Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates

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.

Various solitons induced by relative phase in the nonlinear Schrödinger Maxwell-Bloch system

We study the effect of relative phase on the characteristics of rogue waves and solitons described by rational solutions in the nonlinear Schrödinger Maxwell-Bloch system. We derived the rational rogue wave and soliton solutions with adjustable relative phase and present the parameter range of different types of rogue waves and solitons. Our findings show that the relative phase can alter the distribution of rational solitons and even change the type of rational solitons, leading to a rich array of rational soliton types by adjusting the relative phase. However, the relative phase does not affect the structure of the rogue wave, because the relative phase of the rogue wave changes during evolution. In particular, we confirm that the rational solitons with varying relative phases and the rogue waves at corresponding different evolution positions share the same distribution mode. This relationship holds true for rogue waves or breathers and their stable counterparts solitons or periodic waves in different nonlinear systems. The implications of our study are significant for exploring fundamental excitation elements in nonlinear systems.

Simulation of universal optical logic gates under energy sharing collisions of Manakov solitons and fulfillment of practical optical logic criteria

The universal optical logic gates, namely, nand and nor gates, have been theoretically simulated by employing the energy sharing collision of bright optical solitons in the Manakov system, governing pulse propagation in a highly birefringent fiber. Further, we also realize the two-input optical logic gates, such as and, or, xor, xnor, for completeness of our scheme. Interestingly, our idea behind the simulation naturally satisfies all the criteria for practical optical logic, which in turn displays the strength and versatility of our theoretical simulation of universal optical logic gates. Hence, our approach paves the way for the experimentalists to create a new avenue in this direction if the energy sharing collisions of Manakov solitons are experimentally realized in the future.

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.

Excitation of mirror symmetry higher-order rational soliton in modulation stability regimes on continuous wave background

We study the relationship between the structures of the nonlinear localized waves and the distribution characteristics of the modulation stability regime in a nonlinear fiber with both third-order and fourth-order effects. On the background frequency and background amplitude plane, the modulation stability region consists of two symmetric curves on the left and right and a point on the symmetry axis. We find that the higher-order excitation characteristics are obviously different at different positions in the modulation stability region. Their excitation characteristics are closely related to the modulation instability distribution characteristics of the system. It is shown that asymmetric high-order rational solitons are excited at the left and right stable curves, and the symmetric one is excited at the stable points. Interestingly, the asymmetric higher-order rational solitons on the left and right sides are mirror-symmetrical to each other, which coincides with the symmetry of the modulation instability distribution. These results can deepen our understanding of the relationship between nonlinear excitation and modulation instability and enrich our knowledge about higher-order nonlinear excitations.

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.

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.

Classification of dark solitons via topological vector potentials

Dark solitons are some of the most interesting nonlinear excitations and are considered to be the one-dimensional topological analogs of vortices. However, in contrast to their two-dimensional vortex counterparts, the topological characteristics of a dark soliton are far from fully understood because the topological charge defined according to the phase jump cannot reflect its essential property. Here, similar to the complex extension used in the exploration of the partition-function zeros to depict thermodynamic states, we extend the complex coordinate space to explore the density zeros of dark solitons. Surprisingly we find that these zeros constitute some pointlike magnetic fields, each of which has a quantized magnetic flux of elementary π. The corresponding vector potential fields demonstrate the topology of the Wess-Zumino term and can depict the essential characteristics of dark solitons. Then we classify the dark solitons according to the Euler characteristic of the topological manifold of the vector potential fields. Our study not only reveals the topological features of dark solitons but can also be applied to explore and identify new dark solitons with high topological complexity.

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.