Formation of rogue waves on the periodic background in a fifth-order nonlinear Schrödinger equation
FFormation of rogue waves on the periodic backgroundin a fifth-order nonlinear Schr¨odinger equation
N. Sinthuja a , K. Manikandan a , M. Senthilvelan a, a Department of Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620 024,Tamil Nadu, India
Abstract
We construct rogue wave solution of a fifth-order nonlinear Schr¨odinger equa-tion on the Jacobian elliptic function background. By augmenting Darbouxtransformation and the nonlinearization of spectral problem applicable for thefifth-order nonlinear Schr¨odinger equation, we generate rogue wave solution ontwo different periodic wave backgrounds. We analyze the obtained solution indetail for different eigenvalues and point out certain novel features of our results.
Keywords:
Fifth-order nonlinear Schr¨odinger equation; Rogue waves;Darboux transformation; Nonlinearization of Lax pair; Algebraic method withan eigenvalue.
1. Introduction
During the past two decades, both experimentalists and theoreticians havebeen attracted towards the studies on rogue waves (RWs) because of its impor-tance in their concerned fields [1, 2, 3, 4, 5, 6]. RW is traditionally defined by“ appears from nowhere and disappears without a trace” [7]. The formationof RWs can be related to the modulation instability of the wave background[8, 9, 10]. Based on the theoretical arguements it has been shown that RWs canarise on different backgrounds, including constant, multi-soliton and periodicbackgrounds [11, 12, 13, 14, 15]. On the other hand, in nature, surface of theocean and the waves that appear on it may exhibit periodic phenomena. Hence,the emergence of RWs on top of periodic wave background can be imaginedas one of the naturally happening event in the ocean. To study such a phe-nomenon, very recently, investigations have been made to construct RWs onthe periodic wave background. Initially, the existence of RWs on the periodicbackground was studied in the nonlinear Schr¨odinger (NLS) equation througha numerical scheme [16]. Later, an explicit expression of the RW solution on aperiodic background has been derived analytically for the same NLS equation
Email address: [email protected] (M. Senthilvelan)
Preprint submitted to Elsevier February 24, 2021 a r X i v : . [ n li n . PS ] F e b dn and cn periodic wave background by suitably combining the DT andthe method of nonlinearization of Lax pair. To begin using the latter method,we derive certain differential constraints which involve the potential, admissi-ble eigenvalues and the squared eigenfunctions. By appropriately solving theseconstraints we identify the eigenvalues and the squared eigenfunctions that cor-respond to an elliptic function solution. Substituting the obtained eigenvalues,eigenfunctions and seed/periodic travelling wave solution in the one-fold DTformula, we create the periodic background solution. To generate the RWs onthis periodic wave background, we construct another independent solution ofthe spectral problem for the same eigenvalues. We analyze the obtained RWson the periodic background for different parametric values. We observe that theamplitude of RWs on the periodic background diminishes (enhances) for dn ( cn )when we vary the elliptic modulus ( k ), from a lower value to a higher value. Ourresults also show that the frequency of the periodic background wave increases2n the ( x − t ) plane when we increase the value of the system parameter.We organize our work as follows: In Sec. 2, we consider the FONLS equationto construct RW solutions on the periodic wave background and also we presentthe Lax pair and one-fold Darboux transformation for the considered equation.In Sec. 3, we derive the periodic travelling wave solutions of FONLS equationand determine the eigenvalues and squared eigenfunctions by using the methodof nonlinearization of spectral problem. In Sec.4, we find the second linearlyindependent solution of the spectral problem with the same eigenvalue to achievethe desired form. We summarize our results in Sec. 5.
2. Model and one-fold Darboux Transformation
We consider a FONLS equation [27, 28], ir t + r xx + 2 | r | r − i ( (cid:15) ( r xxxxx + 10 | r | r xxx + 20¯ rr x r xx + 30 | r | r x +10( r | r x | ) x ) − r x ) = 0 , (1)which plays a crucial role in describing the dynamics of nonlinear wave prop-agation in fiber optics [24, 26] and information technology [29]. In Eq. (1), r ( x, t ) represents the complex wave envelope, x and t describes the spatial andtemporal coordinates, subscripts denote partial derivatives with respect to x and t respectively, and (cid:15) is a real constant.Equation (1) possesses a (2 ×
2) Lax pair of the form [27] ϕ x = U ( λ, r ) ϕ, U ( λ, r ) = (cid:18) λ r − ¯ r − λ (cid:19) , (2a) ϕ t = V ( λ, r ) ϕ, V ( λ, r ) = (cid:18) A BC − A (cid:19) , (2b)with A =16 λ (cid:15) + 8 λ (cid:15) | r | − λ (cid:15) ( r ¯ r x − r x ¯ r ) + 2 iλ + 2 λ(cid:15) ( r ¯ r xx + ¯ rr xx − | r x | + 3 | r | )+ λ + (cid:15) (¯ rr xxx − r ¯ r xxx + r x ¯ r xx − ¯ r x r xx + 6 | r | ¯ rr x − | r | r ¯ r x ) + i | r | , (2c) B =16 λ (cid:15)r + 8 λ (cid:15)r x + 4 λ (cid:15) ( r xx + 2 | r | r ) + 2 λ(cid:15) ( r xxx + 6 | r | r x ) + 2 iλr + (cid:15) ( r xxxx + 8 | r | r xx + 2 r ¯ r xx + 4 | r x | r + 6 r x ¯ r + 6 | r | r ) + ir x + r, (2d) C = − ¯ B, (2e)where r is the potential and ¯ r denotes the complex conjugate of r . The FONLSequation (1) emerges from the compatibility condition U t − V x + [ U, V ] = 0 ofthe above pair of the linear equations (2 a ) and (2 b ).In the literature, numerous efforts have been made to construct RW solutionsof nonlinear partial differential equations using DT technique. Let us recall theone-fold Darboux transformation [27] for the Eq. (1)ˆ r ( x, t ) = r ( x, t ) + 2( λ + ¯ λ ) f ¯ g | f | + | g | , (3)3here ϕ = ( f ( x, t ) , g ( x, t )) T is a non-zero solution of the first order Lax pairEqs. (2 a ) and (2 b ) at λ = λ , r ( x, t ) and ˆ r ( x, t ) denotes the seed and firstiterated solution of Eq. (1).
3. Periodic travelling wave solutions of (1)
We consider the following form of periodic wave solution for the function r ( x, t ), that is r ( x, t ) = R ( x, t ) e ibt , (4)with the aim that R ( x, t ) should be determined in terms of elliptic function.Here b is a real parameter. Substituting Eq. (4) into Eq. (1) and separatingreal and imaginary parts, we obtain the following two differential equations forthe function R ( x, t ), namely R xx + 2 R R − bR = 0 , (5) R t − R x − (cid:15) ( R xxxxx + 10 R R xxx + 40 RR x R xx + 30 R R x + 10 R x ) = 0 . (6)Upon integrating Eq. (5) once, we obtain R x + R − bR − d = 0 , (7)where d is the integration constant.A compatible solution for the Eqs. (5) and (6) can be obtained from (7)in terms of Jacobian elliptic functions with suitable parameters. Among thetwo types of elliptic function solution which it admits, one turns out to be thepositive-definite dn -periodic wave R ( x, t ) = √ − k dn ( x − ct, k ) , b = 2 − k , c = (2 − k ) (cid:15), d = − (1 − k ) , (8)and the other one is sign-indefinite cn -periodic wave, R ( x, t ) = k √ − k cn ( x − ct, k ) , b = 2 k − , c = (2 k − (cid:15), d = k (1 − k ) , (9)where the parameter k ∈ (0 ,
1) is the elliptic modulus. It is staightforward toverify that (8) and (9) constitutes the solution of Eq. (1).In Fig. 1 we represent the qualitative profile of dn and cn -periodic waves of(1). To draw these figures, we fix the modulus parameter k as 0 . (cid:15) value. In these figures, 1(a)-(b), we plot the solution r ( x, t ) for two differentvalues of (cid:15) . In Figs. 1(c)-(d), the cn -periodic profiles of (4) with (9) are shownfor (cid:15) = 0 .
75 and (cid:15) = 2 .
5. From these figures, we notice that while the frequencyof the periodic waves increases in the ( x − t ) plane (see Figs. 1(b) and 1(d)) andtheir amplitudes do not change when we increase the value of the parameter (cid:15) from 0 .
75 to 2 .
5. 4 a) (b)(c) (d)
Figure 1:
Periodic profile of (4) using (8) for k = 0 . (cid:15) = 0 .
75 and (b) (cid:15) = 2 . k = 0 . (cid:15) = 0 .
75 and (d) (cid:15) = 2 . In the one-fold DT formula (3), we intend to feed the elliptic function solu-tions, (8) and (9), as seed solutions. To fulfill this task we should know for whateigenvalues ( λ ) these solutions arise? Conventionally, in the DT method, theeigenvalues can be determined by substituting the considered seed solution intothe Lax pair equations in place of r ( x, t ) and solving the underlying first orderODEs (in our case four equations). However, in the present case it is very diffi-cult to integrate the Lax pair equations with the presence of elliptic functions.To overcome this difficulty we seek another technique, namely nonlinearizationof Lax pair [30, 31] and identify the eigenvalues ( λ ) which are associated withthe solutions (8) and (9).In this method, by introducing a Bargmann constraint and appropriatelyusing the Lax pairs given in (2a) and (2b), we derive two differential constraints.Comparing these two differential constraints with Eqs. (5) and (7) we determinethe admissible eigenvalues and the corresponding squared eigenfunctions.We consider the Bargmann constraint between the solution of Eq. (1) andthe squared eigenfunctions in the form r ( x, t ) = f + ¯ g , (10)5here ( f , g ) T is a non-zero solution of the linear system of Eqs. (2a) and (2b)at λ = λ . Substituting Eq. (10) into Eq. (2a), we identify the underlyingequations can be written as a finite-dimensional Hamiltonian system, that is df dx = ∂H ∂g , dg dx = − ∂H ∂f , (11)with H is given by H = λ f g + ¯ λ ¯ f ¯ g + 12 ( f + ¯ g )( ¯ f + g ) . (12)Differentiating Eq. (10) with respect to x and using the Eqs. (2a) and (12), weobtain the following first order ODE, that is r x + 2 irF = 2( λ f − ¯ λ ¯ g ) , (13)where F = i ( f g − ¯ f ¯ g ).Differentiating Eq. (13) with respect to x , and using (12) in it, we obtain r xx + 2 i ( F + iλ − i ¯ λ ) r x + 2 | r | r = 4( | λ | + H + iF ( λ − ¯ λ )) r, (14)where F = λ f g + ¯ λ ¯ f ¯ g + ( | f | + | g | ) . It is straightforward to verifythat dF dx = 0 and dF dx = 0. These two constants of motion ( F and F ) can beconnected to H given in (12) through the relation H = F − F .The second-order differential Eq. (14) admits the following Lax representa-tion [17], that is ddx J ( λ ) = [ U ( λ, r ) , J ( λ )] , λ ∈ C , (15)where U ( λ, r ) and r ( x, t ) are given in (2a) and (10) respectively, and the matrix J ( λ ) is defined by, J ( λ ) = (cid:20) J ( λ ) J ( λ ) J ( − λ ) − J ( − λ ) (cid:21) , (16)in which the components take the form J ( λ ) = 1 − (cid:16) f g λ − λ − ¯ f ¯ g λ +¯ λ (cid:17) and J ( λ ) = (cid:16) f λ − λ − ¯ g λ +¯ λ (cid:17) . With the help of Eqs. (10) and (13), the functions J ( λ ) and J ( λ ) can be identified in the form J ( λ ) = ( λ − λ )( λ + ¯ λ ) + iF ( λ − λ + ¯ λ ) + ( F + | r | ) − F ( λ − λ )( λ + ¯ λ ) ,J ( λ ) = ( λ − λ + ¯ λ + iF ) r + r x ( λ − λ )( λ + ¯ λ ) . (17)One can verify that the upper right component ( J ) that appear in thematrix Eq. (15), yields the same second-order ODE (14).6e consider the eigenvalue in the form λ = ξ + iη , where ξ and η are tworeal parameters. With this choice Eq. (14) can be rewritten in the form r xx + 2 i ( F − η ) r x + 2 | r | r = 4( ξ + η + F − F + 2 ηF ) r. (18)While evaluating the determinant of J ( λ ), we notice that the determinantcontains two simple poles which can be expressed in terms of F and F . Im-posing the constraint J ( − λ ) = J ( λ ) and using the exact forms given in Eq.(17) for J ( λ ) and J ( λ ) we obtain the following expression for detJ ( λ ):det J ( λ ) = − (cid:104) ( λ − λ )( λ + ¯ λ ) + iF ( λ − λ + ¯ λ ) + ( F + | r | ) − F ( λ − λ )( λ + ¯ λ ) (cid:105) + [( λ − λ + ¯ λ + iF ) r + r x ][( λ − λ + ¯ λ + iF )¯ r − ¯ r x ]( λ − λ ) ( λ + ¯ λ ) . (19)The det J ( λ ) has double poles, one at λ = λ and another at λ = − ¯ λ .After removing the double poles at λ = λ or λ = − ¯ λ and making appropriatesimplifications, we identify the following two differential constraints, namely¯ rr x − r ¯ r x =2 i (2 η − F ) | r | + 2 iF ( F + 2 ηF − F ) , (20) | r x | + | r | =4( ξ + η − F + F − ηF ) | r | + 4 ξ F − ( F + 2 ηF − F )(5 F − ηF − F ) . (21)Substituting the expression r ( x, t ) given in Eq. (4) in (20) the left-hand sidebecomes zero for both the periodic waves. The right-hand side of Eq. (20) yields F = 2 η, η ( F − η ) = 0 . (22)Comparing the Eqs. (18) and (21) with the expressions (5) and (7), we can fixthe parameters ( b , c and d ) in the form b = 4( ξ − η + F ) , c = 4 (cid:15) ( ξ − η + F ) ,d = 4[4 ξ η − ( F − η )( F − η )] . (23)Equation (22) yields two conditions, namely (i) η = 0 and (ii) F = 4 η ( η (cid:54) = 0).In the first choice, we find F = 0 and hence the parameters in Eq. (23) arerestricted to b = 4( ξ + F ) , c = 4 (cid:15) ( ξ + F ) , d = − F . (24)For the second choice η (cid:54) = 0, the parameters b, c and d take the form b = 4( ξ − η ) , c = 4 (cid:15) ( ξ − η ) , d = 16 ξ η . (25)We notice that the parameter d has a sign difference in these two caseswhereas the parameters b and c take the same sign. Upon comparing the values7f the parameters which we found through nonlinearization of Lax pair technique(Eqs. (24) and (25)) with the ones obtained through travelling wave reduction(Eqs. (8) and (9)), we conclude that the minus sign result represents the dn periodic wave whereas the plus sign result corresponds to the cn periodic wave.In the following, we determine the admissible eigenvalues and eigenfunctions ofthe periodic wave solutions (8) and (9) of Eq. (1). Upon comparing the Eqs. (8) and (24) with λ = ξ and F = ± √ − k ,we obtain two real eigenvalues for the dn -periodic wave, namely λ ± = 12 (1 ± (cid:112) − k ) . (26)As far as the cn -periodic wave is concerned we compare the Eqs. (9) and (25)with λ = ξ + iη . In this case, we find λ ± = 12 ( k ± i (cid:112) − k ) . (27)Recalling the Eqs. (10) and (13), we obtain the following expressions for thesquared eigenfunctions, namely f = 2 λ r + r x λ + ¯ λ ) , ¯ g = 2¯ λ r − r x λ + ¯ λ ) . (28)Next, we determine the explicit form of the product of the eigenfunctions f and g in terms of r ( x, t ). For the dn -periodic case, we already know η = 0, F = 0. From the identity H = F − F we can fix H = F = ± √ − k .Substituting these relations back in Eq. (12) along with | r | = | f + ¯ g | and ξ = λ = λ + , we obtain f g = − λ [ | r ( x, t ) | + (cid:112) − k ] , (29)where r ( x, t ) = R ( x, t )e ibt in which R ( x, t ) is given in Eq. (8).As far as the cn -periodic wave case is concerned, we identify H = 2 η fromthe relations F = 2 η and F = 4 η . Inserting these functions back in Eq. (12)with λ = ξ + iη , we find f g = − k [ | r ( x, t ) | + ik (cid:112) − k ] , (30)where r ( x, t ) = R ( x, t )e ibt and R ( x, t ) is given in Eq. (9). In the above, wehave determined the necessary eigenvalue and the squared eigenfunctions thatcan generate the first iterated solution of Eq. (1) with the considered seed so-lution. Upon substituting the obtained eigenvalue ( λ ), periodic eigenfunctions( f , ¯ g , f g ) and periodic wave solutions r ( x, t ) in the one-fold DT formula (3),we create the periodic background. As our aim is to construct a RW solutionon the periodic wave background, we move on to construct a second linearlyindependent solution of the spectral problem (2) for the same eigenvalue λ = λ which in turn creates the desired solution.8 . RW solutions on the periodic background We construct a second linearly independent solution ( ϕ = ( ˆ f , ˆ g ) T ) for theEq. (1) with the following two properties: (i) the second solution ϕ = ( ˆ f , ˆ g ) T should also posseses the same eigenvalue λ = λ and (ii) it should exhibit anon-periodic localized profile. Based on these two requirements, we choose thesecond linearly independent solution in the formˆ f = f δ − g | f | + | g | , ˆ g = g δ + 2 ¯ f | f | + | g | , (31)where δ ( x, t ) is an unknown function which is to be determined. By insertingEq. (31) into Eqs. (2a) and (2b) and utilizing the later equations for ϕ =( ˆ f , ˆ g ) T , we obtain the following two first-order partial differential equationsfor the unknown function δ , that is ∂δ ∂x = M := − λ + ¯ λ ) ¯ f ¯ g (cid:16) | ˆ f | + | ˆ g | (cid:17) , (32) ∂δ ∂t = M := 2( ¯ f S + 2¯ g S − ¯ f ¯ g S )( | f | + | g | ) , (33)where S = ir x + 6 (cid:15) | r | r + (cid:15) ( − ¯ r x r xx + ¯ r (6 r x + r xxx ) + r xxxx + 2 λ ( r xxx + 2 r xx λ )+ r x (¯ r xx + 8 λ )) − ¯ λ + 2 (cid:15) ( | r x | − ¯ rr xx )¯ λ + 2( i + 2 (cid:15) ¯ rr x )¯ λ − (cid:15) ¯ λ + 2 r (cid:15) (¯ r xx + ¯ r ( − r x + 4 λ − r ¯ λ )) + r (1 + 6 (cid:15) ¯ r r x + 4 (cid:15) | r x | + 2 iλ − (cid:15) (¯ r xxx − λ + 2¯ r xx ¯ λ + 4¯ r x ¯ λ ) + ¯ r ( i + 8 (cid:15)r xx + 12 (cid:15)r x λ − (cid:15) ¯ λ )) ,S = λ ( (cid:15) ¯ r xxx + 4 | r | ¯ rλ + 2(¯ r xx + 2 (cid:15) ¯ r x λ ) + ¯ r ( − i + 6 (cid:15)r ¯ r x + 8 (cid:15)λ )) − ( (cid:15) ¯ r xxx + ¯ r ( − i + 6 (cid:15)r ¯ r x ))¯ λ − (cid:15) (2 | r | ¯ r + ¯ r xx )¯ λ − (cid:15) ¯ r x ¯ λ − (cid:15) ¯ r ¯ λ ,S = 6 (cid:15) | r | r + 6 (cid:15) ¯ rr x − (cid:15) ¯ r x r xx + (cid:15) ¯ rr xxx + (cid:15)r xxxx + 2 λ + 4 (cid:15) ¯ rr xx λ + 4 iλ + 32 (cid:15)λ + ¯ λ + 2 (cid:15) (¯ rr xx + r xxx )¯ λ + 2( − i + 2 (cid:15)r xx )¯ λ + 16 (cid:15) ¯ λ + 2 (cid:15)r × (¯ r xx + ¯ r ( − r x + 6¯ rλ + 3¯ r ¯ λ + 4¯ λ )) + r x ( i + (cid:15) (¯ r xx + 8¯ r ¯ λ − r ¯ λ + 8¯ λ − r x (2 λ + ¯ λ ))) + r (1 + 6 (cid:15) ¯ r r x + 2 i ¯ λ + (cid:15) (4 | r x | − ¯ r xxx + 4 λ × (¯ r xx − r x λ ) + 2¯ r xx ¯ λ + 4¯ r x ¯ λ + 16¯ λ ) + ¯ r ( i + 4 (cid:15) (2 r xx + 4 λ + 3 r x × ¯ λ + 2¯ λ ))) . The system of partial differential equations (32) and (33) are compatiblewith each other ( M t = M x ) because both the expressions are derived from thecompatible Lax Eqs. (2a) and (2b). These two expressions (32) and (33) canbe solved with the integration formula δ ( x, t ) = (cid:90) xx M ( x (cid:48) , t ) dx (cid:48) + (cid:90) tt M ( x , t (cid:48) ) dt (cid:48) , (34)9here ( x , t ) is arbitrarily fixed. The presence of higher-order derivative termsin (34), enforces us to integrate the underlying integrals numerically using theNewton-Raphson method.Substituting the considered second seed solution ϕ = ( ˆ f , ˆ g ) T of the linearequations (2a)-(2b) with λ = λ in the one-fold DT formula (3), we obtain anew solution to the FONLS Eq. (1) of the formˆ r ( x, t ) = r ( x, t ) + 2( λ + ¯ λ ) ˆ f ¯ˆ g | ˆ f | + | ˆ g | = r ( x, t ) + 2( λ + ¯ λ )[ f ( | f | + | g | ) δ − g ][¯ g ( | f | + | g | )¯ δ + 2 f ] | f ( | f | + | g | ) δ − g | + | ¯ g ( | f | + | g | )¯ δ + 2 f | , (35)where ˆ f and ˆ g are given in (31), the periodic wave solution r ( x, t ) can betaken from (4) in which R ( x, t ) is given in (8) and (9). If we consider theseed solution r ( x, t ) in dn periodic wave form with λ = [1 + √ − k ], thenthe new solution reveals RW structure on the dn periodic wave background.Similarly, if we consider the seed solution in cn periodic wave form with λ = [ k + i √ − k ], then the new solution creates a RW structure on the cn periodicwave background. (a) (b) Figure 2:
Rogue dn-periodic wave profile of (35) with (8) and k = 0 . (cid:15) = 0 .
75 and (b) (cid:15) = 35.
The surface plots of the RW solution on the dn periodic wave backgroundare shown in Figs. 2(a)-(b) using (35) with λ = (1 + √ − k ) for two differentvalues of (cid:15) , namely (cid:15) = 0 . (cid:15) = 35 and k = 0 .
5. The RW attains its highestamplitude at its origin, that is ( x , t ) = (0 , | ˆ r | = 2 .
573 for (cid:15) = 0 .
75 which can be seen from Fig. 2(a).The RW retains its height even while we increase the value of the parameter (cid:15) to 35. A main difference which we observe here is that the frequency of theperiodic background waves increases in the ( x − t ) plane for (cid:15) = 35 as displayedin Fig. 2(b).For the parametric values ( (cid:15) = 0 .
75 and (cid:15) = 55), the surface plots of | ˆ r | of RWs on the dn periodic wave background using the solution (35) for the10 a) (b) Figure 3:
Rogue dn-periodic wave profile of (35) with (8) and k = 0 . (cid:15) = 0 .
75 and (b) (cid:15) = 55 . eigenvalue λ = (1 + √ − k ) with k = 0 . | ˆ r | = 2 .
047 at theirorigin. From this, we notice that the amplitude of the RW decreases when weenhance the k value to 0 .
8. Further, the frequency of the periodic backgroundwaves increases when we increase the value of (cid:15) from 0 .
75 to 55. We concludethat the amplitude of RWs on the periodic background changes when we varythe k value. (a) (b) Figure 4:
Rogue cn -periodic wave profile of (35) with (9) and k = 0 . (cid:15) = 1 and (b) (cid:15) = 10 . Figure 4 shows the surface plots of the RWs on the cn periodic wave back-ground using the solution (35) with (cid:15) = 1, (cid:15) = 10 and k = 0 . λ = (1 + i √ − k ). We observe that the maximum amplitude of RWs for (cid:15) = 1 is found at | ˆ r | = 0 . (cid:15) = 10, the localization of periodic back-ground waves occurs in different orientations and the amplitude of RWs retainsits height as in the choice (cid:15) = 1. Unlike the dn − periodic case, here the fre-quency of the periodic background waves increases in the ( x − t ) plane when weenhance the (cid:15) value to 10. A similar dynamical characteristics is also observed11 a) (b) Figure 5:
Rogue cn -periodic wave profile of (35) with (9) and k = 0 . (cid:15) = 1 and (b) (cid:15) = 10 . while we vary the k value to 0 . | ˆ r | = 0 . | ˆ r | = 1 .
5. Conclusion
In this work, we have constructed RW solutions on the dn and cn peri-odic wave background for the FONLS equation using one-fold DT technique.Utilizing the method of nonlinearization of Lax pair, we have obtained theeigenvalues and their corresponding squared eigenfunctions. Using the obtainedeigenvalues, eigenfunctions and the seed solution choosen, we have created theperiodic background solution. We have constructed the RWs on top of thisperiodic wave background by determining another independent solution (in anon-periodic form) of the spectral problem with the same eigenvalues. We haveanalyzed the RWs on the periodic background for different parametric values.Our results show that while increasing the elliptic modulus value, the amplitudeof RWs on the periodic background decreases (increases) for dn ( cn ). We alsonotice that the frequency of periodic background wave increases in the ( x − t )plane when we increase the value of system parameter. The obtained resultsare potentially useful in optics experimentalists. Acknowledgments
NS thanks the University for providing University Research Fellowship. KMwishes to thank the Council of Scientific and Industrial Research, Govern-ment of India, for providing the Research Associateship under the Grant No.03/1397/17/EMR-II. The work of MS forms part of a research project spon-sored by National Board for Higher Mathematics, Government of India, underthe Grant No. 02011/20/2018NBHM(R.P)/R&D 24II/15064.12 eferencesReferences [1] J.M. Dudley, G. Genty, A. Mussot, A. Chabchoub and F. Dias, Nat. Rev.Phys. , 675-689 (2019).[2] A. Chabchoub, N. Hoffmann and N. Akhmediev, Phys. Rev. Lett. ,204502 (2011).[3] D. R. Solli, C. Ropes, P. Kovnath, and B. Jalali, Nature , 1054-1057(2007).[4] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmedievand J. M. Dudley, Nat. phys. , 790-795 (2010).[5] K. Manikandan, P. Muruganandam, M. Senthilvelan and M. Lakshmanan,Phys. Rev. E. , 062905 (2014).[6] Y. V. Bludov, V. V. Konotop and N. Akhmediev, Phys. Rev. A. , 033610(2009).[7] N. Akhmediev , A. Ankiewicz and M. Taki, Phys. Lett. A. , 675-678(2009).[8] J. Chen, D. E. Pelinovsky and R. E. White, Physica D , 132378 (2020).[9] Y. Ye, J. Liu, L. Bu, C. Pan, S. Chen and D. Mihalache, Nonlin. Dyn. ,1801-1812 (2020).[10] V. E. Zakharov and L. A. Ostrovsky, Physica D , 540 (2009).[11] D. S. Agafontsev and V. E. Zakharov, Nonlinearity , 2791-2821 (2015).[12] G. Mu, Z. Qin and R. Grimshaw, SIAM J. Appl. Math. , 1-20 (2015).[13] D. S. Agafontsev and V. E. Zakharov, Nonlinearity , 3551-3578 (2016).[14] K. Manikandan, P. Muruganandam, M. Senthilvelan and M. Lakshmanan,Phys. Rev. E. , 093202 (2016).[15] J. Chen, D. E. Pelinovsky and R. E. White, Phys. Rev. E , 052219(2019).[16] D. J. Kedziora, A. Ankiewicz and N. Akhmediev, Euro. Phys. J. Spec.Topics , 43-62 (2014).[17] J. Chen and D. E. Pelinovsky, Proc. R. Soc. A , 20170814 (2018).[18] J. Chen and D. E. Pelinovsky, Nonlinearity , 1955 (2018).[19] J. Chen and D. E. Pelinovsky, J. Nonlinear Sci. , 424 (2019).1320] R. Li and X. Geng, J. Appl. Math. Lett. , 106147 (2020).[21] N. Sinthuja, K. Manikandan and M. Senthilvelan, Euro. Phys. J. Plus (Inpress) (2021).[22] W. Q. Peng, S. F. Tian, X. B. Wang and T. T. Zhang, Wave Motion ,102454 (2020).[23] H. Q. Zhang, X. Gao, Z. J. Pei and F. Chen, J. Appl. Math. Letts. ,106464 (2020).[24] Y. Yang, Z. Yan and B. A. Malomed, Chaos. , 103112 (2015).[25] W. R. Sun, B. Tian, H. L. Zhen and Y. Sun, Nonlinear Dyn. , 725-732(2015).[26] Q. M. Wang, Y. T. Gao, C. Q. Su, Y. J. Shen, Y. J. Feng and L.Xue, Z.Naturforsch. , 365-374 (2015).[27] L. L. Feng, S. F. Tian and T. T. Zhang, Rocky Mountain J. Math. ,29-45 (2019).[28] N.Song, H. Xue and X. Zhao, IEEE Access , 99 (2020).[29] B. K. Esbensen, A. Wlotzka, M. Bache, O. Bang and W. Krolikouski, Phys.Rev. A, Gen. Phys. , 05385 (2011).[30] R. G. Zhou, J. Math. Phys , 013510 (2007).[31] R. G. Zhou, Stud. Appl. Math123