Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation

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.

Prediction of phase transition and time-varying dynamics of the (2+1)-dimensional Boussinesq equation by parameter-integrated physics-informed neural networks with phase domain decomposition

A meaningful topic that needs to be explored in the field of nonlinear waves is whether a neural network can reveal the phase transition of different types of waves and novel dynamical properties. In this paper, a physics-informed neural network (PINN) with parameters is used to explore the phase transition and time-varying dynamics of nonlinear waves of the (2+1)-dimensional Boussinesq equation describing the propagation of gravity waves on the surface of water. We embed the physical parameters into the neural network for this purpose. Via such algorithm, we find the exact boundary of the phase transition that distinguishes the periodic lump chain and transformed wave, and the inexact boundaries of the phase transition for various transformed waves are detected through PINNs with phase domain decomposition. In particular, based only on the simple soliton solution, we discover types of nonlinear waves as well as their interesting time-varying properties for the (2+1)-dimensional Boussinesq equation. We further investigate the stability by adding noise to the initial data. Finally, we perform the parameters discovery of the equation in the case of data with and without noise, respectively. Our paper introduces deep learning into the study of the phase transition of nonlinear waves and paves the way for intelligent explorations of the unknown properties of waves by means of the PINN technique with a simple solution and small data set.

Formation of Rogue Waves and Modulational Instability with Zero-Wavenumber Gain in Multicomponent Systems with Coherent Coupling

It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Starting with the Bers-Kaup-Reiman system for three-wave resonant interactions, we identify a specific RW-building mechanism based on MI which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by MI in other multicomponent systems-in particular, in the massive Thirring model.

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.

Wave-packet behaviors of the defocusing nonlinear Schrödinger equation based on the modified physics-informed neural networks

Investigated in this paper is the defocusing nonlinear Schrödinger (NLS) equation, which is used for describing the wave-packet dynamics in certain weakly nonlinear media. With the physics-informed neural networks (PINNs), we modify the corresponding loss function in the existing literature and obtain two types of dark solitons, type-I and type-II solitons. It is demonstrated that the modified loss function presents higher-precision wave-packet behaviors based on fewer initial and boundary data. Taking type-I solitons into consideration, we find that when only a small fraction of initial and boundary data are given, the prediction accuracy of the wave packets will be increased one or two orders of magnitude at least if the modification term of the loss function is introduced. Furthermore, for the inverse problem, the modified loss function provides a better estimate of the nonlinear coefficient of the NLS equation based on fewer observed data of the wave packets. For type-II solitons, we compare the required data and predicted results of the PINNs with those of the conventional time-splitting finite difference (TSFD) method and reveal that achieving the same precision of the wave-packet behavior, the PINNs with the modified loss functions require only one tenth of the amount of the initial and boundary data of the TSFD method. Besides, both unmodified and modified loss functions are exploited for predicting the behaviors of Gaussian wave packets, and it is observed that the predicted result of the modified loss function agrees with the high-precision solution of the time-splitting Fourier pseudospectral method, whereas the unmodified loss function fails.

High-dimensional nonlinear wave transitions and their mechanisms

In this paper, the dynamics of transformed nonlinear waves in the (2+1)-dimensional Ito equation are studied by virtue of the analysis of characteristic line and phase shift. First, the N-soliton solution is obtained via the Hirota bilinear method, from which the breath-wave solution is derived by changing values of wave numbers into complex forms. Then, the transition condition for the breath waves is obtained analytically. We show that the breath waves can be transformed into various nonlinear wave structures including the multi-peak soliton, M-shaped soliton, quasi-anti-dark soliton, three types of quasi-periodic waves, and W-shaped soliton. The correspondence of the phase diagram for such nonlinear waves on the wave number plane is presented. The gradient property of the transformed solution is discussed through the wave number ratio. We study the mechanism of wave formation by analyzing the nonlinear superposition between a solitary wave component and a periodic wave component with different phases. The locality and oscillation of transformed waves can also be explained by the superposition mechanism. Furthermore, the time-varying characteristics of high-dimensional transformed waves are investigated by analyzing the geometric properties (angle and distance) of two characteristic lines of waves, which do not exist in (1+1)-dimensional systems. Based on the high-order breath-wave solutions, the interactions between those transformed nonlinear waves are investigated, such as the completely elastic mode, semi-elastic mode, inelastic mode, and collision-free mode. We reveal that the diversity of transformed waves, time-varying property, and shape-changed collision mainly appear as a result of the difference of phase shifts of the solitary wave and periodic wave components. Such phase shifts come from the time evolution as well as the collisions. Finally, the dynamics of the double shape-changed collisions are presented.

Solitary waves, breathers, and rogue waves modulated by long waves for a model of a baroclinic shear flow

Investigated in this paper is a quasigeostrophic two-layer model for the wave packets in a marginally stable or unstable baroclinic shear flow. We find that the wave packets can be modulated by certain long waves, resulting in different behaviors from those in the existing literature. Via the bilinear method, we construct the modulated Nth-order (N=1,2,...) solitary waves, breathers, and rogue waves for the wave-packet equations. Based on the modulation effects of the long waves, the solitary waves are classified into three types, i.e., Type-I, Type-II, and Type-III solitary waves. Type-I solitary waves, without the modulations, are the bell shaped and propagate with constant velocities; Type-II solitary waves, with the weak modulations, are shape changing within a short time and subsequently return to the bell-shaped state; and Type-III solitary waves, with the strong modulations, show not only the variations of shapes but also the appearances, splits, combinations, and disappearances of certain bulges in the evolution. For the interaction between the two unmodulated solitary waves, two Type-I solitary waves can bring about the oscillations in the interaction zone when they possess different velocities, and bring into being the bound-state, oscillation-state, and bi-oscillation-state solitary waves when they possess the same velocity. For the two interactive modulated solitary waves, bound-state, oscillation-state, and bi-oscillation-state solitary waves with the short-time variations of shapes or appearances of bulges can occur. Due to the modulations of the long waves, breathers and rogue waves are distorted and stretched, mainly in two aspects: one is the evolution trajectories for the breathers; the other is the shape variations for each element of the breathers and rogue waves. Numbers of the peaks and valleys for the rogue waves are adjustable via the modulations. In addition, modulated breathers and rogue waves can degenerate into the M- or W-shaped or multipeak solitary waves under certain conditions.

Frequency-doubling effect in acoustic reflection by a nonlinear, architected rotating-square metasurface

Nonlinear acoustic metamaterials offer the potential to enhance wave control opportunities beyond those already demonstrated via dispersion engineering in linear metamaterials. Managing the nonlinearities of a dynamic elastic system, however, remains a challenge, and the need now exists for new strategies to model and design these wave nonlinearities. Inspired by recent research on soft architected rotating-square structures, we propose herein a design for a nonlinear elastic metasurface with the capability to achieve nonlinear acoustic wave reflection control. The designed metasurface is composed of a single layer of rotating squares connected to thin and highly deformable ligaments placed between a rigid plate and a wall. It is shown that during the process of reflection at normal incidence, most of the incoming fundamental wave energy can be converted into the second harmonic wave. A conversion coefficient of approximately 0.8 towards the second harmonic is derived with a reflection coefficient of <0.05 at the incoming fundamental frequency. The theoretical results obtained using the harmonic balance method for a monochromatic pump source are confirmed by time-domain simulations for wave packets. The reported design of a nonlinear acoustic metasurface can be extended to a large family of architected structures, thus opening new avenues for realistic metasurface designs that provide for nonlinear or amplitude-dependent wave tailoring.

High-power pulse, pulse pair, and pulse train generated by breathers in dispersion exponentially decreasing fiber

Based on the derived rational solutions of the nonautonomous nonlinear Schrödinger equation with varying coefficients, we present a simple scheme to generate a high-power pulse, pulse pair, and pulse train with non-oscillating amplitudes in dispersion exponentially decreasing fiber. Without requiring elimination of the background, the stable pulse train can be generated from the first-order Akhmediev breather, and the high-power pulse and pulse pair can be generated from the second-order Kuznetsov-Ma breather. Moreover, it is found that the characteristics of these pulses can be controlled by adjusting the eigenvalue parameter and fiber parameters. The results presented here are expected to be useful in large-capacity and high-power optical communication systems.

Excitation conditions of several fundamental nonlinear waves on continuous-wave background

We study the excitation conditions of antidark solitons and nonrational W-shaped solitons in a nonlinear fiber with both third-order and fourth-order effects. We show that the relative phase can be used to distinguish antidark solitons and nonrational W-shaped solitons. The excitation conditions of these well-known fundamental nonlinear waves (on a continuous-wave background) can be clarified clearly by the relative phase and three previously reported parameters (background frequency, perturbation frequency, and perturbation energy). Moreover, the numerical simulations from the nonideal initial states also support these theoretical results. These results provide an important complement for the studies on relationship between modulation instability and nonlinear wave excitations, and are helpful for controllable nonlinear excitations in experiments.

Growth rate of modulation instability driven by superregular breathers

We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schrödinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.

Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits

We present one- and two-breather solutions of the fourth-order nonlinear Schrödinger equation. With several parameters to play with, the solution may take a variety of forms. We consider most of these cases including the general form and limiting cases when the modulation frequencies are 0 or coincide. The zero-frequency limit produces a combination of breather-soliton structures on a constant background. The case of equal modulation frequencies produces a degenerate solution that requires a special technique for deriving. A zero-frequency limit of this degenerate solution produces a rational second-order rogue wave solution with a stretching factor involved. Taking, in addition, the zero limit of the stretching factor transforms the second-order rogue waves into a soliton. Adding a differential shift in the degenerate solution results in structural changes in the wave profile. Moreover, the zero-frequency limit of the degenerate solution with differential shift results in a rogue wave triplet. The zero limit of the stretching factor in this solution, in turn, transforms the triplet into a singlet plus a low-amplitude soliton on the background. A large value of the differential shift parameter converts the triplet into a pure singlet.

Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates

To show the existence and properties of matter rogue waves in an F=1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.

Effect of high-order dispersion on three-soliton interactions for the variable-coefficients Hirota equation

The interactions of multiple solitons show different properties with two-soliton interactions. For the difficulty of deriving multiple soliton solutions, it is rare to study multiple soliton interactions analytically. In this paper, three-soliton interactions in inhomogeneous optical fibers, which are described by the variable coefficient Hirota equation, are investigated. Via the Hirota bilinear method and symbolic computation, analytic three-soliton solutions are obtained. According to the obtained solutions, properties and features of three-soliton interactions are discussed by changing the third-order dispersion (TOD) and other relevant coefficients, and some plentiful structure of three-soliton interactions are presented for the first time. The influences of TOD on the intensity and propagation distance of solitons are described, which can be used to realize the soliton control. Besides, the method that can achieve the phase reverse of solitons is suggested, and bound states of three solitons are observed, which have potential applications in the mode-locked fiber lasers. Furthermore, comparing to two-soliton interactions, a novel phenomenon of three-soliton interactions with a strong phase shift at x=0 is revealed, which is potentially useful for optical logic switches.

Breather-to-soliton transformation rules in the hierarchy of nonlinear Schrödinger equations

We study the exact first-order soliton and breather solutions of the integrable nonlinear Schrödinger equations hierarchy up to fifth order. We reveal the underlying physical mechanism which transforms a breather into a soliton. Furthermore, we show how the dynamics of the Akhmediev breathers which exist on a constant background as a result of modulation instability, is connected with solitons on a zero background. We also demonstrate that, while a first-order rogue wave can be directly transformed into a soliton, higher-order rogue wave solutions become rational two-soliton solutions with complex collisional structure on a background. Our results will have practical implications in supercontinuum generation, turbulence, and similar other complex nonlinear scenarios.

Soliton excitations on a continuous-wave background in the modulational instability regime with fourth-order effects

We study the correspondence between modulational instability and types of fundamental nonlinear excitation in a nonlinear fiber with both third-order and fourth-order effects. Some soliton excitations are obtained in the modulational instability regime which have not been found in nonlinear fibers with second-order effects and third-order effects. Explicit analysis suggests that the existence of solitons is related to the modulation stability circle in the modulation instability regime, and they just exist in the modulational instability regime outside of the modulational stability circle. It should be emphasized that the solitons exist only with two special profiles on a continuous-wave background at a certain frequency. The evolution stability of the solitons is tested numerically by adding some noise to initial states, which indicates that they are robust against perturbations even in the modulation instability regime. Further analysis indicates that solitons in the modulational instability regime are caused by fourth-order effects.

Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov's theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

Symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime

We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a single pulse formed by a nonlinear superposition of a periodic wave and a single-peak (W-shaped or antidark) soliton. In particular, a phase diagram for different types of nonlinear excitations on a continuous wave background, including the unusual multipeak soliton, the W-shaped soliton, the antidark soliton, the periodic wave, and the known breather rogue wave, is established based on the explicit link between exact solution and modulation instability analysis. Numerical simulations are performed to confirm the propagation stability of the multipeak solitons with symmetric and asymmetric structures. Further, we unveil a remarkable shape-changing feature of asymmetric multipeak solitons. It is interesting that these shape-changing interactions occur not only in the intraspecific collision (soliton mutual collision) but also in the interspecific interaction (soliton-breather interaction). Our results demonstrate that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.

Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.