Asymptotic analysis of multi-valley dark soliton solutions in defocusing coupled Hirota equations

Ziwei Jiang and Liming Ling

Department of Mathematics,South China University of Technology,Guangzhou 510641,China

Abstract We construct uniform expressions of such dark soliton solutions encompassing both singlevalley and double-valley dark solitons for the defocusing coupled Hirota equation with highorder nonlinear effects utilizing the uniform Darboux transformation,in addition to proposing a sufficient condition for the existence of the above dark soliton solutions.Furthermore,the asymptotic analysis we perform reveals that collisions for single-valley dark solitons typically exhibit elastic behavior;however,collisions for double-valley dark solitons are generally inelastic.In light of this,we further propose a sufficient condition for the elastic collisions of double-valley dark soliton solutions.Our results offer valuable insights into the dynamics of dark soliton solutions in the defocusing coupled Hirota equation and can contribute to the advancement of studies in nonlinear optics.

Keywords: coupled Hirota equation,uniform Darboux transformation,dark soliton solution,asymptotic analysis

1.Introduction

For integrable systems,the nonlinear Schr?dinger equation plays an important role in various fields such as nonlinear optics [1,2],water waves [3,4],plasma [5],and Bose–Einstein condensates [6].In optical fibers,the nonlinear Schr?dinger equation can describe the propagation of a picosecond optical pulse [2,7],but for high-bit-rate transmission systems,higherorder nonlinear and dispersive effects are taken into account,which yields the higher-order nonlinear Schr?dinger equation involving the Hirota equation [8–12].The exact localized wave solutions of the Hirota equation,such as multi-solitons,rogue waves,and breathers,have been extensively studied [13–22].Furthermore,the explicit expressions of the asymptotic analyses of single-valley dark solitons (abbreviated as SVDS) and double-valley dark solitons (abbreviated as DVDS) have been given for the defocusing case,and a sufficient condition for elastic collisions has been obtained [21].Notably,dark solitons with delayed nonlinear response and third-order dispersion,in contrast to those with only second-order dispersion and selfphase modulation,can admit single dark solitons with the same velocity under two different phase shifts identified as DVDSs[23].Moreover,Hirota equations in different physical backgrounds have the characteristics of being multi-component and having variable coefficients [24].And multi-component nonlinear systems are more widely used and possess more abundant dynamic phenomena than one-component systems[25–27].In this work,we mainly study the dark soliton solutions of the defocusing couple Hirota equation,which is completely integrable and admits the following form [28–32]:

where α is the real parameter;q=(q1,q2)?is a two-dimensional complex vector;the superscripts ‘?’ and ‘?’ represent the transposition and conjugate transpose of the matrix,respectively.When α=0,equation (1) is reduced to the coupled nonlinear Schr?dinger equation.

In recent years,some exact solutions of the coupled Hirota equation,such as soliton solutions [13,33],rogue wave solutions [26,34],breather solutions [35],and traveling wave solutions [36] have also been derived.There are transition phenomena in the evolution process between solitons,breathers,and rogue waves in the focusing case [37–39].Additionally,scholars pay attention to the dynamic behavior of the above exact solutions.For instance,elastic collisions are permitted in solutions such as SVDSs of the coupled Hirota equation [31,40,41].Interestingly,in the coupled higher-order nonlinear Schr?dinger equation,there exist the dark double-hump three-soliton solutions with higher order effects generated by the Hirota bilinear method,which admit elastic interactions among each other [41].The soliton with a double-humped shape,or DVDS,has found extensive applications in power amplification processes owing to its wider pulse width and capacity to withstand higher power [42].In fact,in the coupled Hirota equation,a single dark soliton can admit two types of intensity profiles: the dark soliton with a single valley and the dark soliton with double valleys.As far as our current state of knowledge allows us to ascertain that the question of whether there exists solely elastic interaction for DVDSs and SVDSs under the context of the coupled Hirota equation remains an open research field.The above problems in the coupled Hirota equation motivate us to further study the dynamic behaviors of its dark soliton solutions.

The paper is organized as follows: in section 2,with the aid of the uniform Darboux transformation [43],we construct uniform expressions to represent the multi-dark soliton solutions consisting of SVDSs and DVDSs for the coupled Hirota equation.Meanwhile,we propose a sufficient condition for the existence of dark soliton solutions of the coupled Hirota equation by studying the corresponding characteristic equation.In section 3,we explore the intriguing properties of these solutions through asymptotic analysis.It is revealed that the interaction among single dark soliton solutions can be divided into the following two cases: if the single dark soliton solution corresponds to an SVDS,it will inevitably result in an elastic collision.On the other hand,if the single dark soliton solution represents a DVDS,it is more likely to exhibit inelastic collision.The conclusions are given in section 4.

2.The dark soliton solutions for the coupled Hirota equation

The coupled Hirota equation (1) admits the Lax pair σ3=diag(1,-1,-1),λ∈Cis a spectral parameter;q is defined in equation (1).02denotes the 2 × 2 null matrix.Utilizing the compatibility condition Φxt=Φtxof the Lax pair(2),we can obtain the zero curvature equation Ut-Vx+[U,V]=0 with [U,V]=UV -VU,which results in the Hirota equation (1).

And the characteristic equation of matrix U1is as follows:

where μ is the eigenvalue of equation (4) and I3denotes the 3 × 3 identity matrix.The coefficients of the algebraic expression (4) with respect to μ are real-valued if the spectral parameter λ is real,which guarantees that expression (4) possesses either real-valued roots or a set of complex conjugate roots.To get the dark soliton solutions of equation (1),it is necessary to possess a pair of conjugate complex roots μ and μ*of equation (4).It is straightforward to obtain the vector solution of equation (3) by this pair of complex roots,and then substituting the above solution into the transformation Φ=yields

which is the vector solution of equation (2).

Figure 1.The density proflies of intensity square of the dark soliton solution with the parameters ≈ (- 0.5,1,- 0.2513 +1.2203i),a1=1,a2=-0.4,c1=1,c2=1,and α=0.65,which corresponds to an SVDS.(a) The density proflie of.(b) The density proflie of.

We are going to employ the uniform Darboux transformation [43],which is widely used to generate solitonic solutions.Due to the limitations of the classical Darboux transformation,it is not feasible to directly derive the multidark soliton solutions of multi-component systems.Hence,we adopt the uniform Darboux transformation proposed in reference [43] to construct multi-dark soliton solutions of the coupled Hirota equation.According to equation (5),the uniform Darboux transformation can be constructed explicitly as

We select a set of parameters based on equation (7),allowing us to successfully present the density profile of the SVDS,as shown in figure 1.In particular,substituting the parameters a1=1,a2=-0.4,c1=1,c2=1,and λ1=1 into the characteristic equation (4) to yield the complex root μ1≈-0.2513+1.2203i and then substituting all parameters into the above results,we can obtain that: the velocity v1of the dark soliton solution is approximately equal to 4.6289;the valley depths ofandare approximately equal to 0.9601 and 0.5291,respectively;the evolution direction of dark soliton solution is along the trajectory x -v1t -0.5=0,v1≈4.6289.

Figure 2.The density profiles of intensity square of the dark soliton solution with the parameters ns=3,nd=0,a1=0.5,a2=-0.4,c1=1,c2=1,α=0.625,and c ≈(1,0.5,-1,1,1.2,10,0.0686+0.9824i,0.0302+1.2606i,-0.1154+1.0236i),which corresponds to a general multi-dark soliton solution.(a) The density proflie of .(b) The density proflie of.

Theorem 1.The expressions for the multi-dark soliton solutions can be derived by the n-fold uniform Darboux transformation (8):

We select two sets of parameters to construct two types of multi-dark soliton solutions respectively.The multi-dark soliton solution in figure 2 exhibits the dynamics of three SVDSs,whereas the multi-dark soliton solution in figure 3 displays the dynamics of a DVDS and an SVDS.Notably,in contrast to the scalar Hirota equation,the two valleys of the DVDS can remain relatively far away from each other.

Whilst it is true that not all parameters selected can yield a dark soliton solution for the coupled Hirota equation,we shall endeavor to identify the underlying conditions that satisfy the existence of such solutions.Especially,we restrict our attention to the case of a1＞a2and c1=c2in the subsequent proposition.

Proposition 1.If the following conditions (1) or (2) hold:

Proof.In order to construct dark soliton solutions by uniform Darboux transformation,equation (4) ought to admit a pair of conjugate complex roots.Considering that μ serves as an eigenvalue of matrixU1,we identify the discriminant of this equation with respect to μ to obtain

Figure 3.The density profiles of the intensity square of the dark soliton solution with the parameters ns=1,nd=1,a1=-0.2,a2=-0.4,c1=1,c2=1,α=0.625,and c ≈(0.15,0.1,-1,1,1.2,10,0.15 -1.4107i,0.1498 +1.498i,0.1443 +0.8251i),which corresponds to a multi-dark soliton consisting of a symmetric DVDS and an SVDS.(a) The density proflie of.(b) The density proflie of .

We can perform a similar analysis in the absence of the restrictions of a1＞a2and c1=c2,but we are unable to provide an explicit expression of the existence condition of the solution (9).In order to vividly demonstrate the relationship between parameters and the existence of dark soliton solutions,we plot figure 4.It is worth noting that the dark soliton solutions exist solely in the X-type region,with no such solutions being present in other regions.Moreover,the color bar in figure 4 indicates that the velocity of the dark soliton solution varies monotonically within some intervals.This figure agrees with the conditions of the existence of dark soliton solutions for the coupled Hirota equation (1).

3.The asymptotic analysis of the dark soliton solutions

In this section,we primarily employ asymptotic analysis to explore the evolution of the exact solutions for the coupled Hirota equation,which are composed of SVDSs and DVDSs.

Lemma 1.Set the matrices

whereχl,βl,kandδk,lare defined in equation (10).The determinants of matricesA,B,andCaredet(A)=

For convenience,we introduce the following notations:

where χj=λj+μj,the velocity vjis an expression related to λj.Indeed,we express the velocity vjin terms of the parameter μjas specified in equation (7).Notably,since the parameter μjand λjare conjoined via the characteristic equation (4),the velocity vjis inherently linked to λjas well.With the aforementioned notational framework and results established,we are now poised to undertake an asymptotic analysis [44] of the dynamic behavior exhibited by both SVDSs and DVDSs.

Figure 4.The existence and velocity variation of the dark soliton solutions.The parameters are a2=-0.4,c1=c2=1,α=0.625.The white square corresponds to the dark soliton solution in figure 1,the green triangle to the dark soliton solution in figure 2,and the pink pentagram to the special dark soliton solution in figure 3.The parameter selections of the solutions depicted in this figure are consistent with the requirement for the existence of solutions as stipulated in lemma 1.

Proof.The proof of theorem 2 mainly comprises two paragraphs: one is the asymptotic expressions for multi-dark soliton solutions(x,t;c)along the trajectory lj,and the other is along the trajectory Lj.To begin,we perform the asymptotic analysis of the multi-dark soliton solutions along the trajectory lj.The expressions of the multi-dark soliton solutions (9) can be written as

where matricesA,B,Care defined in lemma 1 and

Moreover,the multi-dark soliton solutions can be expressed as

Then we conduct the asymptotic analysis of the multidark soliton solutions along the trajectory Lj.The multi-dark soliton solutions can be further expressed as

implying that the SVDSs keep their shape following a collision with a phase shift where i=1,2,j=1,2,…,n,andis defined in equation (14).Undoubtedly,the interactions for SVDSs are always elastic.Figure 5 depicts an example of observing changes following the collision of two SVDSs.The shapes of the SVDSs do not change after the collision,indicating that the SVDSs admit elastic collisions.Next,we would like to look into the interaction between an SVDS and a DVDS.Following the collision with a DVDS,the SVDS retains its original form,as shown in figure 6,which implies that the collision for the SVDS is still elastic.However,after colliding with the SVDS,the shape of the DVDS changes significantly,implying that the DVDS admits an inelastic collision.

Figure 5.The collision dynamics of two SVDSs.Left panels: dynamical evolution of dark soliton solutionbefore (t=-2,(a)) and after (t=10,(c)) the collision.Right panels: dynamical evolution of dark soliton solutionbefore (t=-2,(b)) and after (t=10,(d)) the collision.The solid red line describes the evolution of the dark soliton solution (9) with ns=2,nd=0.The blue line and green line show the evolution of the solution (24) along the trajectory l1 and the trajectory l2,respectively.The relevant parameters are consistent with those selected in figure 1.

Figure 6.The collision dynamics of an SVDS and a DVDS.Left panels: dynamical evolution of dark soliton solutionbefore (t=-3,(a)) and after (t=5,(c)) the collision.Right panels: dynamical evolution of dark soliton solutionbefore (t=-3,(b)) and after (t=5,(d)) the collision.The solid red line describes the evolution of the dark soliton solution (9) with ns=1,nd=1.The blue line shows the evolution of the solution (30) along the trajectory L2.The green line shows the evolution of the solution (24) along the trajectory l3.The analysis suggests that the collisions for SVDSs are always elastic,whereas the collision of DVDSs can be inelastic.The parameters are c=(-0.1,0.7,-0.7,-0.8,2,5,-0.1-1.1662i,0.1246-1.1093i,-0.2942+1.1603i),a1=1,a2=-0.6,c1=1,c2=1,and α=0.5.

Figure 7.The collision dynamics of two DVDSs.Left panels: dynamical evolution of multi-dark soliton solutionbefore (t=-50,(a))and after (t=50,(c)) the collision.Right panels: dynamical evolution of dark soliton solutionbefore (t=-50,(b)) and after (t=50,(d)) the collision.The solid red line describes the evolution of the dark soliton solution (9) with ns=0,nd=2.The blue line shows the evolution of the solution (30) along the trajectory L2.The green line shows the evolution of the solution (30) along the trajectory L4.The collisions of DVDSs are obviously inelastic.The parameters are a1=-0.6,a2=-0.6,c1=0.7,c2=1,α=0.5,and c=(-0.9,-0.7,-0.6,2,5.2,5,-0.5,0.3–1.0630i,0.3–0.7i,0.3–0.8246i,0.3–1.1136i).

Figure 8.The collision dynamics of an SVDS and a DVDS.Left panels: Dynamical evolution of a multi-dark soliton solutionbefore(t=-1,(a)) and after (t=6,(c)) the collision.Right panels: dynamical evolution of a multi-dark soliton solutionbefore (t=-1,(b))and after (t=6,(d)) the collision.The solid red line describes the evolution of the dark soliton solution (9) with ns=1,nd=1.The blue line shows the evolution of the solution (30) along the trajectory L2.The green line shows the evolution of the solution (24) along the trajectory l3.The profile of the DVDS changes too slightly to be visible after the collision at these parameters.The parameters are the same as in figure 3.

In fact,a plethora of experimental evidence has demonstrated that inelastic collisions occur in most cases for DVDSs,which is consistent with the outcomes we discussed in theorem 2.For example,the two DVDSs in figure 7 do not keep their pre-collision shape after the collision implying that they both exhibit inelastic collisions.

In light of this,we proceed to ascertain the conditions that give rise to elastic behavior in collisions for DVDSs.It should be highlighted that the asymptotic expression of a DVDS before and after the collision differs primarily in terms of the phase shift.Thus,the collision is elastic if the phase differences of the two valleys of the DVDS are equal before and after the collision;otherwise,it is inelastic.From the asymptotic expressions (30) we can also derive the elastic condition for the DVDSs as follows:

where χlis defined in equation (10).Different from figure 6,the two valleys of the DVDS in figure 8 are separated by a relatively wide distance (the displacement difference between the two valleys before and after the collision of the DVDS is much smaller than the initial distance of the two valleys).The interaction between the two valleys is extremely weak in this case,so even if the DVDS in figure 8 has an inelastic collision,the shape change after the collision is easily ignored.

4.Conclusions

In summary,we provide a sufficient condition for the existence of dark soliton solutions and proceed to derive the uniform expressions of such solutions including both SVDSs and DVDSs by means of the uniform Darboux transformation.The analysis indicates that while elastic collisions are a common feature of SVDSs,inelastic collisions are prevalent in most instances for DVDSs.Notably,we also propose a condition that guarantees elastic collisions for DVDSs.The dark soliton solutions derived from the defocusing coupled Hirota equation possess the potential for applications in physical fields such as signal transmission and modulation in the realm of fiber optic communication [32,45].Furthermore,our results also shed new light on the fundamental properties of dark solitons,and may provide a promising avenue for future research in the fields of nonlinear optics and photonics [46,47].

Acknowledgments

Liming Ling is supported by the National Natural Science Foundation of China (No.12 122 105).

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