Exact Breather Solutions Research Articles

Breather Interaction Properties Induced by Self-Steepening and Space-Time Correction* *Supported by the National Natural Science Foundation of China (Grant Nos. 11705145, 11947301, 11434013, and 11425522), the Major Basic Research Program of Natural Science of Shaanxi Province (Grant Nos. 2017KCT-12 and 2017ZDJC-32), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JQ1003).

We study the properties of breather interactions in nonlinear Kerr media with self-steepening and spacetime correction and with either self-focusing or self-defocusing nonlinearity, and present a new family of exact breather solutions via the Darboux transformation with a special-designed quadratic spectral parameter. In contrast to the previous results of the nonlinear Schrödinger equation (NLSE) hierarchy, we show that the relative phase of colliding breathers has a significant effect on the collision manifestation. In particular, only the out-of-phase interactions can generate small amplitude waves at the collision center, which are analogous to the NLSE super-regular breathers. Our results will deepen our understanding of the properties of breather interactions and they will offer the possibility of experimental observations of super-regular breather dynamics in systems with self-steepening and spacetime correction.

Dynamics of breather waves and rogue waves on a soliton background in the coupled Hirota systems

Based on the Darboux‐dressing transformation, the new localized wave solutions of the coupled Hirota systems are constructed with a detailed derivation. Furthermore, by using Taylor series expansions for the trigonometric and exponential functions of our obtained exact breather solution, the N‐order rogue wave solutions are also expressed explicitly. Besides, the dynamics of these rogue wave solutions are illustrated with some vivid graphics.

ПАРНЫЕ ВЗАИМОДЕЙСТВИЯ КОГЕРЕНТНЫХ СТРУКТУР НАПОВЕРХНОСТИ ГЛУБОКОЙ ВОДЫ

The interactions of coherent structures (different types of solitary wave groups) on the surface of deep water is an important nonlinear wave process, which has been studied both theoretically and experimentally (Dyachenko et al., 2013a, b; Slunyaev et al., 2017). At the moment, a complete theoretical description of such interactions is known only for the simplest model – the nonlinear Schrödinger equation (NSE) where exact multi-soliton solutions are found. In the work (Kachulin, Gelash, 2018), the dynamics of pairwise interactions of coherent structures (breathers) on the surface of deep water were numerically investigated in the framework of the Dyachenko-Zakharov model. Significant differences were found in the collision dynamics of breathers of the compact Dyachenko-Zakharov equation when compared to the behavior of the NLSE solitons. It was found that in a more precise model of gravitational surface waves, in contrast to the NLSE, the maximum amplification of the wave field amplitude during the collision process of coherent structures can exceed the sum of the initial amplitudes of the breathers. In addition, the maximum amplification of the wave field amplitude increases with increasing steepness of the interacting breathers and exceeds unity by 20% at the value of the wave steepness m ≈ 0.2. It was revealed that an important parameter determining the dynamics of pairwise collisions of breathers is the relative phase of these objects at the moment of interaction. The interaction of breathers in the non-integrable Dyachenko-Zakharov model leads to the appearance of small radiation, which was discovered earlier in 2013 (Dyachenko et al., 2013a, b). In the work (Kachulin, Gelash, 2018) we demonstrate that the magnitude of the energy losses of the colliding solitons to radiation also depends on their relative phase. Maximum of the energy losses is observed at the same relative phase, at which the amplitude amplification maximum is observed. In addition, depending on the value of the relative phase, solitons can both gain and lose the energy, which results in increase or decrease of their amplitude after a collision. It was found that, in contrast to the NSE model, the spatial shifts of solitons in a more precise model can be both positive and negative. We use the exact breather solutions of the Dyachenko-Zakharov model and the canonical transformation to physical variables (the free surface profile and the potential on the liquid surface) to find approximate solutions in the form of breathers within the framework of exact nonlinear equations for potential incompressible fluid flows. The preliminary results of our numerical experiments in the exact model demonstrate similar dynamics of the interaction of breathers, which indicates that the theoretical picture of the interaction of coherent structures presented here is universal and can be observed in laboratory experiments. The study of the dynamics of breather interactions in the exact model performed by D.I. Kachulin was supported by the Russian Science Foundation (Grant No. 18-71-00079). The work of V.E. Zakharov and A.I. Dyachenko on the dynamics of breather interactions in approximate models was supported by the state assignment “Dynamics of the complex materials”.

Formation mechanism of asymmetric breather and rogue waves in pair-transition-coupled nonlinear Schrödinger equations**Project supported by the National Natural Science Foundation of China (Grant No. 61774001) and the Natural Science Foundation of Hunan Province, China (Grant No. 2017JJ2045).

Based on the developed Darboux transformation, we investigate the exact asymmetric solutions of breather and rogue waves in pair-transition-coupled nonlinear Schrödinger equations. As an example, some types of exact breather solutions are given analytically by adjusting the parameters. Moreover, the interesting fundamental problem is to clarify the formation mechanism of asymmetry breather solutions and how the particle number and energy exchange between the background and soliton ultimately form the breather solutions. Our results also show that the formation mechanism from breather to rogue wave arises from the transformation from the periodic total exchange into the temporal local property.

Exact breather solutions of repulsive Bose atoms in a one-dimensional harmonic trap

Bose–Einstein condensates of repulsive Bose atoms in a one-dimensional harmonic trap are investigated within the framework of a mean field theory. We solve the one-dimensional nonlinear Gross–Pitaevskii (GP) equation that describes atomic Bose–Einstein condensates. As a result, we acquire a family of exact breather solutions of the GP equation. We numerically calculate the number density [Formula: see text] of atoms that is associated with these solutions. The first discovery of the calculation is that at the instant of the saddle point, the density profile exhibits a sharp peak with extremely narrow width. The second discovery of the calculation is that in the center of the trap ([Formula: see text] m), the number density is a U-shaped function of the time [Formula: see text]. The third discovery of the calculation is that the surface plot of the density [Formula: see text] likes a saddle surface. The fourth discovery of the calculation is that as the number [Formula: see text] of atoms increases, the Bose–Einstein condensate in a one-dimensional harmonic trap becomes stabler and stabler.

Electromagnetic Breathers and Periodic Loops in a Ferromagnet with the Uniaxial Anisotropy

The nonlinear magnetization dynamics of an uniaxial anisotropy ferromagnet thin film under an external magnetic field is analytically studied. The celebrated Landau–Lifshitz equation depicts the magnetization dynamics of the anisotropy ferromagnetic thin film and the Maxwell’s equations govern the propagation dynamics of electromagnetic wave along the magnetic anisotropy axis. By means of a reductive perturbation approach, the dynamics equations of the system can be reduced as a modified Korteweg-de Vries equation, which exist a exact breather solution. We find that the excitations of magnetization of the ferromagnetic thin film and the magnetic induction and magnetic field of the electromagnetic wave can be governed by the breather mode. Our results show that the easy axis-component of the magnetization is a highly localized magnetized state in the form of the breather and the other-components of the magnetization are evolving like localized periodic loops when the breather mode is excited, which is very interesting. Up to now, few similar results have been reported.

Optimal frequency conversion in the nonlinear stage of modulation instability.

We investigate multi-wave mixing associated with the strongly pump depleted regime of induced modulation instability (MI) in optical fibers. For a complete transfer of pump power into the sideband modes, we theoretically and experimentally demonstrate that it is necessary to use a much lower seeding modulation frequency than the peak MI gain value. Our experiment shows that, at such optimal modulation frequency, a record 95 % of the output pump power is frequency converted into the comb of sidebands, in good quantitative agreement with analytical predictions based on the simplest exact breather solution of the nonlinear Schrodinger ¨ equation.

On the highest non-breaking wave in a group: fully nonlinear water wave breathers versus weakly nonlinear theory

AbstractIn nature, water waves usually propagate in groups and the open question about the characteristics of the highest possible wave in a group is of significant theoretical and practical interest. We examine the problem of the highest non-breaking wave in a wave group by direct numerical simulations of the exact Euler equations. The main aim of the study is twofold: (i) to describe the highest wave in a group in fully nonlinear setting and find its dependence on parameters; (ii) to examine correspondence between the exact breather solutions of weakly nonlinear analytic theory based on the integrable nonlinear Schrödinger (NLS) equation and their strongly nonlinear analogues. In contrast to weakly nonlinear models the very notion of the highest wave is ill-defined: the maximal crest elevation, the maximal trough-to-crest height and the deepest trough all occur at close but different moments; correspondingly, we have to speak about distinctively different extreme waves. In the simulations small initial perturbation of a uniform wave train were prescribed in a way ensuring that the initial perturbation excites a single breather-type modulation. The ensuing growth results in higher wave magnitudes and takes longer time to develop compared with the NLS theory. The maxima of crest elevation noticeably exceed their weakly nonlinear analogues. The wave with the highest crest differs significantly from the unmodulated wave: the local wavelength contracts considerably, the crest becomes noticeably higher; the vicinity of the crest of such an extreme wave is close to that of the limiting Stokes periodic wave. Thus, the shape of the maximal crest wave is almost universal, i.e. it practically does not depend on the way the wave group evolved, or even whether there was initially more than one group. The evolution of a single NLS breather has been shown to have a qualitatively similar but quantitatively quite different analogue in the fully nonlinear setting. The one-to-one mapping of the NLS breather solutions onto fully nonlinear ones has been constructed. The fully nonlinear breathers are found to be robust, which provides grounds for applying the results for developing short-term deterministic forecasting of rogue waves.

Q-balls, integrability and duality

This paper is concerned with the dynamics and interactions of Q-balls in (1+1)-dimensions. The asymptotic force between well-separated Q-balls is calculated to show that Q-balls can be attractive or repulsive depending upon their relative internal phase. An integrable model with exact multi-Q-ball solutions is investigated and found to be of use in explaining the dynamics in non-integrable theories. In particular, it is demonstrated that the dynamics of small Q-balls in a generic class of non-integrable models tends towards integrable dynamics as the charge decreases. Long-lived oscillations of a single Q-ball can also be understood in terms of a deformation of an exact breather solution in the integrable model. Finally, we show that any theory with Q-ball solutions has a dual description in which a stationary Q-ball is dual to a static kink, with an interchange of Noether and topological charges.

Breathers in a discrete nonlinear Schrödinger-type model: Exact stability results

Following our earlier work [Phys. Rev. Lett. 84, 3570 (2000)] we present an exact linear stability analysis of one-site monochromatic breathers in a piecewise smooth discrete nonlinear Schrödinger-type model. Destabilization of the breather occurs by virtue of a growth rate becoming positive as a stability border is crossed, while above a critical spatial decay rate (lambda(c)) the breather is found to be intrinsically unstable. The model admits of other exact breather solutions, including multisite monochromatic breathers for which the profile variable (phi(n)) crosses a relevant threshold at more than one site. In particular, we consider exact two-site breather solutions with phase difference delta between the two sites above threshold, and present stability results for delta=pi (antiphase breather; the in-phase breather with delta=0 happens to be intrinsically unstable). We obtain a band of extended eigenmodes, together with a pair of localized symmetric modes and another pair of localized antisymmetric ones. The frequencies of the localized modes vary as the parameters characterizing the breather are made to vary, and destabilization occurs through the Krein collision of a quartet of growth rates, leading to temporal growth of a pair of symmetric eigenmodes of nonzero frequency. We clarify the limit N--> infinity (N is the gap length between the sites above threshold) when the two-site breather reduces to a pair of decoupled one-site breathers. The model offers the possibility of obtaining spatially random vortex-type breathers.

Wave group dynamics in weakly nonlinear long-wave models

The dynamics of wave groups is studied for long waves, using the framework of the extended Korteweg–de Vries equation. It is shown that the dynamics is much richer than the corresponding results obtained just from the Korteweg–de Vries equation. First, a reduction to a nonlinear Schrödinger equation is obtained for weakly nonlinear wave packets, and it is demonstrated that either the focussing or the defocussing case can be obtained. This is in contrast to the corresponding reduction for the Korteweg–de Vries equation, where only the defocussing case is obtained. Next, the condition for modulational instability is obtained. It is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term in the extended Korteweg–de Vries equation, and for a high carrier frequency. At the boundary of this parameter space, a modified nonlinear Schrödinger equation is derived, and its steady-state solutions, including an algebraic soliton, are found. The exact breather solution of the extended Korteweg–de Vries equation is analysed. It is shown that in the limit of weak nonlinearity it transforms to a wave group with an envelope described by soliton solutions of the nonlinear Schrödinger equation and its modification as described above. Numerical simulations demonstrate the main features of wave group evolution and show some differences in the behaviour of the solutions of the extended Korteweg–de Vries equation, compared with those of the nonlinear Schrödinger equation.

Discrete Breathers: Exact Solutions in Piecewise Linear Models

Exact breather solutions are constructed in piecewise linear (PWL) versions of the discrete nonlinear Schrodinger and Klein-Gordon equations. These solutions correspond to intersections of stable and unstable manifolds of relevant fixed points in associated 2D mappings, an exact construction of which is possible due to the PWL nature of the models. Such exact solutions give us insight into several aspects of breather properties. The problem of dynamical stability of the breathers is mentioned as an instance, detailed results on which will be presented in a future paper.

Discrete lattice effects on breathers in a spatially linear potential.

summary, we have generalized the Hirota method to include a spatially linear, time dependent external field.We have presented an exact breather solution, which is,in the potential-free case, a discrete generalization of the Satsuma-Yajima continuum solution starting with twoidentical solitons riding on each other with a coinciding center. Unlike the continuum breather in a spatially lin-ear, static potential, which remains the same coherent structure as that in the potential-free case, but with its center executing a parabolic motion in time, the discretebreather will evolve into two spatially separate, coherent structures whose motions are individually bounded. Wepoint out that this breakup is a general discrete eAect onbreathers in the presence of the external potential. [1] For example, Future Directions of Nonlinear Dynamics inPhysical 6 Biological Systems, special issue of Physica 68D (1993), and also in Vol. 312 of NATO ASI Series B, edited by P.L. Christiansen, 3. C. Eilbeck, and