EPJ manuscript No.(will be inserted by the editor)

Rogue waves on the double periodic background in Hirotaequation

N. Sinthuja1, K. Manikandan1 and M. Senthilvelan1, a

Department of Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620024, Tamil Nadu, India.

the date of receipt and acceptance should be inserted later

Abstract. We construct rogue wave solutions on the double periodic background for the Hirota equationthrough one fold Darboux transformation formula. We consider two types of double periodic solutions asseed solutions. We identify the squared eigenfunctions and eigenvalues that appear in the one fold Darbouxtransformation formula through an algebraic method with two eigenvalues. We then construct the desiredsolution in two steps. In the first step, we create double periodic waves as the background. In the secondstep, we build rogue wave solution on the top of this double periodic solution. We present the localizedstructures for two different values of the arbitrary parameter each one with two different sets of eigenvalues.

PACS. XX.XX.XX No PACS code given

1 Inroduction

The nonlinear Schrodinger (NLS) equation describes thewave phenomena in the ocean [1,2,3,4] and the propaga-tion of optical pulse in optical fiber [5,6,7,8,9]. Consider-ing the external factors such as depth of the sea, bottomfriction and viscosity in the ocean and the femtosecondpulse propagation in fibers, several higher order NLS equa-tions have been introduced in the literature [10,11,12,13,14,15]. One such higher-order NLS equation is the Hirotaequation. In dimensionless form, the Hirota equation reads[15],

iψt + α2(ψxx + 2|ψ|2ψ)− iα3(ψxxx + 6|ψ|2ψx) = 0, (1)

where the function ψ = ψ(x, t) denotes the complex waveenvelope, α2 and α3 are arbitrary parameters. The sub-scripts t and x stand for partial derivatives with respect totemporal and spatial variables, respectively. When α2 =1, α3 = 0, Eq. (1) reduces to the standard NLS equation.During the past two decades, several works have been de-voted to construct and analyze the characteristics of lo-calized solutions in the Hirota equation [16,17,18,19,20,21]. Multisolitons, breathers and rogue wave (RW) solu-tions have been constructed for the Hirota equation usingdifferent techniques, see for example Refs. [17,18,22,23].In these studies, RW solutions have been constructed onlyon the constant wave background.

In nature, RWs can arise from a variety of backgrounds,including constant background, multi-soliton backgroundand periodic wave background [24,25,26,27,28]. Recently,interest has been shown to construct RW solutions on the

a e-mail:velan@cnld.bdu.ac.in

periodic background. Initially, the occurrence of RWs onperiodic background has been brought out only throughnumerical schemes. For example, computations of RWs onthe periodic background were performed in [29], where theauthors have computed the solutions of Zakharov-Shabatspectral problem only numerically. Similarly, RWs on thedouble-periodic background were constructed in [30] butonly via a numerical scheme. The resulting wave patternsexhibit periodicity in both space and time. Differing fromthese, very recently, Chen and his collaborators have re-ported an algorithm to construct RW solutions on theperiodic wave background for the focusing NLS equation[31]. The same authors have also derived RW solutionson the double-periodic background for the focusing NLSequation using an effective algebraic method [32]. Subse-quently, Peng et al have studied the characteristics of RWson the periodic background for the Hirota equation [33].However, the authors have reported RWs only for a limit-ing case. The studies on RWs on the periodic backgroundcan be found potential applications both in experimentand theory in their relative fields [34,35].

In this work, we construct RW solutions on the double-periodic background for the Hirota equation. We utilizeone fold Darboux transformation formula and the methodof nonlinearization of Lax pairs to derive these solutions.To begin, we assume a constraint between the potentialand squared eigenfunctions of the Lax system for the Hi-rota equation. Using this constraint we identify Hamiltoni-ans associated with the spatial and temporal parts of theLax pair. We notice that the Hamiltonian derived fromspatial part of the Lax pair of Hirota equation matcheswith the one constructed for the NLS equation. This is be-cause the spatial part of the spectral problem of NLS equa-

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2 N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation

tion and the Hirota equation are one and the same. How-ever, the Hamiltonian associated with the time part of theHirota equation differs from NLS counterpart. Since thespatial part is same, we obtain the same sets of differentialconstraints which connect the potential with eigenvaluesthat are given in Ref. [32]. We construct periodic eigen-functions by solving the differential constraints in termsof known double periodic solutions of the Hirota equation.Substituting these periodic eigenfunctions and the identi-fied set of eigenvalues in the one-fold Darboux transfor-mation formula, we create the double-periodic wave back-ground. We then construct another linearly independentsolution of the same Lax equations for the same set ofeigenvalues which in turn creates the RW solution on topof the double-periodic wave background. Since the timeevolution part in the Lax pair of Hirota equation differsfrom the NLS equation the second independent solutionwhich we construct for the Hirota equation differs fromthe NLS equation. We present surface plots of |r| for twodifferent values of α3 each one with two different sets ofeigenvalues. We find that the amplitude of RWs increasesin the order of eigenvalues, such as λ2 < λ1 < λ3 in the(x− t) plane. When we vary the value of the arbitrary pa-rameter α3, we notice that the amplitude of RWs remainsthe same whereas the double-periodic background wavesare localized in different positions in the (x− t) plane.

We organize our presentation as follows. In Sec. 2,to begin, we present the Lax pair and one-fold Darbouxtransformation formula for the Hirota equation and twotypes of double periodic solutions of the Hirota equation.We also briefly recall the method of constructing RW so-lution on the double periodic background. In Sec. 3, weapply the algebraic method with two eigenvalues to Hi-rota equation and construct the RWs on the desired back-ground. In Sec. 4, we summarize our work.

2 Lax pair and Darboux transformation

The complete integrability of the nonlinear system (1) isguaranteed by the presence of Lax pair. Once the Lax pairis known, one can efficiently exploit the Darboux transfor-mation technique to construct various localized solutions[18,19,20,21] for the concerned equation. With the helpof Darboux transformation method, we can construct anew solution ψ(x, t) = r(x, t) by providing a suitable seedsolution ψ(x, t) = r(x, t). The solution ψ(x, t) = r(x, t)to the Hirota Eq. (1) comes out from the compatibilitycondition (ϕxt = ϕtx) of the pair of linear equations on ϕ.The Lax pair for the Hirota equation is given by [23,33]

ϕx = U(λ, r)ϕ, U(λ, r) =

(λ r−r −λ

), (2)

and

ϕt = V (λ, r)ϕ,

V (λ, r) = λ3(

4α3 00 −4α3

)+ λ2

(2iα2 4α3r−4α3r −2iα2

)+ λ

(2α3|r|2 2α3rx + 2iα2r

2α3rx − 2iα2r −2α3|r|2)

+

(iα2|r|2 − α3(rrx − rrx) iα2rx + α3(rxx + 2|r|2r)iα2rx − α3(rxx + 2|r|2r) −iα2|r|2 + α3(rrx − rrx)

),

(3)

where r is the potential and r denotes the complex con-jugate of r. One can easily verify that the compatibilitycondition Ut − Vx + [U, V ] = 0 gives rise to Eq. (1).

The one-fold Darboux transformation of the Eq. (1)reads [31]

r(x, t) = r(x, t) +2(λ1 + λ1)f1g1|f1|2 + |g1|2

, (4)

where ϕ = (f1(x, t), g1(x, t))T - a non-zero solution ofthe Lax pair (2) and (3) for a fixed eigenvalue λ = λ1and r(x, t) and r(x, t) represents the seed and first iter-ated solutions of Eq. (1), respectively. With the chosenseed solution, the linear Eqs. (2) and (3) can be solvedin a number of ways. For example, it is trivial to inte-grate the Eqs. (2) and (3) with the plane wave background(r(x, t) = ei(kx−ωt)) and generate the first-order RW solu-tion (r(x, t)). However, it is highly nontrivial to integratethe Eqs. (2) and (3) with double periodic solutions. So onehas to adopt a suitable methodology to integrate themwith the genus-2 solutions as background.

Very recently double periodic solutions for the Hirotaequation with free real parameters have been constructedin Ref. [15]. In our present study, we consider two types ofdouble-periodic solutions which are given by the followingrational functions of Jacobian elliptic functions sn, cn, anddn in [15]:

(i) r(x, t) = k

√1 + ksn(Mt, k)− iC(x+ vt)cn(Mt, k)√

1 + k − C(x+ vt)dn(Mt, k)eibt,

(5a)

where the function C(x+ vt) is given by

C(x+ vt) =cn(√

1 + kx, κ)

dn(√

1 + kx, κ), κ =

√1− k1 + k

. (5b)

(ii) r(x, t) =ksn(Mt/k, k)− iS(x+ vt)dn(Mt/k, k)

k(1− S(x+ vt)cn(Mt/k, k))eibt,

(6a)

where the function S(x+ vt) is given by

S(x+ vt) =

√k

1 + kcn

(√2

kx, κ

), κ =

√1− k

2,

(6b)

N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation 3

(a) (b)

(c) (d)

Fig. 1. Double periodic profile of (5) for k = 0.9 with (a) α3 = 0.1667 and (b) α3 = 0.85 and double periodic profile of (6) fork = 0.9 with (c) α3 = 0.1667 and (d) α3 = 0.5.

and k ∈ (0, 1), M = 2α2, v = 4α3, b = 2α2, α2 = 1 andα3 is the arbitrary constant. It follows from (5) and (6)that |r(x, t)|=|r(x+L, t)|=|r(x, t+T )|, (x, t) ∈ R2, with

the fundamental periods L =4K(κ)√

1 + kand T = 2K(k) for

(5) and L = 2√

2K(κ) and T = 4K(k) for (6), whereK(k) represents the complete elliptic integral of the firstkind with the elliptic modulus parameter κ. In Fig. 1,we display the qualitative profiles of double-periodic wavecoming out from the solutions (5) and (6) for the modulusk = 0.9 and α2 = 1 with two different values of α3. In Figs.1(a)-1(b), double-periodic waves of (5) are shown for α3 =1/6 (or) 0.1667 and α3 = 0.85, respectively. Figures 1(c)and 1(d) represent the double-periodic profiles of (6) forα3 = 0.1667 and α3 = 0.5, respectively. From these plots,we observe that double-periodic wave is more localized forlow values of α3 when compared to the higher values ofα3 which can be visualized from Figs. 1(a) and 1(c). Wealso notice that the orientation of double-periodic wavechanges when we vary the value of the arbitrary parameterα3.

We intend to construct RWs on the double periodicprofiles given in (5) and (6). However, these solutions are

travelling wave solutions. We invoke the method of non-linearization of Lax pairs [36,37,38,39] in order to iden-tify the eigenvalues for which these solutions appear. Inthis procedure one usually identifies the Hamiltonian as-sociated with the spatial and temporal parts of the Laxpair by introducing a constraint between the potentialr(x, t) and the eigenfunctions. In our case, the constraintreads r = 〈f, f〉 := f21 + g21 + f22 + g22 , where the functions(f1, g1, f2, g2) are nothing but the solutions of the Lax pairEqs. (2) and (3) and bar denotes the complex conjugateof the respective function.

After introducing this constraint we can proceed toconstruct a set of ordinary differential equations (ODEs)with r(x, t) as the dependent variable. The order and num-ber of these ODEs normally extends depending upon thenumber of unknowns to be determined. In our case, wehave four unknowns, namely (f1, g1, f2, g2) and so we se-quentially go upto fourth order ODE. The last differentialequation turns out to be the fourth-order Lax-Novikovequation in the hierarchy of stationary NLS equation. Thefourth-order complex-valued Lax-Novikov equation is in-tegrable with two complex-valued constants of motion [32].From these integrals of motion, we identify a set of admis-sible eigenvalues λ1 and λ2 which are symmetric about

4 N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation

the imaginary axis. Once the eigenvalues are fixed, sub-stituting the double periodic solution and the obtainedsquared eigenfunctions in the one-fold Darboux transfor-mation formula (4), we can create double periodic solutionas the background for the considered equation. The result-ing solution is periodic both in space and time. We thenproceed to construct RWs on top of this double periodicwaves. To achieve this we consider a second linearly in-dependent solution for the Lax system (2) and (3) withthe same set of eigenvalues and eigenfunctions with an un-known function. We determine this unknown function bysubstituting the considered second independent solutionback in the Lax system and integrating the underlyingequations. Substituting back the second solution in theone-fold Darboux transformation formula (4), we obtainthe RW solution on top of the double-periodic wave back-ground.

3 Demonstration on Hirota equation

In this section, we demonstrate the method of construct-ing RW solutions on the double periodic background forthe Hirota equation. In Ref. [32], the authors have con-structed the rogue wave solutions on the double periodicbackground for the focusing NLS equation. We adopt thesame procedure and derive the RW solution on the dou-ble periodic background to Eq. (1). While implement-ing this procedure, we observe that Hirota equation alsoyields the same results upto section III in Ref. [32] withα2 = 1. While performing the procedure of nonlineariza-tion of Lax pair for the Eqs. (2) and (3) with the constraintr = f21 + g21 + f22 + g22 in the time part, we come acrosscertain differences. For instance, substituting the assumedconstraint into Eqs. (2) and (3), we obtain the followingfinite-dimensional Hamiltonians, namely

dfjdx

=∂H0

∂gj,

dgjdx

= −∂H0

∂fj, j = 1, 2, (7)

where

H0 = 〈Λf,g〉+1

2〈f,f〉〈g,g〉, (8)

and

dfjdt

=∂H1

∂gj,

dgjdt

= −∂H1

∂fj, j = 1, 2. (9)

where

H1 =1

4[i〈f,g〉+

2

3〈f,f〉2〈g,g〉2 + i

1

2〈f,f〉〈Λg,g〉

+ i1

2〈Λf,f〉〈g,g〉+

4

3〈Λf,g〉〈f,f〉〈g,g〉+ 2i〈Λ2f,g〉

+2

3〈Λ2g,g〉〈f,f〉+

2

3〈Λ2f,f〉〈g,g〉+

8

3〈Λ3f,g〉

− i

3〈f,f〉〈f,g3〉 − i

3〈g,g〉〈f3,g〉 − 1

6〈f,f〉2〈g2,g2〉

− 1

6〈g,g〉〈f2, f2〉 − 4

9〈f,f〉〈Λf,g3〉 − 4

9〈g,g〉〈Λf3,g〉].

(10)

In Eqs. (8) and (10), the notations f, g and Λ denotef=(f1, f2, g1, g2)T , g=(g1, g2,−f1,−f2)T andΛ = diag(λ1, λ2,−λ1,−λ2), respectively. One may noticethat the Hamiltonian (10) which comes from the time evo-lution part of Lax pair of the Hirota equation differs fromthe NLS equation [32].

The squared eigenfunctions f21 and g21 which appear inthe Eq. (4) can be obtained by substituting the expressions(5) and (6) and their derivatives in the following equations:

f21 =λ1

4(λ1 + λ1)(λ1 − λ2)(λ1 + λ2)

×[r′′ + 2|r|2r + 4(b+ λ21)r + 2λ1r

′] ,g21 =

λ14(λ1 + λ1)(λ1 − λ2)(λ1 + λ2)

×[r′′ + 2|r|2r + 4(b+ λ21)r − 2λ1r

′] ,f1g1 =− λ1

4(λ1 + λ1)(λ1 − λ2)(λ1 + λ2)

×[r′r − rr′ + 2λ1(2b+ 2λ21 + |r|2)

]. (11)

(since the details of getting (11) have already been givenin Ref. [32] we do not reproduce them here).

Similarly we can determine the expressions of squaredeigenfunctions, namely f22 , g22 and f2g2 from the aboveequations by replacing λ1 by λ2 and λ2 by λ1. We con-sider two different sets of three admissible pairs of eigen-values among ±λ1, ±λ2 and ±λ3 which were found inRef. [32] for the family of double-periodic wave solutionsgiven in Eqs. (5) and (6). Each pair of eigenvalues canbe taken in place of λ1 and λ2. In the first set, all theeigenvalues are considered to be real, that is λ1 = ±√τ1,λ2 = ±√τ2, and λ3 = ±√τ3 where τ1 varies from 0 to 1,τ2 = τ3− τ1 and τ3 = 1. In the second set, it is consideredthat one of the eigenvalues λ1 = ±√τ1 is positive andthe other two eigenvalues are complex-conjugate, namelyλ2 = ±

√β + iγ and λ3 = ±

√β − iγ, where τ1 = 2β and

β varies from 0 to 0.5. Now, substituting the resultantexpressions coming out from (11) and considering threeadmissible pairs of eigenvalues given above in (4), we canset up the double-periodic wave pattern as background forthe Hirota Eq. (1).

To generate RWs on this double periodic background,we construct a second linearly independent solution forthe Lax pair Eqs. (2) and (3) with the same eigenvalues.The second linearly independent solution which we intendto construct should be non-periodic and grow linearly inx and t. We consider the second linearly independent so-

lution ϕ = (f1, g1)T for the Lax pair Eqs. (2) and (3) inthe form

f1 = f1η1 −2g1

|f1|2 + |g1|2, g1 = g1η1 +

2f1|f1|2 + |g1|2

,

(12)

N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation 5

(a) (b)

(c)

Fig. 2. RWs on the background of the double-periodic solution (5) for the Hirota Eq. (1) with k = 0.8 and α3 = 0.1667 forthree different eigenvalues (a) λ1 =

√τ1, (b) λ2 =

√τ2 and (c) λ3 =

√τ3.

where η1 is the unknown function to be determined. TheWronskian between the two solutions (4) and (12) is nonzero,

f1g1 − f1g1 =f1

(g1η1 +

2f1|f1|2 + |g1|2

)− g1

(f1η1 −

2g1|f1|2 + |g1|2

)= 2, (13)

which in turn proves that the second solution is linearlyindependent from the first one.

Substituting (12) into (2) and utilizing (5) for ϕ =

(f1, g1)T , we obtain the following first-order partial differ-ential equation for the unknown function η1, that is

∂η1∂x

= W1 :=− 4(λ1 + λ1)f1g1(|f1|2 + |g1|2

)2 . (14)

Similarly, inserting (12) into (3) and utilizing (3) for

ϕ = (f1, g1)T , we obtain another first-order partial differ-

ential equation for η1 in time derivative, namely

∂η1∂t

= W2 :=2(λ1 + λ1)S1f

21

(|f1|2 + |g1|2)2+

2(λ1 + λ1)S2g21

(|f1|2 + |g1|2)2

− 4(λ1 + λ1)S3f1g1(|f1|2 + |g1|2)2

, (15)

where S1 = 2rxα3+(i+4α3(λ1−λ1))r, S2 = −2rxα3+(i+4α3(λ1−λ1))r and S3 = 2|r|2α3+4α3λ

21+(i−4α3λ1)(λ1−

λ1). The first-order equations (14) and (15) are compat-ible with each other (W1t = W2x). Upon integrating theexpressions (14) and (15), we obtain

η1(x, t) =

∫ x

x0

W1(x′, t)dx′ +

∫ t

t0

W2(x0, t′)dt′, (16)

where (x0, t0) is arbitrarily fixed. It is very difficult tointegrate Eq. (16) and obtain an explicit expression forη1(x, t) analytically. So we evaluate the expression (16)numerically by Newton-Raphson method.

Substituting the double periodic solutions r(x, t) (givenin (5) and (6)) and the eigenfunctions of the second so-

lution ϕ = (f1, g1)T of the linear equations (2)-(3) with

6 N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation

(a) (b)

(c)

Fig. 3. RWs on the background of the double-periodic solution (5) for the Hirota Eq. (1) with k = 0.8 and α3 = 0.85 for threedifferent eigenvalues (a) λ1 =

√τ1, (b) λ2 =

√τ2 and (c) λ3 =

√τ3.

λ = λ1 in the one-fold Darboux transformation formula(4), we obtain a new form of solution (RW solution on thedouble periodic background) to the Hirota Eq. (1) in theform

r(x, t) =r(x, t) +2(λ1 + λ1)f1 ¯g1

|f1|2 + |g1|2, (17)

where f1 and g1 are given in (12). We note here that thissecond solution differs from the one reported for the NLSequation.

In Figs. 2(a)-(c), we show the surface plots of |r| ofRWs on the double periodic background using the solution(5) with k = 0.8 and α3 = 1/6 (or) 0.1667 for three realeigenvalues, namely λ1 =

√τ1, λ2 =

√τ2 and λ3 =

√τ3

with τ1 = 0.8, τ3 = 1 and τ2 = τ3−τ1. In this figure, in allthe cases, RW attains their maximum amplitude at theirorigin, that is (x0, t0) = (0, 0). The amplitude of RWs isdifferent in each case and is found to be |r| ≈ 4.13, 3.236and 4.342 for the eigenvalues λ1, λ2 and λ3, respectively.The outcome is shown in Figs. 2(a), 2(b) and 2(c). Whenwe increase the value of the arbitrary parameter α3 to0.85, the amplitude of RWs remains the same as in the caseof α3 = 0.1667. But we observe that the double-periodic

background waves are localized in different positions inthe (x − t) plane which can be visualized from the Figs.3(a)-(c). For k = 0.95 and α3 = 0.1667, the surface plotsof |r| of RWs on the double periodic background usingthe solution (5) with three real eigenvalues, namely λ1 =√τ1, λ2 =

√τ2 and λ3 =

√τ3 with τ1 = 0.655, τ3 =

1 and τ2 = τ3 − τ1 are displayed in Figs. 4(a)-(c). Forthis set of eigenvalues also, we observe that the amplitudeof RWs exhibits a maximum value at their origins. Theamplitudes of RWs for the eigenvalues λ1, λ2 and λ3 are|r| ≈ 4.015, 3.571 and 4.397 which are shown in Figs. 4(a),4(b) and 4(c), respectively. From these observations, weinfer that the amplitude of RWs increases in the orderof eigenvalues as λ2 < λ1 < λ3 in the x − t plane. Theamplitude of RWs for the eigenvalue λ3 is higher thanthat of the other two eigenvalues (λ1 and λ2).

Figure 5 represents the surface plots of |r| of RWs onthe double periodic background using the solution (6) withk = 0.9 for (a) λ1 =

√τ1, τ1 = 2β and (b) λ2 =

√β + iγ,

β = 0.45 and γ = 0.217945. From Figs. 5(a) and 5(b),one can visualize the amplitude of RWs attains the max-imum value |r| ≈ 4.35 and 3.841 for the real eigenvalueλ1 and the complex eigenvalue λ2, respectively. The sur-

N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation 7

(a) (b)

(c)

Fig. 4. RWs on the background of the double-periodic solution (5) for the Hirota Eq. (1) with k = 0.95 and α3 = 0.1667 andfor eigenvalues (a) λ1 =

√τ1, (b) λ2 =

√τ2 and (c) λ3 =

√τ3.

(a) (b)

Fig. 5. RWs on the background of the double-periodic solution (6) for the Hirota Eq. (1) with k = 0.9 and α3 = 0.1667 forcomplex conjugate eigenvalues (a) λ1 =

√τ1 and (b) λ2 =

√β + iγ.

face plots of |r| for the same set of eigenvalues but for thechoice of the arbitrary parameter, α3 = 0.5 are presentedin Figs. 6(a) and 6(b). Here we notice that the localizationof double-periodic background waves occurs in different

orientations whereas the amplitudes of RWs remain thesame as in the case of α3 = 0.1667. The RWs on the dou-ble periodic background which are shown in Figs. 7(a)-(b)are drawn for the same set of values as given in Fig. 5 with

8 N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation

(a) (b)

Fig. 6. RWs on the background of the double-periodic solution (6) for the Hirota Eq. (1) with k = 0.9 and α3 = 0.5 for complexconjugate eigenvalues (a) λ1 =

√τ1 and (b) λ2 =

√β + iγ.

(a) (b)

Fig. 7. RWs on the background of the double-periodic solution (6) for the Hirota Eq. (1) with k = 0.95 and α3 = 0.1667 forcomplex conjugate eigenvalues (a) λ1 =

√τ1 and (b) λ2 =

√β + iγ.

k = 0.95 and α3 = 0.1667. The maximum amplitudes ofRWs for the eigenvalues λ1 and λ2 are |r| ≈ 4.025 and3.811. This is shown in Figs. 7(a) and 7(b). We concludethat the amplitude of RWs for the case of real eigenvalue(λ1) is higher than that of the complex eigenvalue (λ2).

4 Summary

In this work, we have constructed RW solutions on thedouble-periodic background for the Hirota equation in analgebraic way. One of the efficient tools to derive RWsolution for the given nonlinear partial differential equa-tion is the Darboux transformation method. Through thismethod, one can construct the desired solution by appro-priately choosing the seed solution. However, it is very dif-ficult to integrate the underlying equations arising fromLax pair when we consider double periodic solution asseed solution. To circumvent this difficulty, we consideredanother procedure through which the necessary expres-

sions for the squared eigenfunctions and the eigenvaluesthat appear in the one fold Darboux transformation for-mula can be identified. The procedure which we brought-in to identify the eigenfunctions and eigenvalues is themethod of nonlinearization of Lax pairs introduced longago in a different context in the nonlinear dynamics lit-erature. With the help of this method and the spectraltheory of Lax pairs, we have captured the eigenvalues andthe squared eigenfunctions which appear in the one-foldDarboux transformation formula. With the obtained ex-pressions, we have created the double periodic waves asbackground. In the second step, we have created the RWon top of this background. We have presented the localizedstructures for two different values of the arbitrary param-eter each one with two sets of eigenvalues. Our studiesreveal that the amplitudes of RWs increase in the orderof eigenvalues as λ2 < λ1 < λ3 in the (x − t) plane. Ourresults also show that when we vary the value of the ar-bitrary parameter α3 the amplitudes of RWs remain thesame and localization of the double-periodic background

N. Sinthuja et al.: Rogue waves on the double-periodic background in Hirota equation 9

waves occur in different orientations in the (x− t) plane.The application of this approach to few other evolutionaryequations is under progress.

Acknowledgments

NS thanks the University for providing University Re-search Fellowship. KM wishes to thank the Council ofScientific and Industrial Research, Government of India,for providing the Research Associateship under the GrantNo. 03/1397/17/EMR-II. The work of MS forms part of aresearch project sponsored by National Board for HigherMathematics, Government of India, under the Grant No.02011/20/2018NBHM(R.P)/R&D 24II/15064. The authorsthank Prof. Dmitry E. Pelinovsky for helping to evaluatethe numerical part of this work.

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