Nondegenerate Solitons in Manakov System

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.

Deformation of optical solitons in a variable-coefficient nonlinear Schrödinger equation with three distinct PT-symmetric potentials and modulated nonlinearities

We investigate deformed/controllable characteristics of solitons in inhomogeneous parity-time (PT)-symmetric optical media. To explore this, we consider a variable-coefficient nonlinear Schrödinger equation involving modulated dispersion, nonlinearity, and tapering effect with PT-symmetric potential, which governs the dynamics of optical pulse/beam propagation in longitudinally inhomogeneous media. By incorporating three physically interesting and recently identified forms of PT-symmetric potentials, namely, rational, Jacobian periodic, and harmonic-Gaussian potentials, we construct explicit soliton solutions through similarity transformation. Importantly, we investigate the manipulation dynamics of such optical solitons due to diverse inhomogeneities in the medium by implementing step-like, periodic, and localized barrier/well-type nonlinearity modulations and revealing the underlying phenomena. Also, we corroborate the analytical results with direct numerical simulations. Our theoretical exploration will provide further impetus in engineering optical solitons and their experimental realization in nonlinear optics and other inhomogeneous physical systems.

Fundamental and second-order dark soliton solutions of two- and three-component Manakov equations in the defocusing regime

We present exact multiparameter families of soliton solutions for two- and three-component Manakov equations in the defocusing regime. Existence diagrams for such solutions in the space of parameters are presented. Fundamental soliton solutions exist only in finite areas on the plane of parameters. Within these areas, the solutions demonstrate rich spatiotemporal dynamics. The complexity increases in the case of three-component solutions. The fundamental solutions are dark solitons with complex oscillating patterns in the individual wave components. At the boundaries of existence, the solutions are transformed into plain (nonoscillating) vector dark solitons. The superposition of two dark solitons in the solution adds more frequencies in the patterns of oscillating dynamics. These solutions admit degeneracy when the eigenvalues of fundamental solitons in the superposition coincide.

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.

Interaction dynamics of optical dark bound solitons for a defocusing Lakshmanan-Porsezian-Daniel equation

We investigate the propagation and interaction dynamics of the optical dark bound solitons for the defocusing Lakshmanan-Porsezian-Daniel equation, which is a physically relevant generalization of the nonlinear Schrödinger equation involving the higher-order effects. Explicit N-dark soliton solutions in the compact determinant form are constructed via the binary Darboux transformation method. Bound states of the dark solitons are discussed when the incoherent solitons have the same velocity. We find an interesting phenomenon that dark soliton molecules and double-valley dark solitons (DVDSs) can be obtained by controlling the interval of the bound state dark solitons, and abundant interaction modalities between them can be formed. Moreover, dark soliton molecules always undergo elastic interactions with other solitons, while interactions for the DVDSs are usually inelastic, and special parameter conditions for elastic interaction of DVDSs through asymptotic analysis are obtained. Numerical simulations are employed to verify the stability of the bound state dark solitons. Analytical results obtained in this paper are expected to be useful for the experimental realization of bound-state dark solitons in optical fibers with higher-order effects and a further understanding of their optical transmission properties..

Phase characters of optical dark solitons with third-order dispersion and delayed nonlinear response

Dark soliton is usually seen as one of the simplest topological solitons, due to phase shift across its intensity dip. We investigate phase characters of single-valley dark soliton (SVDS) and double-valley dark soliton (DVDS) in a single-mode optical fiber with third-order dispersion and delayed nonlinear response. Notably, two different phase shifts can produce an SVDS with the same velocity under some conditions, which is not admitted for a dark soliton with only the second-order dispersion and self-phase modulation, whose phase shift and velocity is a one-to-one match. This phase property of SVDS can be used to explain the generation of previously reported DVDS in Hirota equation and make DVDSs show two types of phase profiles. Moreover, the different topological vector potentials underlying the distinct phase profiles have been uncovered. We further explore the collision properties of the DVDSs by analyzing their topological phases. Strikingly, the inelastic collision can lead to the conversion between the two types of phase profiles for DVDS. The results reveal that inelastic or elastic collision can be judged by analyzing virtual topological magnetic fields.

Dynamics of nondegenerate vector solitons in a long-wave-short-wave resonance interaction system

In this paper, we study the dynamics of an interesting class of vector solitons in the long-wave-short-wave resonance interaction (LSRI) system. The model that we consider here describes the nonlinear interaction of long wave and two short waves and it generically appears in several physical settings. To derive this class of nondegenerate vector soliton solutions we adopt the Hirota bilinear method with the more general form of admissible seed solutions with nonidentical distinct propagation constants. We express the resultant fundamental as well as multisoliton solutions in a compact way using Gram-determinants. The general fundamental vector soliton solution possesses several interesting properties. For instance, the double-hump or a single-hump profile structure including a special flattop profile form results in when the soliton propagates in all the components with identical velocities. Interestingly, in the case of nonidentical velocities, the soliton number is increased to two in the long-wave component, while a single-humped soliton propagates in the two short-wave components. We establish through a detailed analysis that the nondegenerate multisolitons in contrast to the already known vector solitons (with identical wave numbers) can undergo three types of elastic collision scenarios: (i) shape-preserving, (ii) shape-altering, and (iii) a shape-changing collision, depending on the choice of the soliton parameters. Here, by shape-altering we mean that the structure of the nondegenerate soliton gets modified slightly during the collision process, whereas if the changes occur appreciably then we call such a collision as shape-changing collision. We distinguish each of the collision scenarios, by deriving a zero phase shift criterion with the help of phase constants. Very importantly, the shape-changing behavior of the nondegenerate vector solitons is observed in the long-wave mode also, along with corresponding changes in the short-wave modes, and this nonlinear phenomenon has not been observed in the already known vector solitons. In addition, we point out the coexistence of nondegenerate and degenerate solitons simultaneously along with the associated physical consequences. We also indicate the physical realizations of these general vector solitons in nonlinear optics, hydrodynamics, and Bose-Einstein condensates. Our results are generic and they will be useful in these physical systems and other closely related systems including plasma physics when the long-wave-short-wave resonance interaction is taken into account.

Rogue and solitary waves in coupled phononic crystals

In this work we present an analytical and numerical study of rogue and solitary waves in a coupled one-dimensional nonlinear lattice that involves both axial and rotational degrees of freedom. Using a multiple-scale analysis, we derive a system of coupled nonlinear Schrödinger-type equations in order to approximate solitary waves and rogue waves of the coupled lattice model. Numerical simulations are found to agree with the analytical approximations. We also consider generic initialization data in the form of a Gaussian profile and observe that they can result in the spontaneous formation of rogue-wave-like patterns in the lattice. The solitary and rogue waves in the lattice demonstrate both energy isolation and exchange between the axial and rotational degrees of freedom of the system. This suggests that the studied coupled lattice has the potential to be an efficient energy isolation, transfer, and focusing medium.

Extreme spectral asymmetry of Akhmediev breathers and Fermi-Pasta-Ulam recurrence in a Manakov system

The dynamics of Fermi-Pasta-Ulam (FPU) recurrence in a Manakov system is studied analytically. Exact Akhmediev breather (AB) solutions for this system are found that cannot be reduced to the ABs of a single-component nonlinear Schrödinger equation. Expansion-contraction cycle of the corresponding spectra with an infinite number of sidebands is calculated analytically using a residue theorem. A distinctive feature of these spectra is the asymmetry between positive and negative spectral modes. A practically important consequence of the spectral asymmetry is a nearly complete energy transfer from the central mode to one of the lowest-order (left or right) sidebands. Numerical simulations started with modulation instability of plane waves confirm the findings based on the exact solutions. It is also shown that the full growth-decay cycle of the AB leads to the nonlinear phase shift between the initial and final states in both components of the Manakov system. This finding shows that the final state of the FPU recurrence described by the vector ABs is not quite the same as the initial state. Our results are applicable and can be observed in a wide range of two-component physical systems such as two-component waves in optical fibers, two-directional waves in crossing seas, and two-component Bose-Einstein condensates.

Nondegenerate Bright Solitons in Coupled Nonlinear Schrödinger Systems: Recent Developments on Optical Vector Solitons

Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical vector solitons has been developed considerably both from theoretical and experimental points of view leading to soliton-based promising potential applications. Mathematically, the dynamics of vector solitons can be understood from the framework of the coupled nonlinear Schrödinger (CNLS) family of equations. In the recent past, many types of vector solitons have been identified both in the integrable and non-integrable CNLS framework. In this article, we review some of the recent progress in understanding the dynamics of the so called nondegenerate vector bright solitons in nonlinear optics, where the fundamental soliton can have more than one propagation constant. We address this theme by considering the integrable two coupled nonlinear Schrödinger family of equations, namely the Manakov system, mixed 2-CNLS system (or focusing-defocusing CNLS system), coherently coupled nonlinear Schrödinger (CCNLS) system, generalized coupled nonlinear Schrödinger (GCNLS) system and two-component long-wave short-wave resonance interaction (LSRI) system. In these models, we discuss the existence of nondegenerate vector solitons and their associated novel multi-hump geometrical profile nature by deriving their analytical forms through the Hirota bilinear method. Then we reveal the novel collision properties of the nondegenerate solitons in the Manakov system as an example. The asymptotic analysis shows that the nondegenerate solitons, in general, undergo three types of elastic collisions without any energy redistribution among the modes. Furthermore, we show that the energy sharing collision exhibiting vector solitons arises as a special case of the newly reported nondegenerate vector solitons. Finally, we point out the possible further developments in this subject and potential applications.

Multivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactions

We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicomponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. We demonstrate that the width-dependent parameters of solitons significantly affect the velocity ranges and phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. For double-valley dark solitons, we find that the phase jump is in the range [0,2π], which is quite different from that of the usual single-valley dark soliton. Based on our results, we argue that the phase jump of an n-valley dark soliton could be in the range [0,nπ], supported by our analysis extending up to five-component condensates. The interaction between a double-valley dark soliton and a single-valley dark soliton is further investigated, and we reveal a striking collision process in which the double-valley dark soliton is transformed into a breather after colliding with the single-valley dark soliton. Our analyses suggest that this breather transition exists widely in the collision processes involving multivalley dark solitons. The possibilities for observing these multivalley dark solitons in related Bose-Einstein condensates experiments are discussed.

Nondegenerate solitons and their collisions in Manakov systems

Recently, we have shown that the Manakov equation can admit a more general class of nondegenerate vector solitons, which can undergo collision without any intensity redistribution in general among the modes, associated with distinct wave numbers, besides the already-known energy exchanging solitons corresponding to identical wave numbers. In the present comprehensive paper, we discuss in detail the various special features of the reported nondegenerate vector solitons. To bring out these details, we derive the exact forms of such vector one-, two-, and three-soliton solutions through Hirota bilinear method and they are rewritten in more compact forms using Gram determinants. The presence of distinct wave numbers allows the nondegenerate fundamental soliton to admit various profiles such as double-hump, flat-top, and single-hump structures. We explain the formation of double-hump structure in the fundamental soliton when the relative velocity of the two modes tends to zero. More critical analysis shows that the nondegenerate fundamental solitons can undergo shape-preserving as well as shape-altering collisions under appropriate conditions. The shape-changing collision occurs between the modes of nondegenerate solitons when the parameters are fixed suitably. Then we observe the coexistence of degenerate and nondegenerate solitons when the wave numbers are restricted appropriately in the obtained two-soliton solution. In such a situation we find the degenerate soliton induces shape-changing behavior of nondegenerate soliton during the collision process. By performing suitable asymptotic analysis we analyze the consequences that occur in each of the collision scenario. Finally, we point out that the previously known class of energy-exchanging vector bright solitons, with identical wave numbers, turns out to be a special case of nondegenerate solitons.

Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates

We investigate nondegenerate bound-state solitons systematically in multicomponent Bose-Einstein condensates, through developing the Darboux transformation method to derive exact soliton solutions analytically. In particular, we show that bright solitons with nodes correspond to the excited bound states in effective quantum wells, in sharp contrast to the bright solitons and dark solitons reported before (which usually correspond to ground state and free state, respectively). We further demonstrate that bound-state solitons with nodes are induced by incoherent superposition of solitons in different components. Moreover, we reveal that the interactions between these bound-state solitons are usually inelastic, caused by the incoherent interactions between solitons in different components and the coherent interactions between solitons in the same component. Additionally, the detailed spectral stability analysis demonstrates the stability of nondegenerate bound-state solitons. The bound-state solitons can be used to study many different physical problems, such as beating dynamics, spin-orbit coupling effects, quantum fluctuations, and even quantum entanglement states.