Kuznetsov-Ma Breather Research Articles

New rational and breather solutions of a higher-order integrable nonlinear Schrödinger equation

In this letter, we investigate the general rational solutions and breather solutions of a higher-order integrable nonlinear Schrödinger (NLS) equation based on Darboux transformation (DT). The rational solutions including W-shape traveling wave solution that is not reported and rogue wave solution are constructed. Time-periodic Kuznetsov–Ma breather, space-periodic Akhmediev breather and time–space periodic breather solutions are obtained. Besides, the interaction properties of two breather solutions are also displayed through numerical simulation. The results exhibit the new dynamical properties in extended nonlinear integrable physical models.

Interaction region of magnon-mediated spin torques and novel magnetic states* *Project supported by the National Natural Science Foundation of China (Grant No. 61774001), the Natural Science Foundation of Hebei Province of China (Grant No. F2019202141), and the Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, China (Grant No. KF201906).

We determine the region in which the magnon-mediated spin torques exist. This region can be controlled by the spin waves. In terms of stability analysis of magnetization dynamics based on the spin-wave background, we obtain the instability conditions of spin waves. With these results, we find the relationship between unstable regions and the formation of Akhmediev breather, Kuznetsov–Ma breather and rogue waves. We establish the phase diagram of some novel magnetic excitaions.

Ion-acoustic solitary, breathers, and freak waves inadegenerate quantum plasma

In this investigation, the characteristics of ion acoustic (IA) solitary, breathers, and freak waves have been examined by using spin evolution quantum hydrodynamic model in a degenerate quantum plasma consisting of inertial non-degenerate ions and degenerate electrons with spin-up and spin-down states. The Korteweg–de Vries (KdV) equation is derived by employing reductive perturbation technique and its solution is determined to see the existence of IA solitary waves. Only positive potential solitary structures are formed. Further, using appropriate transformation, the KdV equation is transformed into nonlinear Schrödinger equation (NLSE) and its different order solutions are used to study the characteristics of breathers such as the Akhmediev breather, the Kuznetsov–Ma breather and freak waves. Further, the formation of super order of freak waves has also been explored. The influence of different plasma parameters including density polarization ratio has been investigated on the characteristics of IA solitary waves and different types of breathers. It is observed that the density polarization ratio has a strong influence on the characteristics of different types of nonlinear structures. The present investigation may be of great importance to provide physical insight to understand nonlinear dynamical processes in dense astrophysical regions such as white dwarfs and neutron stars.

Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation

We numerically realize a breather gas for the focusing nonlinear Schrödinger equation. This is done by building a random ensemble of N∼50 breathers via the Darboux transform recursive scheme in high-precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma, and Peregrine breathers as elementary quasiparticles of the respective gases. The interaction properties of the constructed breather gases are investigated by propagating through them a "trial" generic (Tajiri-Watanabe) breather and comparing the mean propagation velocity with the predictions of the recently developed spectral kinetic theory [El and Tovbis, Phys. Rev. E 101, 052207 (2020)2470-004510.1103/PhysRevE.101.052207].

Controllable generation of several nonlinear waves in optical fibers with third-order dispersion

We propose a method to controllably generate six kinds of nonlinear waves on continuous waves, including the one- and multi-peak solitons, the Akhmediev, Kuznetsov-Ma, and Taijiri-Watanabe breathers, and stable periodic waves. In the nonlinear fiber system with third-order dispersion, we illustrate their generation conditions by the modified linear stability analysis, and numerically generate them from initial perturbations on continuous waves. We implement the quantitative control over their dynamical features, including the wave type, velocity, periodicity, and localization. Our results may provide an effective scheme for generating optical solitons on continuous waves, and it can also be applied for wave generations in other various nonlinear systems.

Dispersion managed generation of Peregrine solitons and Kuznetsov-Ma breather in an optical fiber

Optical rogue waves and its variants have been studied quite extensively in the context of optical fiber in recent years. It has been realized that dispersion management in optical fiber is experimentally much more feasible compared to its nonlinear counterpart. In this work, we report Kuznetsov-Ma (KM)-like breathers from the first three orders of rational solutions of the nonlinear Schrödinger equation with periodic modulation of the dispersion coefficient along the fiber axis. The breather dynamics are then controlled by proper choice of modulating parameters. Additionally, the evolution of new one-peak and two-peak breather-like solutions has been displayed corresponding to the second-order rational solution. Direct numerical simulations based on modulational instability has also been executed which agree well with the analytical results, thereby making the proposed system more feasible for experimental realization.

Breather Structures and Peregrine Solitons in a Polarized Space Dusty Plasma

In this theoretical investigation, we have examined the combined effects of nonthermally revamped polarization force on modulational instability MI of dust acoustic waves DAWs and evolution of different kinds of dust acoustic (DA) breathers in a dusty plasma consisting of negatively charged dust as fluid, Maxwellian electrons, and ions obeying Cairns’ nonthermal distribution. The nonthermality of ions has considerably altered the strength of polarization force. By employing the multiple-scale perturbation technique, the nonlinear Schrödinger equation NLSE is derived to study modulational MI instability of dust acoustic waves DAWs. It is noticed that influence of the polarization force makes the wave number domain narrow where MI sets in. The rational solutions of nonlinear Schrödinger equation illustrate the evolution of DA breathers, namely, Akhmediev breather, Kuznetsov–Ma breather, and Peregrine solitons (rogue waves). Further, the formation of super rogue waves due to nonlinear superposition of DA triplets rogue waves is also discussed. It is analyzed that combined effects of variation in the polarization force and nonthermality of ions have a comprehensive influence on the evolution of different kinds of DA breathers. It is remarked that outcome of present theoretical investigation may provide physical insight into understanding the role of nonlinear phenomena for the generation of various types of DA breathers in experiments and different regions of space (e.g., the planetary spoke and cometary tails).

Modulation instability and rogue wave spectrum for the generalized nonlinear Schrödinger equation

We discuss modulation instability for the generalized nonlinear Schrödinger equation based on nonzero background wave frequency. First of all, we analyze the existence condition of modulation instability under different perturbed frequency. The influences of the background amplitude, background frequency and perturbed frequency on the modulation instability gain are researched, respectively. Also, we obtain the correspondences between several nonlinear excitations (Kuznetsov-Ma breather, general breather, rogue wave, bright soliton and plane wave) and modulation instability according to new parameters. Furthermore, by the Fourier transformation method, we perform spectrum analyses of the first-order and second-order rogue waves. The perturbed frequency of the rogue wave can affect the location and profile of the spectrum. And we find that the spectrum of the second-order rogue wave is jagged due to the collision of the rogue waves. These results would help us further understand the dynamics of rogue wave in complex systems.

The Hybrid Finite Difference and Moving Boundary Methods for Solving a Linear Damped Nonlinear Schrödinger Equation to Model Rogue Waves and Breathers in Plasma Physics

In this paper, the dissipative and nondissipative modulated pulses such as rogue waves (RWs) and breathers (Kuznetsov-Ma breathers and Akhmediev breathers) that can exist and propagate in several fields of sciences, for example, plasma physics, have been analyzed numerically. For this purpose, the fluid dusty plasma equations with taking the kinematic dust viscosity into account are reduced to the linear damped nonlinear Schrödinger equation using a reductive perturbation technique. It is known that this equation is not integrable and, accordingly, does not have analytical solution. Thus, for modelling both dissipative RWs and breathers, the improved finite difference method is introduced for this purpose. It is found that FDM is a good numerical technique for small time interval but for large time interval it becomes sometimes unacceptable. Therefore, to describe these waves accurately, the new improved numerical method is considered, which is called the hybrid finite difference method and moving boundary method (FDM-MBM). This last and updated method gives an accurate and excellent description to many physical results, as it was applied to the dust plasma results and the results were good.

Solitons and breathers for a generalized nonlinear Schrödinger equation via the binary Bell polynomials

In this paper, we investigate a generalized nonlinear Schrödinger equation with two certain terms in the single-mode optical fibers. Via the binary Bell polynomials, a more general bilinear form and analytic solutions are obtained. On the basis of those solutions, we present the parametric regions for the existence of the solitons and breathers on a nonzero background. From the soliton solutions, we prove the soliton interaction to be elastic through the asymptotic analysis. The interactions can be observed between (i) two dark solitons, (ii) two anti-dark solitons, and (iii) one dark and one anti-dark solitons, and relevant parametric conditions of the three types of interactions are given. From the breather solutions, the general breathers, Akhmediev breathers and Kuznetsov–Ma breathers can be respectively derived with the different parametric conditions. Besides, we obtain some rational solutions by taking the limit of the breather solutions. Based on those rational solutions, the parametric conditions for the soliton interactions and rogue waves are given.

Breather-induced quantised superfluid vortex filaments and their characterisation

We study and characterise the breather-induced quantised superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations. Such vortex filaments, emerging from an otherwise perturbed helical vortex, exhibit intriguing loop structures corresponding to the large amplitude of breathers due to the dual action of bending and twisting of the vortex. The loop induced by the Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by the super-regular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of the super-regular breather. In particular, we identify explicitly the generation conditions of these loop excitations by introducing a physical quantity—the integral of the relative quadratic curvature—which corresponds to the effective energy of breathers. Despite the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size. These results will deepen our understanding of breather-induced vortex filaments and be helpful for controllable ring-like excitations on the vortices.

Breather phase transitions for a three-component transient stimulated Raman scattering system in nonlinear optics

A three-component transient stimulated Raman scattering system in nonlinear optics is investigated. By applying the Lax-Darboux scheme, analytical breather solutions are constructed. A detailed investigation was performed for the breathers of the off-diagonal density-matrix element, Stokes wave and pump wave. The system possesses both general and Kuznetsov-Ma breathers. Furthermore, an exact link is established between the phase states and spectrum parameter. Via adjusting the spectrum parameter, the breathers exhibit novel phase transitions. The amplitudes and densities of the wave envelopes also vary at the different phase states.

Phase transitions of breather of a nonlinear Schrödinger equation in inhomogeneous optical fiber system

In this work, the breather solution of an inhomogeneous optical system govern by nonlinear Schrödinger equation is studied. Three types of analytical breathers, namely, general, Akhmediev and Kuznetsov-Ma breathers are obtained via Darboux transformation (DT) technique. Furthermore, novel phase transitions of the breather are observed by adjusting the spectrum parameter. During the phase transitions, the down trough of the breather can split from one into two, while the amplitudes of the breather will increase.

Rogue waves and periodic solutions of a nonlocal nonlinear Schrödinger model

In the present work, a nonlocal nonlinear Schr\"odinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations, by considering them as fixed points in {\it space-time}. This methodology allows to perform a continuation of well-known solutions of the local NLS model to the nonlocal case. Four different examples of this type are presented, namely (a) the rogue wave in the form of the Peregrine soliton, (b) the generalization thereof in the form of the Kuznetsov-Ma breather, as well as two spatio-temporally periodic solutions in the form of elliptic functions. Importantly, all four waveforms can be continued in intervals of the parameter controlling the nonlocality of the model. The first two can be continued in a narrower interval, while the periodic ones can be extended to arbitrary nonlocalities and, in fact, present an intriguing bifurcation whereby they merge with (only) spatially periodic structures. The results suggest the generic relevance of rogue waves and related structures, as well as periodic solutions, in nonlocal NLS models.

Breathers and breather-rogue waves on a periodic background for the derivative nonlinear Schrödinger equation

In this paper, we give the solutions on a periodic background in terms of the determinant form for the derivative nonlinear Schrödinger equation. Because its rogue wave on a periodic background has been studied, we investigate only the breather and breather-rogue wave on a periodic background for the derivative nonlinear Schrödinger equation. We obtain Kuznetsov–Ma breather, Akhmediev breather and spatio-temporal breather on a periodic background for this equation. In addition, we mainly focus on three types of the breather-rogue wave on a periodic background: (1) the interaction between a Peregrine soliton and a breather; (2) the interaction between a Peregrine soliton and two breathers; (3) the interaction between a second-order rogue wave and a breather. For the first type, we analyse the effects of the free parameters on its dynamical behaviour. The second type is described as ‘rogue wave quanta’ on a periodic background. The third type has two spatial-temporal distribution structures: the fundamental structure and the triangular structure.

Numerical analysis of the Hirota equation: Modulational instability, breathers, rogue waves, and interactions.

In this paper, we focus on the integrable Hirota equation, which describes the propagation of ultrashort light pulses in optical fibers. First, we numerically study spectral signatures of the spatial Lax pair with distinct potentials [e.g., solitons, Akhmediev-Kuznetsov-Ma (AKM) and Kuznetsov-Ma (KM) breathers, and rogue waves (RWs)] of the Hirota equation. Second, we discuss the RW generation by using the dam-break problem with a decaying initial condition and further analyze spectral signatures of periodized wavetrains. Third, we explore two kinds of noise-derived modulational instabilities: (i) the one case is based on the initial condition (one plus a random noise) such that the KM and AKM breathers, and RWs can be generated, and they agree well with analytical solutions; (ii) another case is to consider another initial condition (one plus a Gaussian wave with a random noise phase) such that some RWs with higher amplitudes can be found. Moreover, we also investigate the spectral signatures of corresponding periodic wavetrains. Finally, we find that the interactions of two waves can also generate the RW phenomena with higher amplitudes. These obtained results will be useful to understand the RW generation in the third-order nonlinear Schrödinger equation and other related models.

Space–time evolution of optical breathers and modulation instability patterns in metamaterial waveguides

We present numerical and theoretical investigations of the spontaneous emergence of noise-driven modulation instability patterns in a metamaterial waveguide, which involves the generation of optical breather waves such as the Peregrine soliton, Akhmediev breathers and Kuznetsov–Ma breathers. We show that the intrinsic properties of the metamaterial waveguide, e.g. self-steepening and the magnetooptic effects, offer the potential to control the formation and subsequent spectral and temporal dynamics of these localized nonlinear waves. Such internal or external perturbations break the symmetry of the spectrum of nonlinear waves, thus leading to the existence of a controllable characteristic group velocity in their space–time evolution.

Higher-order solutions of a matrix AKNS system associated with a Hermitian symmetric space

In this paper, we study a matrix Ablowitz–Kaup–Newell–Segur (AKNS) system associated with a Hermitian symmetric space as a follow-up study of an earlier paper. A multi-fold generalized Darboux transformation of the matrix AKNS system associated with a Hermitian symmetric space is constructed by means of determinants. Subsequently, we derive various higher-order solutions for this system, including fan-shaped rogue wave and (truncated) Kuznetsov–Ma breather solutions. Specifically, we show the fusion and fission processes for two truncated Kuznetsov–Ma breathers by taking different free parameters.

On the quintic time-dependent coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics

Under investigation in this paper is a quintic time-dependent coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or a medium with the negative refractive index. A gauge transformation is found to obtain the equivalent form of the equation. With respect to the wave envelope for the free water surface displacement or envelope of the electric field, Painleve integrable condition, different from that in the existing literature, is derived, with which the bilinear forms and N-soliton solutions are constructed. Asymptotic analysis illustrates that the interactions between the bright and bound solitons as well as between the bright solitons and Kuznetsov–Ma breathers are elastic with certain conditions, while some other interactions are inelastic under other conditions. Propagation paths and velocities for the solitons are both affected by the dispersion coefficient function when the relations among the coefficients are linear, or affected by the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions when the relations among the coefficients are nonlinear. Under different conditions, bell-shaped solitons can evolve into the bound solitons or Kuznetsov–Ma breathers, respectively. Interactions between the bright and parabolic (or hyperbolic) solitons are related to the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions. Compression effect on the propagation paths of the solitons, caused by the dispersion coefficient, is observed.

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.