breathers and rogue waves: demonstration with coupled

PRAMANA c© Indian Academy of Sciences Vol. 84, No. 3— journal of March 2015

physics pp. 339–352

Breathers and rogue waves: Demonstration with couplednonlinear Schrödinger family of equations

N VISHNU PRIYA, M SENTHILVELAN∗ and M LAKSHMANANCentre for Nonlinear Dynamics, School of Physics, Bharathidasan University,Tiruchirappalli 620 024, India∗Corresponding author. E-mail: [email protected]

DOI: 10.1007/s12043-015-0937-4; ePublication: 21 February 2015

Abstract. Different types of breathers and rogue waves (RWs) are some of the important coher-ent structures which have been recently realized in several physical phenomena in hydrodynamics,nonlinear optics, Bose–Einstein condensates, etc. Mathematically, they have been deduced in non-linear Schrödinger (NLS) equations. Here we show the existence of general breathers, Akhmedievbreathers, Ma soliton and rogue wave solutions in coupled Manakov-type NLS equations and cou-pled generalized NLS equations representing four-wave mixing. We deduce their explicit formsusing Hirota bilinearization procedure and bring out their exact structures and important properties.We also show the method to deduce the various breather solutions from rogue wave solutions usingfactorization form and the so-called imbricate series.

Keywords. Rogue wave; breather; coupled nonlinear Schrödinger-type equations.

PACS Nos 02.30.Ik; 42.65.−k; 47.20.Ky; 05.45.Yv

1. Introduction

Rogue waves (RWs) are waves which occasionally appear in the ocean that can reachamplitudes more than twice the value of significant wave height [1]. These wavescause destruction to oil tanks and cruise ships. Unlike tsunami waves, these wavesare unpredictable. A well-known description of RWs is that they appear from nowhereand disappear without a trace [2]. RWs were first recorded by satellite in the 1990sat Draupner oil platform in North Sea [1]. In analogy with oceanic RWs, RWs havebeen observed in optical fibres in the year 2007 by Solli et al [3] when analysing super-continuum generation. This is an important confirmation of the existence of RWs inphysical ystems. This subject has rapidly been gaining importance since 2007 with manypublished works in different fields such as Bose–Einstein condensates [4], plasmas [5],superfluid helium [6], capillary waves [7], etc. These RWs may arise from the instabilityof a certain class of initial conditions that tend to grow exponentially and thus have

Pramana – J. Phys., Vol. 84, No. 3, March 2015 339

N Vishnu Priya, M Senthilvelan and M Lakshmanan

the possibility of increasing up to very high amplitudes, due to modulation instability[2].

Recently, efforts have been made to explain the RW excitation through a nonlinear pro-cess. It has been found that the rational solution of nonlinear Schrödinger (NLS) equationcan describe many characteristics of the RWs. The first-order rational solution of NLSequation was given by Peregrine [8] as early as 1983 which is nothing but a fundamen-tal RW. The next order rational solution, i.e., second-order RW solution, based on [9],was recently presented in [2] as a possible explanation for RWs with higher amplitudes.Higher-order RW solutions of NLS equation was presented by Akhmediev et al usingmodified Darboux transformation method [10]. Recently, Nth-order RW solution for theNLS equation has been constructed by Guo et al using the generalized Darboux transfor-mation method [11]. Subsequently, attempts have been made to construct RW solutionfor the higher-order NLS equations and derivative generalizations of NLS equations. Oneway of obtaining RW solution or Peregrine soliton for a given system is to first constructa breather solution, either Akhmediev breather (AB) (spatially periodic and temporallylocalized breather) [12] or Ma soliton (MS) (temporally periodic and spatially localizedbreather) [13]. From these breather solutions, the RW solution can be deduced in anappropriate limit.

From the literature one can see that the RW solution has been constructed frombreathers [14]. An obvious question arising is that whether one can construct a AB orMS from a RW solution? The answer to this question has been given by the earlier workof Tajiri and Watanabe [15] for the case of scalar NLS equation. We extend the workof Tajiri and Watanabe to the case of coupled NLS equations and derive breather solu-tion from RW solutions. To achieve this goal the RW solution is rewritten in a factorizedform and then this is generalized to an imbricate series expression [16–18] with certainunknown parameters. We fix these unknown parameters by substituting it in the cou-pled NLS equations and solving the resultant equations. With three different forms of theimbricate series we derive the AB, MS and general breather (GB) solutions from the RWsolution. The details of this work may be found in [19]. In this paper, we demonstratethis procedure and critically review the properties of breathers and RWs for two importantnonlinear evolution equations, namely (i) Manakov system and (ii) coupled generalizedNLS (CGNLS) equations which also include four-wave mixing effects. We also point outhere that the four-wave mixing parameter influences breather and RW profiles.

This paper is organized as follows. In §2, we construct explicit forms of the GB and RWsolutions of Manakov system through Hirota’s bilinearization method. We then explainthe method of deriving AB, MS and GB solutions from RW solution for this system. In§3, we demonstrate the construction of GB and RW solutions from two periodic solitonsolution for CGNLS equations. We also analyse the role of four-wave mixing. We thenconstruct AB, MS and GB solutions of CGNLS system from RW solution. We devote §4for conclusions.

2. CNLS system

To begin with, we consider the integrable Manakov system [20]

iqjt + qjxx + 2μ(|q1|2 + |q2|2)qj = 0, j = 1, 2, (1)

340 Pramana – J. Phys., Vol. 84, No. 3, March 2015

Demonstration of breathers and rogue waves

where qj ’s, j = 1, 2, are wave envelopes, x and t are space and time variables, respec-tively, μ is a real constant and subscripts denote partial derivatives with respect to thecorresponding variables. Equation (1) represents the propagation of an optical pulse in abirefringent optical fibre and in wavelength division multiplexed system [20]. The com-plete integrability of this system of coupled nonlinear Schrödinger equations (NLSEs)was first established by Manakov [20] and later it has been discussed widely in the lite-rature (see e.g., [20–26] and references therein). In this work we derive breather and RWsolutions of (1).

2.1 General breathers and AB/MS/RW from GB

To start with, we bilinearize eq. (1) through the transformation q1 = g/f and q2 = h/f ,where g and h are complex functions and f is a real function. The resultant bilinearequations read (iDt + 2ikDx +D2

x)g ·f = 0, (iDt + 2ikDx +D2x)h ·f = 0, (D2

t + 2�1)

f · f − 2μ(|g|2 + |h|2) = 0, where �1 = μ(τ 21 + τ 2

2 ) and τj , j = 1, 2,are real constants,Dt and Dx are Hirota’s bilinear operators. We then solve the bilinearized equations byassuming the functions f , g and h suitably and obtain the two-soliton solution in the form

qj = τj eiθ g

f, θ = kx − ωt, j = 1, 2, (2)

where

g = h = 1 + eη1+2iφ1 + eη2+2iφ2 + ϑeη1+η2+2iφ1+2iφ2 ,

f = 1 + eη1 + eη2 + eη1+η2 , ηj = pjx − j t + η0j ,

ϑ =(

sin 12 (φ1 − φ2)

sin 12 (φ1 + φ2)

)2

, ω = k2 −2�1, pj = 2i√

�1 sin φj ,

j = 2kjpj −p2j cot φj , η

0j and φj , j = 1, 2,

are complex parameters.We then restrict the parameters η1 = η∗

2 ≡ η, φ2 = φ∗1 ± π , η = ηR + iηI and

φ1 = φR + iφI, in the two-soliton solution and rewrite the latter in the form

qj =τj cos 2φRei(θ+2φR)

[1+ 1√

ϑ cosh(ηR + σ)+ cos ηI

((cos 2φI

cos 2φR−1

)cos ηI

+i

(√ϑ tan 2φR sinh(ηR + σ) − sinh 2φI

cos 2φRsin ηI

))], j = 1, 2, (3)

where

ηR = pRx − Rt + η0R, ηI = pIx − It + η0

I ,

p1 = pR + ipI, 1 = R + iI,



η0R, η0

I and σ are constants with

pR = −2√

�1 cos φR sinh φI, pI = 2√

�1 sin φR cosh φI,

R = 2kpR − (p2R − p2

I ) sin 2φR + 2pRpI sinh 2φI

cosh 2φI − cos 2φR

and

I = 2kpI − (p2R − p2

I ) sinh 2φI + 2pRpI sin 2φR

cosh 2φI − cos 2φR.

Equation (3) is the GB solution of (1). This periodic envelope solution (3) satisfies theboundary conditions |qi |2 → τ 2

i , i = 1, 2, as x → ±∞. The breather solution (3) isdepicted in figure 1. The GB solution is periodic both in space and time.

We can derive the spatially-periodic breather (AB) and time-periodic breather (MS)from the GB (3). To derive AB, one should consider φI = 0 and φR �= 0 in the GB.The other choice φR = 0 and φI �= 0 provides MS. Exact expressions of AB and MSsolutions are given below (vide eqs (8) and (13)). We can also deduce RW solution from(3) by choosing the parameters φR and φI suitably. For example, restricting η2 = η∗

1 andφ2 = φ∗

1 + π and making the Taylor expansion at ε → 0 we find that the expressions pR,pI, R, I, ϑ all can be expressed as polynomials in ε. Substituting these expressionsalong with φR = εγ and φI = ερ, where γ and ρ are constants and ε is a small parameter,in GB solution (3) and taking the limit ε → 0 in the resultant expression we can captureRW solution of the form

qj = τj eiθ (1 − Q), Q = 4 + 16i�1t

1 + 4�1(x − 2kt)2 + 16�21t

2, j = 1, 2. (4)

The RW solution is localized both in space and time. A typical evolution of the RW isshown in figure 1b.

2.2 Breathers from RW

Now we consider RW solution as the starting point and derive AB, MS and GB solutionsfrom it in the reverse direction.

(a) (b)

Figure 1. (a) General breather profile of q1 for the parametric choices τ1 = 2, τ2 =1.5, φR = 3, φI = 2, η0

I = 0.8, η0R = 0.5, k = 0.2, μ = 1. (b) Rogue wave profile of

q1 for τ1 = 2, τ2 = 1.5, μ = 1, k = 0. Similar profile occurs for q2 also (not shownhere).



2.2.1 AB from RW. We factorize the RW solution (4) as

qj = τj exp(i(kx − (k2 − 2�1)t))

(1 + 1

F1+

)(1 − 1

F1−

), j = 1, 2, (5)

where

F1± = 2i�1t ± 1

2

√1 + 4�1(x − 2kt)2

and then it is considered in a more general form with certain unknowns, i.e.,

qj = τj exp(i(σ t+φ))

(1+ϒ

∞∑n=−∞

1

F2+

)(1+ϒ

∞∑n=−∞

1

F2−

), j = 1, 2,

(6)

where F2± = iαt ± v(x) + n, ϒ is a constant, α, σ and v(x) need to be determined andin the previous equation we have assumed k = 0 for mathematical simplicity. Replacingthe infinite series by cot function and using the identity [27]

cot πx = 1

πx+ x

π

∞∑n=−∞

1

n(x − n), n �= 0,

expression (6) can be brought to the following form, i.e.,

qj = τj exp(i(σ t+φ))[1+ϒπ cot(πF2+)][1−ϒπ cot(−πF2−)], j = 1, 2.

(7)

To determine the unknowns α, σ and v(x) we replace the cot functions in (7) arereplaced as cos(πv(x)± iπαt)/ sin(πv(x)± iπαt) and substitute them in (1) and rewritethe equations in terms of sin(πv(x)± iπαt) and cos(πv(x)± iπαt) and their powers andproducts. We then simplify these equations using suitable trigonometric identities andrearrange the resultant expressions in the variables cos(iπαt) sin(iπαt) and their powers.Doing so, we obtain a set of equations for the unknowns α, σ and v(x). Solving them weobtain

σ = 2�1(1 + π2ϒ2)2,

v(x) = 1

2πarccos

(1√

1 + π2ϒ2cos(

√2π2αϒx + v0)

)and

α = 2ϒ�1(1 + π2ϒ2),

where v0 is a constant of integration. With these explicit forms, eq. (7) provides the ABsolution of the form

qj = τj (1 + π2ϒ2) exp(i(2�1t + φ))

(1 − 2πϒ

1 + π2ϒ2M

),

M = πϒ cosh 2παt + i sinh 2παt

cosh 2παt − (1/√

1 + π2ϒ2) cos(√

2π2αϒx + v0), j = 1, 2. (8)

The solution (8) is periodic in the spatial direction and localized in temporal direction.After reaching the maximum amplitude at a specific time, it decays exponentially againto the constant background. The solution plot is given in figure 2a.



(a) (b)

Figure 2. (a) Akhmediev breather profile of q1 for τ1 = 3, τ2 = 1, μ = 1, ϒ = 0.5,φ = 0. (b) Ma breather profile of q1 for τ1 = 2, τ2 = 1, μ = 1, h = 0.5, φ = 0.Similar profile occurs for q2 also (not shown here).

2.2.2 MS from RW. To achieve this goal we rewrite the RW solution, vide eq. (4), in aslightly different factorized form, as

qj = τj exp(i(kx−(k2 −2�1)t))

(1 + i

G1+

)(1 + i

G1−

), j = 1, 2, (9)

where

G1± = −2�1t ± i

2

√1 + 4�1(x − 2kt)2.

We then consider (9) in a more general form,

qj = τj exp(i(ζ t+φ))

(1+ih

∞∑n=−∞

1

G2+

)(1+ih

∞∑n=−∞

1

G2−

), j =1, 2,

(10)

where G2± = κt ± i�(x) + n, with three unknowns, namely �(x), κ and ζ . Here also wehave assumed k = 0 for simplicity. As in the previous case we identify the infinite serieswith the cot hyperbolic function [27],

coth πx = 1

πx− ix

π

∞∑n=−∞

1

n(x − in), n �= 0,

and rewrite the above expression as

qj =τj exp(i(ζ t+φ))[1+hπ coth(iπG2+)][1−hπ coth(−iπG2−)], j =1, 2.

(11)

We then follow the same procedure given in the previous case to determine theunknowns ζ , κ and �(x). Substituting these values back in (11) we can get the MSsolution of the form

qj = τj (1 − π2h2) exp(i(2�1t + φ))

(1 + 2πh

1 − π2h2M

), (12)

M = πh cos 2πκt − i sin 2πκt

cos 2πκt − (1/√

1 − π2h2) cosh(√−2π2κhx + c)

, j = 1, 2. (13)

This solution is periodic in the temporal direction and localized in space. It grows anddecays periodically in constant background as in the case of NLS equation [15]. The Mabreather solution of CNLS equations for a set of parametric values is shown in figure 2b.



2.2.3 GB from RW. To derive GB from RW we consider the absolute square of themodulus of RW solution of (1) and factorize it in the form |qj |2 = τ 2

j (1 − Q)(1 − Q∗),where Q is given in (4). It is then rewritten as the second derivative of a logarithmicfunction

|qj |2 = τ 2j − τ 2

j

�1

∂2

∂x2ln

(1

S+× 1

S−

),

whereS± =

(1

2

√1 + 4�1(x − 2kt)2 ± 2i�1t

)2

.

Now considering the functions S± in the form of infinite series, we get

|qj |2 = τ 2j − τ 2

j

�1

∂2

∂x2ln

[( ∞∑n=−∞

1

H 2+

) ( ∞∑n=−∞

1

H 2−

)], j = 1, 2, (14)

where H± = φ(x, t)± iψ(x, t)−n, φ(x, t) and ψ(x, t) are arbitrary functions of x and t

which need to be determined, and rewriting the expression in a more compact form usingtrigonometric identities, we arrive at

|qj |2 = τ 2j + τ 2

j

�1

∂2

∂x2ln[cosh 2πψ − cos 2πφ], j = 1, 2. (15)

We can also rewrite the absolute square of the modulus of the GB solution as the secondderivative of a logarithmic function

|qj |2 = τ 21 + τ 2

1

�1

∂2

∂x2ln[√ϑ cosh(pRx − Rt + σ) − cos(pIx − It + θ)],

j =1, 2, (16)

whereσ = η0

R + 1

2ln a

and

θ = η0I + π.

At this stage both the expressions, GB and RW, are written in the same form. Com-paring the arguments inside the logarithmic function in two eqs (15) and (16), the exactexpression of the unknown arbitrary functions ψ and φ can be fixed as

ψ = 1

2πln

[√ϑ cosh(pRx − Rt + σ)+

√ϑ cosh2(pRx − Rt + σ)−1

],

φ = 1

2π(pIx − It + θ) (17)

andψ = − 1

2π(pRx −Rt +σ), φ = 1

2πarccos

(1√ϑ

cos(pIx − It + θ)

).

(18)An exact imbricate series of RW solution for breather solutions of CNLS equations canbe displayed by substituting (17) or (18) in (14). Very recently higher-order RW solutionshave also been constructed for the Manakov system. To derive second-order RW solution,using the parameters adopted here, one has to construct the four-soliton solution for eq. (1)which in principle is possible, though tedious.



3. Coupled generalized nonlinear Schrödinger (CGNLS) equations

In this section, we consider another physically interesting system of two coupledgeneralized nonlinear Schrödinger (CGNLS) type equations, namely [28]

iqjt + qjxx + 2(a|q1|2 + c|q2|2 + bq1q∗2 + b∗q2q

∗1 )qj = 0, j = 1, 2, (19)

where qj ’s, j = 1, 2, are amplitudes of slowly-varying pulse envelopes, a and c are realconstants corresponding to self-phase modulation and cross-phase modulation effects,respectively. Here b is a complex constant corresponding to four-wave mixing and theasterisk denotes complex conjugation. When a = c ≡ μ and b = 0 the above equationreduces to the Manakov system (1). When a = −c and b = 0 it reduces to the mixedcoupled nonlinear Schrödinger equation [29]. The nonlinear system (19) has also beenproved to be completely integrable for arbitrary values of the system parameters a, b

and c through Weiss–Tabor–Carnevale (WTC) algorithm [30]. N-bright soliton formulafor system (19) is constructed in [28]. Here we derive GB and RW solutions for thenonlinear evolutionary eq. (19) and analyse how the solution profiles vary with respect tothe four-wave mixing parameter b.

3.1 General breathers

Similar to the case of Manakov eq. (1), to derive the GB solution we construct a two-periodic soliton solution for the CGNLS eq. (19) of the form

qj = τj ei(kx−ωt)

(1 + eη1+2iφ1 + eη2+2iφ2 + ϑeη1+η2+2iφ1+2iφ2

1 + eη1 + eη2 + ϑeη1+η2

), (20)

where

ηj = pjx − j t + η0j , ω = k2 − 2�2,

�2 = aτ 21 + cτ 2

2 + (b + b∗)τ1τ2, pj = 2i√

�2 sin φj ,

and all other parameters, j , ϑ , φj and η0j match exactly with the ones obtained for the

Manakov system in the previous section. Similar to Manakov system, here also we restrictη1 = η∗

2 ≡ η and φ2 = φ∗1 ± π and consider η = ηR + iηI and φ1 = φR + iφI. With these

restrictions we can rewrite (20) also in terms of trigonometric and hyperbolic functionsas

qj =τj cos 2φRei(θ+2φR)

[1+ 1√

ϑ cosh(ηR+σ)+cos ηI

((cosh 2φI

cos 2φR−1

)cos ηI

+i

(√ϑ tan 2φR sinh(ηR + σ) − sinh 2φI

cos 2φRsin ηI

))], j = 1, 2, (21)

where the parameters ηR, ηI, pR, pI, R, I and ϑ are the same as that of the Manakovsystem. The main difference here is that the expression �1 should be replaced by a newparameter �2. One can see that the four-wave mixing parameter b is included in thequantity �2 which in turn differentiates the CGNLS case from the Manakov solution



profile as shown subsequently. Equation (21) constitutes the GB solution for the CGNLSeq. (19). Figures 3a and 3b illustrate the behaviour of this solution, which is periodic bothin space and time.

The GB solution of CGNLS system, unlike the Manakov equation, contains the four-wave mixing parameter b. So the intention here is to investigate how this parameterinfluences the GB profile. In particular, we wish to investigate whether the number ofpeaks increases or decreases and if the direction gets changed or not when the parameterb is varied. We can confirm the results analytically. To do so, we identify two successivemaximum points of the GB solution and show that the distance between these two maxi-mum points is proportional to the real part of the four-wave mixing parameter b, i.e., Reb. We find that at

(x, t) =(

π

pI+ I

pI

(pRπ + pIσ

pIR − pRI

),

pRπ + pIσ

pIR − pRI

), (22)

the GB solution has a maximum. The next maximum occurs at

(x, t) =(

3π

pI+ I

pI

(3pRπ + pIσ

pIR − pRI

),

3pRπ + pIσ

pIR − pRI

). (23)

The distance between these two maximum points along the x direction is

d(x) = 2π

pI+ I

pI

(2pRπ

pIR − pRI

)(24)

while the time interval between the two peaks is

d(t) = 2pRπ

pIR − pRI. (25)

From the expressions d(t) and d(x), it is clear that the distance and time interval betweenthe two adjacent peaks are inversely proportional to the square root of Re b. Hence, if thevalue of Re b is increased, the distance between peaks decreases and so the number ofpeaks increases. We have also derived an expression for the angle ϕ between the x and t

axes as

cos ϕ = π

pI

(pIR − pRI

pRπ + pIσ

)+ I

pI.

(a) (b)

Figure 3. General breather profiles of q1 for τ1 = 2, τ2 = 0.5, a = 1, c = 1,φR = 0.5, φI = −0.4, η0

I = 1, η0R = 1.5, k = 0.3 with different four-wave mixing

parameters (a) b = 0.5 + i, (b) b = 2 + i. Similar profile occurs for q2 also (notshown here).



This expression reveals that the propagation direction of the profile also depends on thereal part of the four-wave mixing parameter b. The peak increase and the manner ofdirection changes are illustrated in figure 3.

3.2 RW solution

Restricting the parameters η2 and φ2 suitably and implementing a Taylor expansion atε → 0, we can obtain RW solution from the GB solution (the procedure is similar to theone given for Manakov system). The final form reads as

qj = τj eiθ

(1 − 4 + 16i�2t

1 + 4�2(x − 2kt)2 + 16�22t

2

), j = 1, 2, (26)

where �2 is defined earlier. The solution (26) is localized both in space and time. Inthis too, the solution depends on the parameter Re b. One can show that the full-width athalf-maximum of RW solution φw has the form

φw = 3√aτ 2

1 + cτ 22 + (b + b∗)τ1τ2

(27)

which is inversely proportional to the square root of the real value of the four-wave mixingparameter b. In other words, when the value of Re b increases the width of the pulse getsdecreased which is demonstrated in figure 4.

3.3 AB from RW

To derive AB from RW solution we factorize the RW solution (26) in the form

qj =τj exp(i(kx−(k2−2�2)t))

(1+ 1

Q1+R1

)(1 + 1

Q1 − R1

), j = 1, 2,

(28)

where

Q1 = 2i�2t, R1 = 1

2

√1 + 4�2(x − 2kt)2,

(a) (b)

Figure 4. RW profile of q1 for τ1 = 2, τ2 = 1, k = 0, a = 0.8, c = 0.8 with differentfour-wave mixing parameters (a) b = 0.5 + i, (b) b = 5 + i. Similar profile occursfor q2 also (not shown here).



and generalize the outcome in terms of imbricate series and then rewrite the resultantexpressions in terms of trigonometric functions to obtain an equation as in (7). Sub-stituting this expression in (19) and solving the resultant system of equations, we findexpressions for σ , v(x) and α as in Manakov system with the exception that �1 shouldbe replaced by �2. By substituting the expressions for σ , α and v(x) in the general form(7) and rewriting the resultant expressions suitably, we end up with the AB solution of theform

qj = τj (1 + π2ϒ2) exp(i(2(aτ 21 + cτ 2

2 +(b+b∗)τ1τ2)(1 + π2ϒ2)2t + φ))

×⎛⎝1 − (2πϒ)(πϒ cosh 2παt + i sinh 2παt)

(1 + π2ϒ2)(cosh 2παt − 1√1+π2ϒ2 cos(

√2π2αϒx + v0))

⎞⎠ ,

(29)

where j = 1, 2. A typical AB solution for a suitable set of parametric values is shownin figures 5a and 5b. In these figures one can see that when Re b increases, the widthbetween the peaks in the AB profile decreases and number of peaks increases.

3.4 MS from RW

To derive Ma soliton from RW solution we rewrite the RW solution (26) in a slightlydifferent factorized form as

qj =τj exp(i(kx−(k2−2�2))

(1+ i

Q2 + R2

) (1+ i

Q2 − R2

), j = 1, 2,

(30)

where

Q2 = −2�2t, R2 = i

2

√1 + 4�2(x − 2kt)2.

We rewrite eq. (30) in terms of more general trigonometric forms as in eq. (11). Com-bining this expression in (19) and solving the resultant system of equations, after a verylengthy calculation, we find the same expressions for ζ , κ , �(x) as in the Manakov case

(a) (b)

Figure 5. Akhmediev breather profile of p for τ1 = 2, τ2 = 0.5, a = 1, c = 1,ϒ = 1, v0 = 0 with different four-wave mixing parameters (a) b = 0.2 + i,(b) b = 3.5 + i. Similar profile occurs for q2 also (not shown here).



with the only exception that �1 should be replaced by �2. With these expressions, thegeneral form (11) now becomes

qj = τj (1 − π2h2) exp(i(2�2(1 − π2h2)2t + φ))

×(

1+ (2πh)(πh cos 2πκt − i sin 2πκt)

(1−π2h2)(cos 2πκt−(1/√

1−π2h2) cosh(√−2π2κhx+v0))

),

(31)

which is the same as the Ma breather solution. This solution is periodic in the temporaldirection and localized in space. The Ma breather solution of CGNLS equations for a setof parametric values is shown in figures 6a and 6b. We can observe from these figuresthat when Re b increases the width between the peaks in the MS profile decreases and sothe number of peaks increases.

3.5 GB from RW

Now we construct imbricate series of GB solution from RW. We first rewrite the absolutesquare of the RW solution (26) as the second derivative of a logarithimic function:

|qj |2 = τ 2j − τ 2

j

2�2

∂2

∂x2ln

(1

S+× 1

S−

), j = 1, 2, (32)

where

S± =(

1

2

√1 + 4�2)(x − 2kt)2 ± 2i(�2)t

)2

.

We then consider eq. (32) in a more general form in order to superpose RWs in both spaceand time directions, i.e.,

|qj |2 = τ 2j − τ 2

j

2�2

∂2

∂x2ln

( ∞∑n=−∞

1

H 2+×

∞∑n=−∞

1

H 2−

), j = 1, 2, (33)

(a) (b)

Figure 6. Ma breather profile of q1 for τ1 = 2, τ2 = 0.5, a = 1, c = 1, h = 0.8with different four-wave mixing parameters (a) b = 0.5 + i, (b) b = 2.5 + i. Similarprofile occurs for q2 also (not shown here).



where H± is as defined in (14), φ(x, t) and ψ(x, t) are arbitrary functions of x and t

which need to be determined. Using the trigonometric identities, the above expressionscan be brought to the form

|qj |2 = τ 2j + τ 2

j

�2

∂2

∂x2ln[cosh 2πψ − cos 2πφ]. (34)

Now we rewrite the GB solution (21) in the form

|qj |2 = τ 2j + τ 2

j

�2

∂2

∂x2ln[√ϑ cosh(pRx−Rt+σ)−cos(pIx−It+θ)], (35)

where j = 1, 2, σ = η0R + 1

2 ln ϑ and θ = η0I + π .

Comparing the expressions (34) and (35), we find that the forms of ψ and φ are exactlythe same as given in eqs (17) and (18). Substituting one of these two cases in (33) weobtain the imbricate series form of GB solution.

4. Conclusion

RW solutions have been studied intensively in different physical contexts. Few appli-cations have also been proposed in optical fibres, e.g., to enhance supercontinuumgeneration. Conventionally the RW solutions can be constructed from the breather solu-tions. In this work, we have constructed a class of breather solutions, like GB, AB, MSfrom the RW solutions for two important nonlinear evolution equations, namely CNLSand CGNLS equations. The main difference between these two equations is: CGNLSthe system contains four-wave mixing parameters and the breathers and RW profiles ofCGNLS system strongly depend on the parameter b. When we increase the value of Re b

in the CGNLS system, the number of peaks in the breather profile increases and the widthof each peak gets shrunk. One notable behaviour which we have observed in GB is thatthe direction of this profile also changes when we increase the value of Re b. As far as theRW profile is concerned the width of the peak becomes very thin when we increase thevalue of b. We have also proved all these observations with exact mathematical expres-sions. Our study on the coupled NLSEs can be useful in the study of RWs in birefringentoptical fibres, multicomponent Bose–Einstein condensates, multicomponent plasmas, etc.

Acknowledgements

MS and ML wish to thank the organizers of the PNLD-2013 meeting for their excellenthospitality. NVP wishes to thank the University Grants Commission (UGC-RFSMS),Government of India, for providing a Research Fellowship. The work of MS forms partof a research project sponsored by NBHM, Government of India and while the work ofML forms part of an IRHPA project and a Ramanna Fellowship project of ML, sponsoredby the Department of Science and Technology (DST), Government of India. ML alsoacknowledges the financial support under a DAE Raja Ramanna Fellowship.



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