Stability and dynamics of two-soliton molecules

Temporal and spatiotemporal soliton molecules in ultrafast fibre lasers

Ultrafast fibre lasers, characterized by ultrashort pulse duration and broad spectral bandwidth, have drawn significant attention due to their vast potential across a wide range of applications, from fundamental scientific to industrial processing and beyond. As dissipative nonlinear systems, ultrafast fibre lasers not only generate single solitons, but also exhibit various forms of spatiotemporal soliton bunching. Analogous to molecules composed of multiple atoms in chemistry, soliton molecules (SMs) - alias bound states - in ultrafast fibre lasers are a key concept for gaining a deeper understanding of nonlinear interaction and hold a promise for advancing high-capacity fibre-optic communications. SMs are particularly notable for their high degree of controllability, including their internal temporal separation, and relative phase differences, thereby suggesting new possibilities for manipulating multi-pulse systems. In this review, we provide a comprehensive overview of recent advancements in the studies of SMs with the multidimensional parameter space in ultrafast fibre lasers. Owing to the flexibility afforded by mode-locking techniques and dispersion management, various types of SMs - with diverse values of the soliton number, relative phase, pulse separation, carrier frequencies, and even modal dispersion - have been experimentally demonstrated. We also discuss other basic nonlinear optical phenomena observed in fibre lasers, including the formation, spatiotemporal pulsations, and interaction dynamics of SMs. Furthermore, we explore the multidimensional control of SMs through approaches such as gain modulation, polarization control, dispersion management, and photomechanical effects, along with their applications to optical data encoding. Finally, we discuss challenges and future development of multidimensional technologies for the manipulation of SMs.

Investigations on diverse dynamics of soliton triplets in mode-locked fiber lasers

Optical soliton molecules exhibiting behaviors analogous to matter molecules have been the hotspot in the dissipative system for decades. Based on the dispersion Fourier transformation technique, the real-time spectral interferometry has become the popular method to reveal the internal dynamics of soliton molecules. The rising degrees of freedom in pace with the increased constitutes of soliton molecules yield more intriguing sights into the internal motions. Yet the soliton molecules with three or more pulses are rarely investigated owing to the exponentially growing complexity. Here, we present both experimental and theoretical studies on the soliton molecules containing three solitons. Different assemblies of the constitutes are categorized as different types of soliton triplet akin to the geometric isomer, including equally-spaced triplet and unequally-spaced triplet. Typical soliton triplets with different dynamics including regular internal motions, hybrid phase dynamics and complex dynamics involving separation evolution are experimentally analyzed and theoretically simulated. Specifically, the energy difference which remains elusive in experiments are uncovered through the simulation of diverse triplets with plentiful dynamics. Moreover, the multi-dimensional interaction space is proposed to visualize the internal motions in connection with the energy exchange, which play significant roles in the interplays among the solitons. Both the experimental and numerical simulations on the isomeric soliton triplets would release a larger number of degrees of freedom and motivate the potentially artificial configuration of soliton molecules for various ultrafast applications, such as all-optical buffering and multiple encoding for telecommunications.

Soliton molecules and their scattering by a localized P T-symmetric potential in atomic gases

We propose a physical scheme to study the formation of optical soliton molecules (SMs), consisting of two solitons bound together with a π-phase difference, and the scattering of SMs by a localized parity-time (P T)-symmetric potential. In order to stabilize SMs, we apply an additional space-dependent magnetic field to introduce a harmonic trapping potential for the two solitons and balance the repulse interaction induced by the π-phase difference between them. On the other hand, a localized complex optical potential obeying P T symmetry can be created through an incoherent pumping and spatial modulation of the control laser field. We investigate the scattering of optical SMs by the localized P T-symmetric potential, which exhibits evident asymmetric behavior and can be actively controlled by changing the incident velocity of SMs. Moreover, the P T symmetry of the localized potential, together with the interaction between two solitons of the SM, can also have a significant effect on the SM scattering behavior. The results presented here may be useful for understanding the unique properties of SMs and have potential applications in optical information processing and transmission.

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.

Singular soliton molecules of the nonlinear Schrödinger equation

We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that this solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthermore, we obtain an exact solution to the nonlocal NLSE using the same method and seed solution. The solution in this case corresponds to an elastic collision of a soliton, a breather soliton on flat background, and a breather soliton on a background with linear ramp. Finally, we consider an NLSE which is nonlocal in time rather than space. Although we did not find a Lax pair to this equation, we derive three exact solutions.

Solitons beyond binary: possibility of fibre-optic transmission of two bits per clock period

Optical telecommunication employs light pulses travelling down optical fibres; in a binary format logical Ones and Zeroes are represented by the presence or absence of a light pulse in a given time slot, respectively. The fibre's data-carrying capacity must keep up with increasing demand, but for binary coding it now approaches its limit. Alternative coding schemes beyond binary are currently hotly debated; the challenge is to mitigate detrimental effects from the fibre's nonlinearity. Here we provide proof-of-principle that coding with solitons and soliton molecules allows to encode two bits of data per clock period. Solitons do not suffer from nonlinearity, rather, they rely on it; this endows them with greater robustness. However, they are universally considered to be restricted to binary coding. With that notion now refuted, it is warranted to rethink future systems.

Binding energy of soliton molecules in time-dependent harmonic potential and nonlinear interaction

We calculate the binding energy of soliton molecules of an integrable nonlinear Schro[over ̈]dinger equation with time-dependent harmonic potential and cubic nonlinearity. Through a scaling transformation, an exact formula for the binding energy can be derived from that of the free soliton molecules in a homogeneous background. In the special case of oscillatory time dependence, sharp resonances occur at some integer and fractional multiples of the natural frequency of the molecule. Enhanced binding is obtained at these resonances and over some finite continuous range of low frequencies.

Interaction forces among two-dimensional bright solitons and many-soliton molecules

We consider two-dimensional bright matter-wave solitons in two-dimensional Bose-Einstein condensates. From the asymptotic form of their wave function, we derive an analytic expression for the force of interaction between solitons in the large separation limit, which turns out to decay with solitons separation Δ as F(Δ)∝exp(-Δ)/√Δ. Simulating the dynamics of two solitons using the relevant Gross-Pitaevskii equation, we obtain the force of the interaction for the full range of Δ, which turns out to be of molecular type. We show that many-soliton molecules can exist as a result of such a molecular-type of interaction. These include string-shaped, ring-shaped, or regular-lattice-shaped soliton molecules. By calculating their binding energy, we investigate the stability of these structures. Contrary to one-dimensional soliton molecules, which have no binding energy, two-dimensional molecules of a lattice of solitons with alternating phases are robust and have a negative binding energy. Lattices of size larger than 2 × 2 solitons have many discrete equilibrium values of the separation between two neighboring solitons.

Scattering of a matter-wave single soliton and a two-soliton molecule by an attractive potential

Scattering of a matter-wave single soliton and two-soliton molecule incident on the modified Pöschl-Teller potential well has been studied by means of a collective coordinate approach and numerical simulations of the Gross-Pitaevskii equation. Despite the attractive nature of the potential we observe total reflection of solitons in particular ranges of parameters, which is the signature of quantum behavior displayed by the matter-wave soliton. For other particular sets of parameters unscathed transmission of solitons and molecules through the potential well has been identified. A specific feature of this process is that the soliton passing through the potential well overtakes the freely propagating counterpart; i.e., its mean position appears to have been advanced in time. An array of such potentials makes the "time advance" effect even more pronounced, so that scattered solitons move well ahead of nonscattered ones, fully preserving their initial shape and velocity. A possible application of the obtained results is pointed out.