatures.

The nondegenerate solitons solutions for the generalized coupled higher-order nonlinear Schrödinger equations with variable coefficients via the Hirota bilinear method

In this paper, the generalized coupled higher-order nonlinear Schrödinger equations (GCHNLSEs) with variable coefficients, describing the propagation of femtosecond pulse with high peak power in birefringence fibers and inhomogeneous media, are researched by the Hirota bilinear method. The nondegenerate two-solitons and nondegenerate N-solitons solutions for these equations are obtained successfully for the first time. Then, we attain the v-shape, u-shape and wave-type double-hump solitons by adjusting the group velocity dispersion, cross-phase modulation and group velocity effect. Finally, through selecting the appropriate complex wave parameters, we change the distances between solitons and analyze the dynamic behaviors of the collisions.

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

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.

Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Optical fiber communication plays a crucial role in modern communication. In this work, we focus on the higher-order matrix nonlinear Schrödinger equation with negative coherent coupling in a birefringent fiber. For the slowly varying envelopes of two interacting optical modes, we construct a binary Darboux transformation using the corresponding Lax pair. With vanishing seed solutions and the binary Darboux transformation, we investigate vector degenerate soliton and exponential soliton solutions. By utilizing these soliton solutions, we demonstrate three types of degenerate solitons and double-hump bright solitons. Furthermore, considering non-vanishing seed solutions and applying the binary Darboux transformation, we obtain vector breather solutions, and present the vector single-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, double-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, and vector degenerate beak-type breathers. Additionally, we take the limit in the breather solutions and derive vector rogue wave solutions. We illustrate the beak-type rogue waves and bright-dark rogue waves. Humps of these vector double-hump waves can separate into two individual humps. The results obtained in this work may potentially provide valuable insights for experimentally manipulating the separation of two-hump solitons, breathers, and rogue waves in optical fibers.

Nondegenerate solitons of the (2+1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optical fibers

We derive the multi-hump nondegenerate solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations with propagation distance dependent diffraction, nonlinearity and gain (loss) using the developing Hirota bilinear method, and analyze the dynamical behaviors of these nondegenerate solitons. The results show that the shapes of the nondegenerate solitons are controllable by selecting different wave numbers, varying diffraction and nonlinearity parameters. In addition, when all the variable coefficients are chosen to be constant, the solutions obtained in this study reduce to the shape-preserving nondegenerate solitons. Finally, it is found that the nondegenerate two-soliton solutions can be bounded to form a double-hump two-soliton molecule after making the velocity of one double-hump soliton resonate with that of the other one.

Solitons in fourth-order Schrödinger equation with parity-time-symmetric extended Rosen-Morse potentials

We investigate the fourth-order nonlinear Schrödinger equation modulated by parity-time-symmetric extended Rosen-Morse potentials. Since the imaginary part of the potentials does not vanish asymptotically, any slight fluctuations in the field can eventually cause the nonlinear modes to become unstable. Here we obtain stable solitons by adding the constraints of coefficients, which make the imaginary part of the potentials component vanish asymptotically. Furthermore, we get other fundamental stable single-hump and double-hump solitons by numerical methods. Then we consider excitations of the soliton via adiabatical change of system parameters. The results we obtained in this work provide a way to search for stable localized modes in parity-time-symmetric extended Rosen-Morse potentials with fourth-order dispersion.

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.

Riemann–Hilbert problem and [formula omitted]-soliton solutions for the [formula omitted]-component derivative nonlinear Schrödinger equations

The n-component derivative nonlinear Schrödinger (DNLS) equations are discussed by the Riemann–Hilbert approach. By developing an improved Riemann–Hilbert problem, explicit soliton solutions of the n-component DNLS equations are obtained. Different from soliton solutions in many literatures, our expression does not include unfriendly integrals. An invariant and its general formula of the single-soliton solution for any n-component DNLS equations are derived. For the 2-component DNLS equation, we obtain one type of interesting solution — double hump soliton. The results play important roles in the vector solitons in many nonlinear physical systems, such as Bose–Einstein condensates, nonlinear optical fibers, etc.

Soliton formation and dynamics in the quintic nonlinear media with [formula omitted]-invariant harmonic-Gaussian potential

In this paper, we introduce a PT-symmetric harmonic-hyperbolic-Gaussian (HHG) potential in the nonlinear Schrödinger (NLS) equation with quintic nonlinearity. Firstly, we present the critical thresholdless PT-phase symmetry breaking in the linear regime. Secondly, we give both exact and numerical single-/double-hump solitons of the quintic NLS equations with the PT-symmetric HHG potential, and show their stabilities by studying the spectrum stability analysis and direct wave propagations. Finally, we investigate the interactions between the single-/double-hump solitons and the exotic hyperbolic secant function, as well as the adiabatic excitations of the solitons with the operation of an adiabatic switch function. All these obtained results provide a theoretical guidance for the applications of the PT-symmetric theory into diverse areas of physics.

Formation of double-hump solitons and their applications in optical communication systems

Optical communication system with optical solitons has the characteristics of low relay cost, low bit error rate and long relay distance. The modern optical communication system is developing towards large capacity, long distance and high speed. The width of optical pulse has been increased to femtosecond. With the increase of signal frequency, the effects of dispersion and nonlinear involved in it become more and more complex. Therefore, using femtosecond soliton communication to overcome these problems has urgent practical significance. In this paper, starting from the coupled nonlinear Schrödinger equation describing soliton transmission, we use the bilinear method to study the transmission characteristics and interaction of double-hump solitons under the interaction of dispersion and nonlinear effects, solve the problem of mutual interference between adjacent optical solitons, and realize the non-interaction transmission between adjacent solitons. This conclusion lays a theoretical foundation for the optical soliton propagation without distortion and interference in optical communication systems.

Non-degenerate high-order solitons of the coupled nonlinear Schrödinger equation

In this paper, we establish the Riemann–Hilbert problem for the non-degenerate high-order solitons of the coupled nonlinear Schrödinger equation. The dynamics of these two components q1 and q2 behave quite differently. By choosing some proper parameters, one component will have a double-hump soliton but the other one will not. Especially, this kind of high-order solitons will not satisfy the scalar nonlinear Schrödinger equation under the SU(2) symmetry. Moreover, we also give the asymptotic analysis of large t for these different components and check the asymptotic solutions and the exact solutions numerically.

Data-driven solutions and parameter discovery of the Sasa–Satsuma equation via the physics-informed neural networks method

Physics-informed neural networks (PINNs) can be used to predict various solutions of nonlinear partial differential equations in diverse physical fields. We use the PINNs to predict the (single-) double-hump soliton, multi-hump breather, anti-dark soliton, Mexican-hat soliton, twisted rogue-wave pairs and rational W-shaped soliton of the Sasa–Satsuma equation in optical fibers and deep water wave. The L2 relative error of the prediction results can achieve high accuracy even with only less training data and training time while the results show that the prediction error will increase with the increase of prediction time. In addition, we employ the PINNs to investigate the parameter discovery of the Sasa–Satsuma equation via the Mexican-hat soliton solution. For the anti-dark soliton, we compare the predicted results and the amount of data for the PINNs method and the traditional split-step Fourier technique, and show that the PINNs method requires less data with the same accuracy of prediction results.

Generation and transformation of dark solitons, anti-dark solitons and dark double-hump solitons

Generation and transformation of dark solitons, anti-dark solitons and dark double-hump solitons

Impact of external potential and non-isospectral functions on optical solitons and modulation instability in a cubic quintic nonlinear media

We investigate the non-isospectral generalized higher order NLS equation (GHNLS) with external potential which is unswervingly represents the ultrashort optical soliton transmission in inhomogeneous non-Kerr media with combined effects of external potentials. Lax pair is constructed for GHNLS equation and three soliton solutions obtained with the aid of Darboux transformation. For the first time, we include the effects of inhomogeneous non-isospectral parameters on soliton dynamics. Using two soliton solutions, interaction among double hump solitons and using three soliton solutions, interaction characteristics are studied analytically. Furthermore, in order to understand the stability mechanism, linear stability analysis also has been performed. Effects of non-isospectral parameters and variable coefficients on soliton dynamics are investigated graphically. The predicted results may be observed in the future experiments on nonlinear fiber optics and applications to femtosecond soliton management.

Interaction properties of double-hump solitons in the dispersion decreasing fiber

Interaction properties of double-hump solitons in the dispersion decreasing fiber

Stable dynamics and excitations of single- and double-hump solitons in the Kerr nonlinear media with $$\varvec{\mathcal {PT}}$$-symmetric HHG potentials

Stable dynamics and excitations of single- and double-hump solitons in the Kerr nonlinear media with $$\varvec{\mathcal {PT}}$$-symmetric HHG potentials

Formation, stability, and adiabatic excitation of peakons and double-hump solitons in parity-time-symmetric Dirac-δ(x)-Scarf-II optical potentials.

We introduce a class of physically intriguing PT-symmetric Dirac-δ-Scarf-II optical potentials. We find the parameter region making the corresponding non-Hermitian Hamiltonian admit the fully real spectra, and present the stable parameter domains for these obtained peakons, smooth solitons, and double-hump solitons in the self-focusing nonlinear Kerr media with PT-symmetric δ-Scarf-II potentials. In particular, the stable wave propagations are exhibited for the peakon solutions and double-hump solitons from some given parameters even if the corresponding parameters belong to the linear PT-phase broken region. Moreover, we also find the stable wave propagations of exact and numerical peakons and double-hump solitons in the interplay between the power-law nonlinearity and PT-symmetric potentials. Finally, we examine the interactions of the nonlinear modes with exotic waves, and the stable adiabatic excitations of peakons and double-hump solitons in the PT-symmetric Kerr nonlinear media. These results provide the theoretical basis for the design of related physical experiments and applications in PT-symmetric nonlinear optics, Bose-Einstein condensates, and other relevant physical fields.

Stable transmission characteristics of double-hump solitons for the coupled Manakov equations in fiber lasers

The fiber laser has become an ideal ultrashort pulse source because of its cheap structure, high integration, convenient and controllable output direction, which greatly promotes the development and application of ultrafast optics. This paper mainly focuses on the control and amplification of double-hump solitons in fiber lasers theoretically. The bilinear forms and soliton solutions of the coupled Manakov equations are presented, and the transmission of double-hump solitons is discussed. The factors affecting the stable transmission of double-hump solitons are analyzed. The relevant conclusions have important guiding significance for understanding the generation of stable double-hump solitons in fiber lasers.

The collision dynamics between double-hump solitons in two mode optical fibers

In this paper, the coupled Manakov equations with variable coefficients, governing the orthogonally polarized pulses transmission in two mode optical fibers, are studied via the Hirota method. The double-hump one- and two-soliton solutions of the Manakov equations are obtained for the first time. Furthermore, the v-type, parabolic, s-type double-hump solitons and the double-hump solitons with phase shift are found. we reveal that the decrease of the distance between two peaks results in the attenuation and amplification. The double-hump solitons have different collision dynamics in different modes. The methods to regulate the phase shift, amplification, collision intensity and splitting of double-hump solitons are investigated. These dynamic analyses will be valuable to increase the capacity of optical fiber communication system and reduce the bit error rate. We also hope that it is helpful for the development of phase shifter, optical amplifier and other optical logic devices.