Stationary nonlinear waves, superposition modes and modulational instability characteristics in the AB system
Abstract
We study the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultrashort pulses in nonlinear optics. We show that the breather can be converted into different types of stationary nonlinear waves on constant backgrounds, including the multipeak soliton, Mshaped soliton, Wshaped soliton and periodic wave. We also investigate the nonlinear interactions between these waves, which display some novel patterns due to the nonpropagating characteristics of the solitons: (1) Two antidark solitons can produce a Wshaped soliton instead of a higherorder antidark one; (2) The interaction between an antidark soliton and a Wshaped soliton can not only generate a higherorder antidark soliton, but also form a Wshaped solion pair; (3) The interactions between an oscillation Wshaped soliton and an oscillation Mshaped soliton show the multipeak structures. We find that the transition occurs at a modulational stability region in a low perturbation frequency region.
Solitons, breathers, and rogue waves have been observed widely in nonlinear optics and fluid mechanics. Recent studies have revealed the intricate relation between the soliton and breather (or rogue wave) solutions of certain higherorder and coupled nonlinear evolution equations. Breathers or rogue waves may be converted into various nonlinear waves on constant backgrounds. The interactions among these transformed waves show some novel characteristics. Hereby, we will consider the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultrashort pulses in nonlinear optics. Several types of transformed stationary nonlinear waves will be demonstrated under some special conditions. Due to the nonpropagating characteristic, the nonlinear superpositions include some interesting phenomena. Further, the transformed nonlinear waves are associated with a modulational stability region in a low perturbation frequency region.
labelfont=bf
1. Introduction
Echoing soliton concepts that flourished in the multidisciplinary since a few decades ago, rogue wave has attracted recently the attention of researchers in various physical settings, e.g., in hydrodynamics [[1]], capillary waves [[2]], plasma physics [[3]], nonlinear optics [[4]], and BoseEinstein condensation [[5]]. Due to their harm to various hydrotechnic constructions, the investigation of rogue waves becomes a very important problem [[6]]. Rogue waves, which have a peak amplitude generally more than twice the significant wave height, are thought to appear from nowhere and disappear without a trace in the ocean and most experimental optical systems [[7]]. Among various models describing such waves, the focusing nonlinear Schrödinger (NLS) equation [[8]] is the most accepted one. The NLS equation admits a type of rational solution that is localized in both space and time, i.e., the Peregine soliton [[9]]. This simplest rational solution of the NLS equation, which was first predicted as far as 30 years ago [[9]], is frequently used as a model of a rogue wave [[10]]. Breather solutions are presently regarded as potential prototypes for the rogue waves in many fields of physics [[11]]. Breathers develop due to the instability of small amplitude perturbations that may grow in size to disastrous proportions [[12]]. There are two types of breathers, namely, the KuznetsovMa breathers (KMBs) [[13]] and Akhmediev breathers (ABs) [[14]]. The KMBs are periodic in space and localized in time while the ABs are periodic in time and localized in space [[13], [14]]. Taking the period of both solutions to infinity leads to a firstorder doubly localized Peregrine soliton.
The modulational instability (MI) is generally considered to be one of factors which may give rise to roguewave excitation [[9]]. As a fundamental characteristic of many nonlinear dispersive systems, MI is related to the growth of periodic perturbations on an unstable continuouswave background [[15]]. A rogue wave may be the result of MI, but not every type of MI results in rogue wave generation [[16]]. One of theoretical researches has revealed the close relationship between rogue waves and baseband MI, i.e., MI whose bandwidth includes components of arbitrarily low frequency [[17], [18]]. Another possible explanation presented by Zhao and Ling is that rogue wave comes from MI under the “resonance” perturbation with continuous wave background [[19]].
Recent studies have revealed the intricate link between the rogue waves (or breathers) and solitons of a certain class of nonlinear evolution equations. When the eigenvalues meet a certain locus on the complex plane, Akhmediev et al. have discovered that the breather solutions of the Hirota [[20]] equation and fifthorder NLS [[21]] equation can be converted into soliton solutions on a background, which does not exist in the standard NLS equation. He et al. have reported that the rational solution of a mixed NLS equation can describe five soliton states, including a paired brightbright soliton, a single soliton, a paired brightgrey soliton, a paired brightblack soliton, and a rogue wave state [[22]]. They have found that the state transition among these five states is induced by tuning the effects of selfsteepening and selfphase modulation. Liu et al. have shown that the breathers can be converted into different types of nonlinear waves in the coupled NLSMB system, including the multipeak soliton, periodic wave, antidark soliton, and Wshaped soliton [[23]]. In particular, they have indicated that the transition between the rogue waves and Wshaped solitons of the Hirota and coupled Hirota equations occurs as a result of the attenuation of MI growth rate to vanishing in the zerofrequency perturbation region [[24]].
In this paper, we study the AB system [[25], [26]],
(1)  
where is the amplitude of the wave packet and is a quantity measuring the correction of the basic flow due to the wave packet, and denote the time and space variables, respectively, and stand for two group velocities of the underlying linear problem, and represents a parameter measuring the state of the basic flow. When the basic flow is supercritical, , and when the basic flow is subcritical, . The parameter reflects the interaction of the wave packet and the meanflow, and it is always positive. System (1) describes marginally unstable baroclinic wave packets in geophysical fluids and also ultrashort pulses in nonlinear optics.
As shown in Ref. [[27]], System (1) can be reduced to a simpler form
(2) 
with
(3) 
by the transformations
(4) 
Recently, certain properties of System (2) have been investigated. System (2) can be transformed to the SineGordon equation when is the real value and to the selfinduced transparency system when is the complex value [[25], [26]]. Lax pair and some periodic solutions of System (2) have been derived in Ref. [[26]]. MI and breather dynamics of System (2) have been discussed in Ref. [[28]]. Ref. [[29]] has studied the envelope solitary waves and periodic waves of System (2). The higherorder rogue wave solutions of System (2) have been found via the modified Darboux transformation (mDT) in Ref. [[30]]. The semirational solutions have been derived in Ref. [[18]] and the link between the baseband MI and the existence condition of these rogue waves has been revealed. Recently, the rogue waves and MI have been demonstrated for a coupled AB system, i.e., a wavecurrent interaction model describing baroclinic instability processes in geophysical flows [[27]].
The aim of the present paper is to study the breathersoliton dynamics of System (2). We present intriguing different kinds of nonlinear localized and periodic waves, including the multipeak soliton, Mshaped soliton, Wshaped soliton and periodic wave. Interestingly, these waves are stationary nonlinear waves with respect to axis. Further, due to the nonpropagating characteristic, the nonlinear interactions show some novel properties. The breathertosoliton transition is related to a special type of MI analysis that involves a MI stability region in a low perturbation frequency region.
The arrangement of the paper is as follows: In Sec. 2, we will present different types of stationary nonlinear waves of System (2), including the multipeak soliton, Mshaped soliton, Wshaped soliton and periodic waves. And the transition condition will be analytically given. The properties of interactions between different types of nonlinear waves will be graphically studied in Sec. 3. The connection between the MI growth rate and transition condition will be revealed in Sec. 4. Finally, Sec. 5 will be the conclusions of this paper.
2. Different types of stationary nonlinear waves
In this section, we mainly study the transitions between the firstorder breather and various nonlinear waves for System (2). The firstorder breather solution of System (2) is given [[18]]
(5)  
with
From the above expression, one can find that the breather solution (5) comprises the hyperbolic functions () and the trigonometric functions (), where and are the corresponding velocities. The hyperbolic functions and trigonometric functions, respectively, characterize the localization and the periodicity of the transverse distribution of those waves. The nonlinear wave described by the solution (5) can be deemed to a nonlinear combination of a soliton and a periodic wave with the velocities and . In the following, we show that the solution (5) can describe different kinds of nonlinear wave states depending on the values of velocity difference .
When (or ), the solution (5) characterizes the localized waves with breathing behavior on constant backgrounds (i.e., the breathers and rogue waves). Further, if , we have the ABs with , the KMBs with , and the Peregrine soliton with . Those solutions have been obtained in Refs. [[18], [28], [30]].
Conversely, if , the soliton and a periodic wave in the solution (5) have the same velocity . Meanwhile, we should point out that the case is equivalent to the following condition
(6) 
i.e.,
(7) 
Eq. (6) means the extrema of trigonometric and hyperbolic functions in the solution (5) is located along the same straight lines in the plane, which results in the transformation of the breather into a continuous soliton. Additionally, the case is also equivalent to . Then, the parameters , , and vanish, and the solution (5) only depends on and is independent of . Thus, the nonlinear waves described by the solution (5) possess the nonpropagating characteristic.
Under the transition condition (7), we demonstrate several kinds of transformed nonlinear waves on constant background for , including the multipeak solitons, Mshaped soliton, periodic waves and Wshaped soliton. We omit the results in the component, since it only describes the plane wave in the case of . Fig. 1 shows a multipeak soliton on constant background that does not propagate along direction. Such structure is composed of a soliton and periodic wave. Adding the absolute value of real part to the eigenvalue, , we find that the peak numbers of multipeak localized structure decrease, as depicted in Fig. 2. From the crosssectional views, it is observed that both of those multipeak solitons have one main peak which is located at . The maximum amplitude of at can be given analytically
(8) 
Eq. (8) indicates that the imaginary part of eigenvalue () has real effects on the maximum amplitude of at .
Further, the structure in Fig. 2 (we call it an oscillation Wshaped soliton) corresponds to the case with the appropriate value of (). In other words, the coordinate origin is a local maximum of . Nevertheless, when the value of exceeds a certain range, is less than zero, which means that the coordinate origin is a local minimum of . In this case, the oscillation Wshaped soliton translates into a Mshaped soliton shown in Fig. 3. It is observed that this structure has two main peaks with identical amplitudes. In order to reveal the effect of the value of on , we plot Fig. 4 with and . If , namely, , the solution (5) describes the Mshaped soliton whereas it describes the Wshaped one.



Next, we derive two special nonlinear waves from the solution (5), i.e., the antidark soliton and periodic wave. The former exists in isolation when vanishes, while the latter independently exists when vanishes. Therefore, the antidark soliton and periodic wave are shown in forms of exponential and trigonometric functions respectively. Specifically, the analytical expressions read as, for the soliton,
(9) 
with
(10)  
and for the periodic wave,
(11) 
with
(12)  


Fig. 5 describes a soliton that does not propagate along direction. It is shown that this soliton lies on a planewave background with the peak . This kind of wave is referred to the antidark soliton which was firstly reported in the scalar NLS system with the thirdorder dispersion [[31]]. Recent studies on the NLSMB system have also presented the similar structures [[23]]. Further, for the value of becomes zero, this soliton will be converted into a standard bright soliton. Fig. 6 exhibits the periodic wave with the period . It is interesting, as it looks like higherorder wave but appears from the same solution.

In particular, as the period is close to zero, namely, , the periodic wave will turn into the Wshaped soliton, as shown in Fig. 7. In this case, the solution (5) is transformed into
(13) 
The maximum height () of the Wshaped wave is nine times the background intensity while the minimum is zero. By comparison with the Wshaped soliton in Fig. 2, the soliton in Fig. 7 has two different features: (1) it shows a singlepeak structure without oscillating tails; (2) it has a rational expression.
We further discuss the effects of the coefficients , and on the transformed solitons. The coefficients , and , which are related to the parameters , , and , are defined by the equation (3). Figs. (8a) and (8b) indicate that the amplitudes of the multipeak solitons decrease with increasing the values of and . In addition, increasing the values of and leads to stronger localization and a smaller oscillation period for the multipeak soliton. This means that we can control the amplitudes of the transformed nonlinear waves by adjusting the two group velocity coefficients and , and the parameter that reflects the interaction of the wave packet and the meanflow. However, as shown in Fig. (8c), changing the value of doesn’t affect the characteristics of the multipeak soliton.
3. Nonlinear wave interactions
In this section, we study the characteristics of interactions between the nonlinear waves presented above. Due to the diversity of the transformed waves (in fact, there are many kinds of nonlinear wave interactions), we only exhibit several typical nonlinear superposition patterns that derive from the twobreather solutions.
By virtue of the fold DT [[28], [18]], the twobreather solution of System (2) is given as
(14) 
with
Fig. 9 displays the elastic interaction between two breathers of System (2). It is found that the twobreather interaction produces a secondorder central rogue wave in () plane. Similar to the case in Section 2, to convert the twobreather solutions into the twosoliton ones on constant backgrounds, the parameter also needs to meet the transition condition (7), namely, . For different kinds of nonlinear superpositions, we can choose the corresponding real () and imaginary () parts of eigenvalues () in the solution (14). The values of and () control the types of the transformed nonlinear waves. In the following, we present six kinds of nonlinear superposition patterns.
The oscillation Wshaped soliton. We first consider the nonlinear superposition of two stationary Wshaped solitons, which is exhibited in Fig. 10. The two eigenvalues and meet the conditions of the Wshaped solitons. Interestingly, instead of a higherorder Wshaped soliton, these two localized waves form an oscillation Wshaped soliton with higher intensity (). by assuming , which gives the asymptotic plane. Due to the nonpropagating characteristic of the nonlinear waves, the nonlinear superposition could show some novel features. The interaction between two same type waves generates a different one.
The Wshaped soliton. Next, we demonstrate the nonlinear interaction between two antidark solitons in Fig. 11. The two eigenvalues and are set to be and , respectively. Surprisingly, we find that the interaction between the antidark solitons doesn’t produce a secondorder antidark soliton, but instead generates a Wshaped soliton.
The antidark soliton and Wshaped soliton pair. We then study the interaction between two different types of nonlinear waves, for example, the antidark soliton and a Wshaped soliton. The eigenvalue is for the antidark soliton while for the Wshaped one. As shown in Fig. 12, an intriguing phenomenon is that the interaction forms an antidark soliton. By adjusting the values of eigenvalues (still for antidark soliton) and (still for Wshaped soliton), we observe a Wshaped soliton pair in Fig. 13. Each component in the Wshaped soliton pair has the same intensity. In spite of the similar single wave as the seeds, the nonlinear interactions can produce different patterns due to the choices of different eigenvalues and . This means the mechanism of nonlinear superposition is controlled by the two eigenvalues. A complete and analytical study of the control mechanism of such composite solutions is an issue that requires special investigation.
The multipeak solitions. Finally, we display some multipeak structures. Fig. 14 describes the nonlinear interactions between the oscillation Wshaped soliton and oscillation Mshaped soliton. The superposition leads to the formation of a threepeak soliton that corresponds to the case . Using different eigenvalues, we plot a fourpeak soliton () also produced by the oscillation Wshaped soliton and oscillation Mshaped soliton, as depicted in Fig. 15. A higherorder multipeak soiton, which is composed of two firstorder multipeak solitons, is plotted in Fig. 16.
4. MI characteristics
In this section, we reveal the explicit relation between the transition and the distribution characteristics of MI growth rate for System (2). Firstly, let us recall the results of MI analysis on System (2). As shown in Ref. [[18]], System (2) admits the following continuous wave solutions,
(16)  
where , , and are real parameters. A perturbed nonlinear background can be expressed as
(17)  
Taking Eq. (17) into System (2) yields the evolution equation for the perturbations as
(18)  
Noting the linearity of Eq. (18) with respect to and , we introduce
(19)  
which is characterized by the wave number and frequency . Using Eq. (19) into Eq. (18) gives a linear homogeneous system of equations for and :
(20) 
(21) 
From the determinant of the coefficient matrix of Eqs. (20)(21), the dispersion relation for the linearized disturbance can be determined as
(22) 
Solving the above equation, we have
(23) 
Fig. 17 shows the characteristics of MI on the () plane. It is found that the MI exists in the region . Ref. [[18]] has found that the rogue wave existence condition of System (2) is strictly related to baseband MI, namely, the MI whose bandwidth includes arbitrary small frequencies. In this case, we obtain the parameter condition [[18]]. Hereby, we discover that the MI growth rate distribution is symmetric with respect to
(24) 
i.e.,
(25) 
The line (red dashed line) in Fig. 17 corresponds to a modulational stability (MS) region where the growth rate is vanishing in the low perturbation frequency region. More interestingly, one can find that the MS condition (24) [or (25)] is completely consistent with the condition (7) which converts breathers into stationary nonlinear waves. Our finding suggests that the transition between breathers and stationary nonlinear waves can occur in the MS region with the low frequency perturbations.
5. Conclusions
In summary, we have investigated the transition between breathers and stationary nonlinear waves for System (2). Some intriguing different types of stationary nonlinear waves, including multipeak solitons, Mshaped soliton, periodic wave, antidark soliton and Wshaped soliton, have been demonstrated graphically and analytically. The reasons and the conditions for transition have been presented explicitly. We have found that the real part of eigenvalue has effects on the peak numbers of the multipeak solitons while the imaginary part controls the transition between the oscillating Wshaped soliton and Mshaped soliton. Further, we have studied the interactions between those transformed nonlinear waves. Due to the nonpropagating characteristic, the nonlinear superposition of two types of nonlinear waves have exhibited some novel features (see Figs. 1016). Finally, we have shown that this transition is strictly associated with the MI analysis that involves a MS region.
Acknowledgements
We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant (Nos. 11305060 and 61505054), by the Fundamental Research Funds of the Central Universities (Project No. 2015ZD16), by the Innovative Talents Scheme of North China Electric Power University, and by the higherlevel item cultivation project of Beijing Wuzi University (No. GJB20141001).
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