Degenerate Soliton Research Articles

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.

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.

The stability of degenerate solitons for derivative nonlinear Schrödinger equations

In this paper, we consider the following nonlinear Schrödinger equation with derivative:i∂tu+∂xxu+i|u|2∂xu+b|u|4u=0,(t,x)∈R×R,b≥0. For the case b=0, the original DNLS, Kwon and Wu [14] proved the conditional orbital stability of degenerate solitons including scaling, phase rotation, and spatial translation with a non-smallness condition, ‖u(t)‖L66>δ. In this paper, we remove this condition for the non-positive initial energy and momentum, and we extend the stability result for b≥0.

Data-driven vector degenerate and nondegenerate solitons of coupled nonlocal nonlinear Schrödinger equation via improved PINN algorithm

Data-driven vector degenerate and nondegenerate solitons of coupled nonlocal nonlinear Schrödinger equation via improved PINN algorithm

Taking into consideration a fifth-order nonlinear Schrödinger equation in an optical fiber

In this paper, symbolic computation on a fifth-order nonlinear Schrödinger equation is done, for the attosecond pulses propagation in an optical fiber. With respect to the complex amplitude of the optical pulse envelope, we work out a Lax pair and derive the modified generalized Darboux transformation. Then, we give the semirational solutions via the modified generalized Darboux transformation method. By means of such solutions, we graphically discuss the properties for three types of the degenerate solitons.

Dynamical and statistical features of soliton interactions in the focusing Gardner equation.

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.

Optical nondegenerate solitons in a birefringent fiber with a 35 degree elliptical angle.

In this paper, we investigate the optical nondegenerate solitons in a birefringent fiber with a 35 degree elliptical angle. We derive the nondegenerate bright one- and two-soliton solutions by solving the coupled Schrödinger equation. The formation of nondegenerate solitons is related to the wave numbers of the solitons, and we further demonstrate that it is caused by the incoherent addition of different components. We note that the interaction between two degenerate solitons or a nondegenerate soliton and a degenerate soliton is usually inelastic. This is led to the incoherent interaction between solitons of different components and the coherent interaction between solitons of the same component. Through the asymptotic analysis, we find that the two degenerate solitons are elastic interactions under certain conditions, and analyzed the influence of the Kerr nonlinear intensity coefficient γ and the second-order group velocity dispersion β2 in this system on solitons: the velocity and amplitude of the solitons are proportional to |β2|, while the amplitude of the solitons is inversely proportional to γ. Two nondegenerate solitons are elastic interactions, but the phase of the soliton can be adjusted to make it inelastic. Furthermore, regardless of the situation mentioned above, total intensities of the solitons before the interaction are equal to that after the soliton interaction.

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.

Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background

Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background

Bright soliton solutions to the nonlocal Manakov equations of reverse-space type

Utilizing the bilinear method combined with the KP hierarchy reduction technique, bright multi-soliton solutions in terms of determinant to the nonlocal Manakov equations of reverse-space type are derived in the focusing case. Our results show that these solutions only allow an even number of solitons and each paired soliton exhibits the head-on collision with the same velocity. In the dynamic behaviors illustrated, the fundamental paired soliton supports valley and hump at the center of collision, as well as one degenerate soliton with position shifts. The higher-order soliton solutions exhibit the interaction of paired solitons, where the particular bound states include the mixed line-breathing patterns and the single breather with position shifts.

Two-component generalized nonlinear Schrödinger equations and their soliton and breather solutions

In this paper, a cluster of two-component generalized nonlinear Schrödinger equations is generated in terms of zero-curvature equation and polynomial expansion of the spectral parameter, which can be used to illustrate effects on various nonlinear phenomena. After that, explicit representations of classical Darboux transformation and generalized perturbation (n,M)-fold Darboux transformation of the above equations are constructed. Localized wave solutions including soliton solutions, degenerate soliton solutions, soliton solutions interacting with degenerate soliton solutions, breather solutions and degenerate breather solutions are subsequently acquired via the classical Darboux matrix. Meanwhile, degenerate soliton solutions are acquired by the generalized perturbation (n,M)-fold Darboux transformation. Analyses of these solutions are shown through a series of figures ultimately.

Interference phenomenon of nondegenerate solitons for nonlocal CLL equation

We derive nondegenerate one-soliton and two-soliton solutions of the nonlocal derivative nonlinear Schrödinger equation using Hirota method. We discuss in detail the temporal and spatial interference phenomena generated by the interaction between nondegenerate two-solitons. Moreover, we provide a detailed explanation and analysis of the conditions affecting the generation of the interference phenomena of nondegenerate soliton, and further compare the results of the interference phenomena of degenerate solitons. This provides potential applications for the interactions of degenerate and nondegenerate soliton, and related interference phenomena in nonlocal wave models.

N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics

N-fold generalized Darboux transformation and asymptotic analysis of the degenerate solitons for the Sasa-Satsuma equation in fluid dynamics and nonlinear optics

Localized wave solutions and mixed interaction structures in the AB system

For AB system, in the first place we calculate the modulational instability for the possible reason of the formation of localized wave. Secondly, based on the generalized (n,N−n)-fold Darboux transformation, we construct order-n degenerate soliton solutions and their interaction solutions sitting on zero background of this system. Specifically, we discuss the three cases of degenerate soliton with interaction solutions obtained above in detail: (a) For order-1 degenerate soliton solutions, we show that they can be decomposed into two order-1 solitons with variable “phase shift”. (b) For higher-order degenerate soliton solutions, we achieve the bright-dark transformation of these solutions by changing the positive and negative of eigenvalues λi. Then a number of the corresponding structural maps are clearly displayed. (c) For interaction solutions with soliton and degenerate soliton solutions, we study the collision under various structures, and these collision characteristics are significantly different from those before. At last, we also sum up various mathematical features of degenerate soliton and mixed interaction solutions.

Degenerate and bound-state solitons of a novel Kundu-nonlinear Schrödinger equation based on generalized Darboux transformation

In this paper, we derive a novel Kundu-nonlinear Schrödinger equation through a 2 × 2 matrix spectral problem, which can be used to describe the phenomena in optical propagation, and the determinant expression of generalized (n,N−n)-fold Darboux transformation for the Kundu-nonlinear Schrödinger equation is given by solving algebraic equations. In addition, through the generalized (n,N−n)-fold Darboux transformation, we obtain three different types of solutions, (i) soliton solutions, (ii) degenerate soliton solutions, (iii) interaction solutions between solitons and degenerate solitons, which are divided into elastic interaction and bound-state. At the same time, the conditions for their generation are discussed. The properties of the aforementioned solutions are analyzed graphically ultimately.

Linear interference of nonlinear waves-Multispeed vector solitons.

The dynamics of envelope solitons in a system of coupled anharmonic chains are addressed. Mathematically, the system is equivalent to the vector soliton propagation model in a single-mode fiber with low birefringence in the presence of coherent and incoherent interactions. It is numerically and analytically shown that multi-component soliton entries can behave as free scalar solitons with arbitrary velocities and amplitudes. The appropriate exact multi-soliton solutions are provided. They can be presented as a linear interference of degenerate vector solitons known before. Furthermore, the interference idea is transferred to other vector integrable systems, including the Manakov model.

Complex Kraenkel-Manna-Merle system in a ferrite: N-fold Darboux transformation, generalized Darboux transformation and solitons

Ferromagnetic materials such as the ferrites are used in the electronic and energy industries. Here, we concentrate on a complex Kraenkel-Manna-Merle system in a ferrite, under some coefficient constraints. An N-fold Darboux transformation of that system is presented via an existing Lax pair, where N is a positive integer. An N-fold generalized Darboux transformation, which admits one spectral parameter, is proposed through a limit procedure. One-, two- and three-soliton solutions of that system are determined via that N-fold Darboux transformation. The second-order and third-order degenerate soliton solutions of that system are derived via that N-fold generalized Darboux transformation. Those solitons are graphically represented for the magnetization and external magnetic field related to a ferrite.

Degenerate and non-degenerate vector solitons and their interactions in the two-component long-wave–short-wave model of Newell type

The focus of this paper is on the dynamics of degenerate and non-degenerate vector solitons and their collision scenarios in the two-component long-wave–short-wave (2-LS) model of Newell type. The model equation that we study here governs the resonance interactions between two capillary waves (short waves) with a common gravity wave (long wave) for the surface wave of deep water. These degenerate and non-degenerate soliton solutions of the 2-LS model of Newell type are constructed via the bilinear KP-hierarchy reduction method and are expressed in forms of determinants. The degenerate solitons having only single-hump or single-valley profiles can undergo both elastic and inelastic collisions. Two particular types of degenerate solitons are also obtained, i.e., bound state solitons and V/Y-shaped solitons. The non-degenerate solitons and their collision scenarios have more interesting properties in contrast with the degenerate ones. The obvious one is that the non-degenerate one solitons possess symmetric or asymmetric double-hump and double-valley profiles if they propagate with identical velocities in all short-wave (SW) and long-wave (LW) components. Additionally, the soliton wavenumbers in the SW components and the LW component are unequal if the ones in the two SW components propagate with different velocities. The non-degenerate two solitons can undergo two different types of elastic collision scenarios in both SW and LW components: shape-preserving and shape-changing scenarios. We also study the collisions of non-degenerate solitons with degenerate ones, and find some intriguing properties. For instance, if the degenerate one soliton in one of the two SW components vanishes into background, then there is only one fundamental non-degenerate soliton in the corresponding SW component. However, this fundamental non-degenerate soliton still alters its profile from a symmetric double-hump waveform to an asymmetric double-hump one during its propagation process without interacting with other solitons. Furthermore, it is found a new class of soliton solutions referred to as (1,2,3) solitons. In these new types of soliton solutions, one SW component has only a single soliton, the other SW component has two solitons, whereas the LW component has three solitons.

Generalized Darboux transformation and solitons for the Ablowitz–Ladik equation in an electrical lattice

In this paper, the Ablowitz–Ladik equation, which describes an electrical lattice employing the inductors and nonlinear capacitors in a transmission line, is investigated. With respect to the complex field amplitude of the electrical lattice, we construct a generalized Darboux transformation in which the multiple spectral parameters are involved. Expressions of the one-soliton solutions are derived. Soliton velocities, amplitudes and widths are presented. Then, solitons, degenerate solitons and interaction among the soliton and degenerate solitons are investigated. We find that when the multi solitons interact with each other/one another, the corrugated regions are generated in the interaction areas. Interactions among the one soliton and degenerate solitons are shown to be elastic.

Dynamics of degenerate and nondegenerate solitons in the two-component nonlinear Schrödinger equations coupled to Boussinesq equation

The paper studies the dynamics of degenerate and nondegenerate bright solitons and their collisions in the two-component nonlinear Schrödinger equations coupled to Boussinesq equation. The degenerate solitons that have only single-hump profiles, exhibit both elastic and inelastic collisions, and their velocities are identical in the two short wave (SW) components and in the long wave (LW) component. The nondegenerate single solitons have two different forms: one of them has double- or single-hump profiles and identical velocities in all SW and LW components, and the other one only admits single-hump profiles and has unequal velocities in the two SW components. The collisions of nondegenerate solitons cannot result in the redistribution of soliton intensities. Three different types of collisions of nondegenerate two-soliton solutions are studied in detail.