Dynamics of breathers and rogue waves in scalar and multicomponent nonlinear systems

Weiying Wang and Xiubin Wang

1 School of Economics,Harbin University of Commerce,Harbin 150028,China

2 Department of Mathematics,Harbin Institute of Technology,Harbin 150001,China

Abstract In this paper,we propose a new method,the variable separation technique,for obtaining a breather and rogue wave solution to the nonlinear evolution equation.Integrable systems of the derivative nonlinear Schr?dinger type are used as three examples to illustrate the effectiveness of the presented method.We then obtain a family of rational solutions.This family of solutions includes the Akhmediev breather,the Kuznetsov-Ma breather,versatile rogue waves,and various interactions of localized waves.Moreover,the main characteristics of these solutions are discussed and some graphics are presented.More importantly,our results show that more abundant and novel localized waves may exist in the multicomponent coupled equations than in the uncoupled ones.

Keywords:rogue waves,breather waves,the variable separation technique

1.Introduction

As we all know,many nonlinear systems of physical interest support solitons,which are localized waves that arise from a balance between nonlinearity and dispersion and can propagate steadily for a long time.It has recently been found that another class of rational solutions,i.e.,breathers,is also of great importance.In addition,because of their localization properties,breathers have been recognized as models of rogue waves,which have recently become a popular research topic[1–3].They also often appear in many fields,such as optics,Bose–Einstein condensates,plasma physics,hydrodynamics,photonics,finance,etc[4–10].

The standard nonlinear Schr?dinger equation(NLSE)is completely integrable[11],and many kinds of exact solution have been found.In particular,the Peregrine soliton[12],the Akhmediev breather(AB)[1,13],and the Kuznetsov-Ma(KM)breathers[14,15]have been associated with rogue waves as a potential outcome of the modulational instability(MI)of a plane wave.In particular,a recent work has examined the relationship between extraordinary MI in optics and hydrodynamics and the generation of large-amplitude periodic wave trains[16].Earlier,in 1993,some scholars applied the powerful analytic method developed in the abovementioned paper by Akhmedievet alto the normaldispersion regime[17–20].Because the Peregrine soliton is localized in both time and space,it is recognized as a rogue wave prototype and reveals the main features of rogue waves[21].More importantly,a Peregrine soliton of the NLSE has been experimentally observed in water wave tanks[22]and nonlinear fiber optics[23,24].

To show the effectiveness of the variable separation technique in this paper,we focus on the following derivative nonlinear Schr?dinger-type equations:

where

Many studies of localized waves in nonlinear science have been carried out recently[25–45].In addition,there have been some developments in the field of the variable separation technique and soliton structures as well as in its application.For example,[39]presents a system with controllable parameters that describes the evolution of polarization modes in nonlinear fibers.In[40],Daiet alinvestigated scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials.In addition,[41,42]studied wick-type stochastic multi-solitons,soliton molecules,and fractional soliton solutions of the NLSE.In[43],vector breathers for the coupled fourth-order NLSE were investigated.In[44],explicit soliton–cnoidal wave interaction solutions for the(2+1)-dimensional negative-order breaking soliton equation were discussed in detail.These results help us to study the soliton structures associated with nonlinear differential equations in the field of mathematical physics.It is also known that the variable separation technique is a powerful method for the derivation of soliton solutions.This concept,to the best of the authors’knowledge,has never been reported before.The chief purpose of this work is to employ a variable separation technique to investigate breathers and rogue wave solutions of the derivative NLSEs.In addition,the dynamic behaviors of the localized wave solutions are also considered by selecting suitable parameters.

2.A variable separation technique

In this section,we introduce a variable separation technique for constructing rogue wave solutions to nonlinear wave equations.A general integrable NLSE has the following Lax pair:

where

To determine nonlinear wave solutions explicitly,we take the following six steps:

Step 1

We consider the following plane wave solution as a seed solution for the general integrable NLSE:

wherea1,a2,…,anare free parameters.

Step 2

We then seek a family of the solutions of the Lax system(2)in the following form:

and

Step 3

Substituting(4)into(2)yields

By combining conditions(6)and(7),we obtain Θ and Ω.

Step 4

Using the mathematical software Maple,we rewrite the exponential matrix R,? in(5)as a matrix function whose elements can be expressed by trigonometric functions and exponential functions.

Step 5

Taking the solution(3)as the seed solution in the Darboux transformation(DT),we can obtain a periodic solution of the general integrable NLSE composed of trigonometric functions and exponential functions.

Step 6

Let the period of the periodic wave go to infinity in the breather solution; the functionthen becomes a combination of exponential and polynomial functions ofxandt.Thus,we get the first-order rogue wave.

3.Example 1

In this section,as an example,we consider the famous derivative derivative NLSE.As a fundamental and important nonlinear physical model,the derivative nonlinear Schr?dinger equation(DNLSE)

has several physical applications,such as weak nonlinear electromagnetic waves in ferromagnetic[46],antiferromagnetic[47],and dielectric[48]systems subjected to external magnetic fields.It accurately describes the propagation of small-amplitude Alfvén waves in a low-β plasma[49,50]and the evolution of large-amplitude magnetohydrodynamic waves in a high-β plasma[51,52]; and also excellently models the transmission of ultra-short optical pulses in single-mode optical fibers[53,54].

3.1.Darboux transformation

Equation(8)is completely integrable.Its Lax pair is

3.2.Exact breather solutions

3.3.Rogue wave solution

where

This is a first-order rogue wave solution.Figure 2 is plotted for the rogue wave |q| for equation(8)using suitable parameters,it is localized both in time(t)and space(x),thus revealing the usual rogue wave features.In particular,in figures 1 and 2,we observe that a rogue wave can arise from the extreme behaviour of a breather wave.

Figure 1.AB and KM breathers in equation(8)(|q|)for the following parameters: a=1,ρ=1, z1=1, z2=1.(a):λ=1.1(1+i)/2,(b):λ=0.9(1+i)/2.

Figure 2.First-order rogue wave obtained from equation(8)(|q|)for the parameters:ρ =1,.

Figure 3.Hybrid solution to equation(15)for the parameters: a1=0, a2=1,λ=0.6(1+i),μ1=1,μ2=1.(a)μ3=5,(b)μ3=25,(c)μ3=80.

Figure 4.Hybrid solution to equation(15)for the parameters: a1=1, a2=-1,λ=0.6(1+i),μ1=μ2=μ3=1.

Figure 5.Hybrid solution to equation(15)for the parameters: a1=0, a2=1,2λ=0.9(1+i),μ1=μ2=μ3=1.

4.Example 2

If we choose the following matrix function:

then system(1)can be reduced to the coupled derivative NLSE(CDNLSE)

Equation(15)is important in plasma physics and the ultrashort pulse field.

4.1.Darboux transformation

Equation(15)is completely integrable; its Lax pairs are:

where

4.2.Exact breather solutions

4.3.Rogue wave solution

Ifa1=a2=1,μ1=0,μ2=μ3=1,the first-order hybrid solution degenerates to an eye-shaped rogue wave in two components.Here,we omit these figures.

If the parameters μj≠0,and the backgrounds are all nonvanishing,we see that a first-order rogue wave interacts with a breather in two components(see figure 8).By choosing different parameters forajand μj,we can obtain various arrangements of the two components,such as a first-order rogue wave and an amplitude-varying soliton,a first-order rogue wave,and a breather,etc.Here,we discuss the dynamics of the first-order hybrid solution.

Figures 6(a)–(c)demonstrate that a hybrid solution between a rogue wave and a soliton exists in theq1component.In particular,if the value of μ3in theq1component is increased,the first-order rogue wave cannot be easily observed,since it appears from the zero plane background.When the value μ1is changed,it reveals that a hybrid solution between a first-order rogue wave and an amplitude-varying soliton exists in theq2component,and a hybrid solution between a first-order rogue wave and a bright soliton exists in theq1component shown in figure 7.Figures 7(d)–(f)show that the soliton in theq2component is an anti-dark soliton ift<0 and becomes a dark soliton ift>0.In particular,this kind of amplitude-varying soliton is annihilated whent=0.If μ1becomes larger,the distance between a first-order rogue wave and a soliton increases.When the value ofa1selected in figure 8 is one,instead of zero in figures 6 and 7,the first-order rogue wave can merge with a breather.In figure 8,we observe that a first-order rogue wave appears with a breather in two components.In addition,we can see that μ1can control the distance between a first-order rogue wave and a breather.In particular,asa1→0,the breather wave yields a bright soliton in theq1component(see figures 6(a)and 8(a))and the breather wave disappears in theq2component(see figures 6(d)and 8(d)).

Figure 6.Hybrid solution of equation(15)for the parameters: a1=0, a2=1,μ1=1,μ2=1.(a),(d)μ3=2,(b),(e)μ3=10,(c),(f)μ3=50.

Figure 7.Hybrid solution of equation(15)for the parameters: a1=0, a2=1,μ2=1,μ3=1.(a),(d)μ1=2,(b),(e)μ1=10,(c),(f)μ1=40.

Figure 8.Hybrid solution of equation(15)for the parameters: a1=1, a2=1,μ2=1,μ3=1.(a),(d)μ1=1,(b),(e)μ1=50,(c),(f)μ1=100.

Figure 9.AB wave in equation(20)for the parameters: c1=0, c2=1, z1=z2=z3=z4=1,2λ=1.1(1+i).

Figure 10.KM breathers in equation(20)for the parameters: c1=0, c2=1, z1=z2=z3=z4=1,2λ=0.9(1+i).

If we do not consider the different arrangements of the two components,the interactions of the localized waves in the coupled system(15)can be completely classified into three types.Our results provide evidence of some obvious interactions between rogue waves and solitons or breathers.To the best of our knowledge,these types of dynamic patterns in the coupled system(15)have never emerged in the corresponding uncoupled systems.In this section,we generalized Baronio’s results[60]to obtain these kinds of mixed interactions of localized waves.In addition,we constructed mixed interactions of localized waves in the two-component system(15)through a variable separation technique.However,these mixed interactions cannot be obtained in single-component systems using the variable separation technique.Therefore,a conclusion can be drawn that these kinds of mixed interactions of localized waves can only be obtained by the variable separation technique in nonlinear systems with more than two components with the corresponding Lax pair including matrices larger than 2×2.

5.Example 3

If the matrix function U is chosen to be a symmetric matrix

the above system(1)becomes the Hermitian symmetric space derivative NLSE(HSS-DNLSE)[56–59]

5.1.Breather wave solution

5.2.Rogue wave solution

In what follows,we will display the propagation characteristics of the novel rogue wave using two images.Figures 11–13 present the dynamics of the novel rogue wave;this is made possible by the selection of suitable parameters which are helpful for enriching the dynamical behaviors of the nonlinear wave solutions.Interestingly,a wave with three peaks emerges without valleys in theq1component;in theq0component,a wave with a peak emerges without valleys.In particular,a wave with two peaks and two valleys emerges(also called a four-petaled rogue wave)in theq-1component.To the best of the authors’ knowledge,the same dynamic patterns have not emerged from the scalar NLS equation and the Hirota equation to date.

Figure 11.Rogue waves obtained from equation(20)for the parameters: c1=0, c2=1,ν1=ν2=ν3=ν4=1.

Figure 12.Density plot of figure 12.

Figure 13.The same parameters as those used in figure 12,except for c1=1.

6.Conclusions

In this work,we have shown how to construct breathers,rogue waves,and mixed interactions in the derivative NLSEs using the DT combined with an asymptotic expansion.These obtained solutions can be explicitly expressed in a‘separation of variables’ form.These solutions exhibit a range of interesting and complicated dynamics,discussed by varying the available parameters.These include the AB,KM breathers,the Peregrine soliton,breathers and rogue waves that interact with dark and bright solitons,a first-order rogue wave that interacts with a breather,a three-peaked rogue wave without valleys,a one-peaked rogue wave without valleys,a fourpetaled rogue wave,etc.These new spatiotemporal patterns reveal the potential rich dynamics in rogue wave solutions.Our results show that multicomponent coupled systems admit more abundant dynamical behaviors than the scalar case,which further helps us to explore different dynamics in related fields,such as Bose–Einstein condensates,optical fibers,superfluids,etc.Reference[60]presents the experimental conditions used to observe the mixed interactions of localized waves in vector NLSEs.Therefore,we expect that the new spatiotemporal patterns obtained in this work will be verified and observed in physical experiments in the near future.

Some scholars have studied the breathers and rogue wave solutions of the derivative NLSEs using DT formation[55,59].In contrast to their work,we obtained these solutions for the derivative NLSEs by adopting a variable separation and Taylor expansion technique.Through comparison,we find that the differences between these works are mainly reflected by two aspects:(i)the other papers that used DT to study the derivative NLSEs did not consider a variable separation technique,as we did;(ii)our work contains many new phenomena that are different from those of the other papers that using DT; these phenomena greatly enrich the properties of the derivative NLSEs.

Acknowledgments

We would like to thank the editor and referees for the valuable suggestions and comments that improved the manuscript.