Rogue waves on the double periodic background in Hirota equation
aa r X i v : . [ n li n . PS ] A ug Rogue waves on the double periodic background inHirota equation
N. Sinthuja a , K. Manikandan a , M. Senthilvelan a, a Department of Nonlinear Dynamics, School of Physics, Bharathidasan University,Tiruchirappalli 620 024, Tamil Nadu, India
Abstract
We construct rogue wave (RW) solutions on the double periodic backgroundfor the Hirota equation through one fold Darboux transformation formula. Weconsider two types of double periodic solutions as seed solution. We identifythe squared eigenfunctions and eigenvalues that appear in the one fold Darbouxtransformation formula through an algebraic method with two eigenvalues. Af-ter determining the eigenfunctions and eigenvalues we construct the desiredsolution in two steps. In the first step, we create double periodic waves as thebackground. In the second step, we build RW solution on the top of this doubleperiodic solution. We present the localized structures for two different doublyperiodic backgrounds.
Keywords:
Hirota equation; Rogue waves; Darboux transformation; Doubleperiodic solutions; Nonlinearization of Lax pair; Integrable systems.
1. Inroduction
The nonlinear Schr¨odinger (NLS) equation describes the wave phenomenain ocean [1, 2, 3] and the propagation of optical pulse in optical fiber [4, 5, 6].Considering the external factors such as depth of the sea, bottom friction andviscosity in the ocean and the femtosecond pulse propagation in fibers, severalhigher order NLS equations have been introduced in the literature [7, 8]. Onesuch higher-order NLS equation is the Hirota equation. In dimensionless form,the Hirota equation reads [9], iψ t + α ( ψ xx + 2 | ψ | ψ ) − iα ( ψ xxx + 6 | ψ | ψ x ) = 0 , (1)where the function ψ = ψ ( x, t ) denotes the complex wave envelope. The sub-scripts t and x stand for partial derivatives with respect to temporal and spatialvariables, respectively. When α = 1 , α = 0, Eq. (1) reduces to the standard Email address: [email protected] (M. Senthilvelan)
Preprint submitted to Elsevier August 28, 2020
LS equation. During the past two decades, several works have been devotedto construct and analyze the characteristics of localized solutions in the Hirotaequation [11, 12, 13, 14, 15]. Multisolitons, breathers and RW solutions havebeen constructed for the Hirota equation using different techniques, see for ex-ample Refs. [12, 13, 16, 17, 18, 19]. In these studies, RW solutions have beenconstructed only on the constant wave background.In nature, RWs can arise from a variety of backgrounds, including constantbackground, multi-soliton background and periodic wave background [20, 21,22]. Recently, interest has been shown to construct RW solutions on the periodicbackground. Initially, the occurrence of RWs on periodic background has beenbrought out only through numerical schemes. For example, computations ofRWs on the periodic background were performed in [23], where the authors havecomputed the solutions of Zakharov-Shabat spectral problem only numerically.Similarly, RWs on the double-periodic background which was constructed in[24] has been executed only through the numerical scheme. The resulting wavepatterns exhibit periodicity in both space and time. Differing from these, veryrecently, Chen and his collabroators have reported an algorithm to construct RWsolutions on the periodic wave background for the focusing NLS equation [25].An attempt has also been made by the same authors to derive RW solutions onthe double-periodic background for the focusing NLS equation using an effectivealgebraic method [26]. Subsequently, Peng et al have studied the characteristicsof RWs on the periodic background for the Hirota equation [27]. However, theauthors have reported RWs only for a limiting case. The studies on RWs onthe periodic background can be found potential applications both in experimentand theory in their relative fields [28, 29].In this work, we construct RW solutions on the double-periodic backgroundfor the Hirota equation. We utilize one fold Darboux transformation formulaand the method of nonlinearization of Lax pairs to derive this solution. Tobegin, we assume a constraint between the potential and squared eigenfunctionsof the Lax system of the considered equation. Using this constraint we identifyHamiltonians associated with the spatial and temporal parts of the Lax pair.We notice that the Hamiltonian derived from spatial part of the Lax pair ofHirota equation matches with the one constructed for the NLS equation. Thisis because the spatial part of the spectral problem of NLS equation and theHirota equation are the same. However, the Hamiltonian coming out fromthe time part of the Hirota equation differs from NLS counterpart. Since thespatial part is same we obtain the same sets of differential constraints whichconnect the potential and eigenvalues that are given in Ref. [26]. We constructperiodic eigenfunctions by solving the differential constraints in terms of knowndouble periodic solutions of the Hirota equation. Substituting these periodiceigenfunctions and the identified set of eigenvalues in the one-fold Darbouxtransformation formula, we create the double-periodic wave background. Wethen construct another linearly independent solution for the same Lax equationsfor the same set of eigenvalues which in turn creates the RW solution on thetop of the double-periodic wave background. Since the time evolution partin the Lax pair of Hirota equation differs from the NLS equation the second2ndependent solution which we construct differs from the NLS equation. Wedemonstrate surface plots of | r | for two different sets of eigenvalues.We organize this letter as follows. In Sec. 2, to begin, we present the Lax pairand one-fold Darboux transformation formula for the Hirota equation and twotypes of double periodic solutions of the Hirota equation. We also briefly recallthe method of constructing RW solution on the double periodic background. InSec. 3, we apply the method to Hirota equation and construct the RWs on thedesired background. In Sec. 4, we summarize our work.
2. Lax pair and Darboux transformation
The complete integrability of the nonlinear system (1) is guaranteed by thepresence of Lax pair. A solution ψ = r to the Hirota Eq. (1) is a compatibilitycondition ( ϕ xt = ϕ tx ) of the pair of linear equations on ϕ . The Lax pair for theHirota equation is given by [9] ϕ x = U ( λ, r ) ϕ, U ( λ, r ) = (cid:18) λ r − ¯ r − λ (cid:19) , (2)and ϕ t = V ( λ, r ) ϕ, V ( λ, r ) = λ (cid:18) α − α (cid:19) + λ (cid:18) iα α r − α ¯ r − iα (cid:19) + λ (cid:18) α | r | α r x + 2 iα r α ¯ r x − iα ¯ r − α | r | (cid:19) + (cid:18) iα | r | − α ( r ¯ r x − ¯ rr x ) iα r x + α ( r xx + 2 | r | r ) iα ¯ r x − α (¯ r xx + 2 | r | ¯ r ) − iα | r | + α ( r ¯ r x − ¯ rr x ) (cid:19) , (3)where r is the potential and ¯ r denotes the complex conjugate of r . One caneasily verify that the compatibility condition U t − V x + [ U, V ] = 0 gives rise toEq. (1).The one-fold Darboux transformation for the Eq. (1) reads [25]ˆ r ( x, t ) = r ( x, t ) + 2( λ + ¯ λ ) f ¯ g | f | + | g | , (4)where ϕ = ( f ( x, t ) , g ( x, t )) T - a non-zero solution of the Lax pair (2) and (3)for a fixed eigenvalue λ = λ and r ( x, t ) and ˆ r ( x, t ) represents the seed andfirst iterated solutions of Eq. (1), respectively. With the chosen seed solution,the linear Eqs. (2) and (3) can be solved in a number of ways. For example,it is trivial to integrate the Eqs. (2) and (3) with the plane wave background( r ( x, t ) = e i ( kx − ωt ) ) and generate the first-order RW solution (ˆ r ( x, t )). However,it is highly nontrivial to integrate the Eqs. (2) and (3) with double periodicsolutions. So one has to adopt a suitable methodology to integrate them withthe genus-2 solutions as background.Very recently double periodic solutions for the Hirota equation with free realparameters have been constructed [9]. In our studies, we consider two types of3ouble-periodic solutions which are given by the following rational functions ofJacobian elliptic functions sn, cn, and dn in [9]:( i ) r ( x, t ) = k √ k sn( M t, k ) − iC ( x + vt )cn( M t, k ) √ k − C ( x + vt )dn( M t, k ) e ibt , (5a)where the function C ( x ) is given by C ( x ) = cn( √ kx, κ )dn( √ kx, κ ) , κ = r − k k . (5b)( ii ) r ( x, t ) = k sn( M t/k, k ) − iS ( x + vt )dn( M t/k, k ) k (1 − S ( x + vt )cn( M t/k, k )) e ibt , (6a)where the function S ( x ) is given by S ( x ) = r k k cn r k x, κ ! κ = r − k , (6b)and k ∈ (0 , M = 2 α , v = 4 α + k α , b = 2 α + k α and α n = n ! are the arbitrary constants. It follows from (5) and (6) that | r ( x, t ) | = | r ( x + L, t ) | = | r ( x, t + T ) | , ( x, t ) ∈ R , with the fundamental periods L = 4 K ( κ ) √ k and T = 2 K ( k ) for (5) and L = 2 √ K ( κ ) and T = 4 K ( k ) for (6), where K ( k )represents the complete elliptic integral of the first kind with the elliptic modulusparameter κ . The qualitative profiles of double-periodic wave are plotted using(5) and (6) for the modulus k = 0 . Figure 1:
Double periodic profile of (5) and (6) with k = 0 . We intend to construct RWs on the double periodic profiles given in (5) and(6). However, these solutions are travelling wave solutions. To identify for whateigenvalues these solutions arise we invoke the method of nonlinearization of Laxpairs introduced in Refs. [30, 31, 32, 33]. In this procedure one usually identifiesthe Hamiltonian associated the spatial and temporal parts of the Lax pair by4ntroducing a constraint between the potential r ( x, t ) and the eigenfunctions. Inour case, the constraint reads r = h f , f i := f + ¯ g + f + ¯ g , where the functions( f , g , f , g ) are nothing but the solutions of the Lax pair (2) and (3).After introducing this constraint we can proceed to construct a set of or-dinary differential equations (ODEs) with r ( x, t ) as the dependent variable.The order and number of these ODEs extend depending upon the numberof unknowns to be determined. In our case, we have four unknowns, namely( f , g , f , g ) and so we sequentially go upto fourth order ODE. The last dif-ferential equation turns out to be the fourth-order Lax-Novikov equation in thehierarchy of stationary NLS equation. The fourth-order complex-valued Lax-Novikov equation is integrable with two complex-valued constants of motion[26]. From these integrals of motion, we identify a set of admissible eigenvalues λ and λ which are symmetric about the imaginary axis. Once the eigenvaluesare fixed, substituting the double periodic solution and the obtained squaredeigenfunctions in the one-fold Darboux transformation formula (4) we createdouble periodic solution as the background for the considered equation. The re-sulting solution is periodic both in space and time. We then proceed to constructRWs on top of this double periodic waves. We achieve this by constructing asecond linearly independent solution of the Lax system (2) and (3) with thesame set of eigenvalues and eigenfunctions earlier with an unknown function.We determine this unknown function by substituting this solution in the Laxsystem and integrating the underlying equations. The second solution which weconstruct is non-periodic one and grows linearly with x and t . Substituting thesecond solution in one-fold Darboux transformation formula (4), we obtain theRW solution on the top of the double-periodic wave background.
3. Demonstration on Hirota equation
In Ref. [26], the authors have constructed the rogue wave solutions on thedouble periodic background for the focusing NLS equation. We adopt the sameprocedure and derive the RW solution on the double periodic background tothe Eq. (1). While implementing this procedure we observe that the Hirotaequation also yields the same results upto section III described in Ref. [26]. Forour convenience we fix α = and α = throughout the manuscript. Whileperforming the procedure of nonlinearization of Lax pair with the constraintfor the Eqs. (2) and (3), we come across differences only in the time part. Forinstance, substituting the assumed constraint into Eqs. (2) and (3), we obtainthe following finite-dimensional Hamiltonian systems, namely df j dx = ∂H ∂g j , dg j dx = − ∂H ∂f j j = 1 , , (7)where H = h Λ f,g i + 12 h f,f ih g,g i (8)5nd df j dt = ∂H ∂g j , dg j dt = − ∂H ∂f j j = 1 , . (9)where H = 14 [ i h f,g i + 23 h f,f i h g,g i + i h f,f ih Λ g,g i + i h Λ f,f ih g,g i + 43 h Λ f,g ih f,f ih g,g i + 2 i h Λ f,g i + 23 h Λ g,g ih f,f i + 23 h Λ f,f ih g,g i + 83 h Λ f,g i − i h f,f ih f , g i − i h g,g ih f , g i − h f,f i h g , g i− h g,g ih f , f i − h f,f ih Λ f , g i − h g,g ih Λ f , g i ] . (10)In the above, the notations f , g and Λ denote f =( f , f , ¯ g , ¯ g ) T , g =( g , g , − ¯ f , − ¯ f ) T and Λ = diag( λ , λ , − ¯ λ , − ¯ λ ), respectively. One may notice that the Hamil-tonian (10) which comes from the time evolution part of Lax pair of Hirotaequation differs from the NLS equation which was studied in [26].The squared eigenfunctions f , g and f ¯ g which appear in the Eq. (4) canbe obtained by substituting the expressions (5) and (6) and their derivatives inthe following equations: f = λ λ + ¯ λ )( λ − λ )( λ + ¯ λ ) (cid:2) r ′′ + 2 | r | r + 4( b + λ ) r + 2 λ r ′ (cid:3) ,g = λ λ + ¯ λ )( λ − λ )( λ + ¯ λ ) (cid:2) ¯ r ′′ + 2 | r | ¯ r + 4( b + λ )¯ r − λ ¯ r ′ (cid:3) ,f g = − λ λ + ¯ λ )( λ − λ )( λ + ¯ λ ) (cid:2) r ′ ¯ r − r ¯ r ′ + 2 λ (2 b + 2 λ + | r | ) (cid:3) . (11)(since the details of getting (11) has already been given in Ref. [26] we do notreproduce them here).Similarly we can determine the expressions of squared eigenfunctions, namely f , g and f g from the above equations by replacing λ by λ and λ by λ .We consider two different sets of three admissable pair of eigenvalues among ± λ , ± λ and ± λ which were found in Ref. [26] for the family of double-periodic wave solutions given in Eqs. (5) and (6). Each pair of eigenvalues canbe taken in place of λ and λ . In the first set, all the eigenvalues are real, thatis λ = ±√ τ , λ = ±√ τ , and λ = ±√ τ = ±√ τ + ±√ τ where τ variesfrom 0 to 0 . τ = τ − τ and τ = τ + τ = 1. In the second set, one of theeigenvalues λ = ±√ τ is positive while the other two eigenvalues are complex-conjugate, namely λ = ±√ β + iγ and λ = ±√ β − iγ , where τ = 2 β , β variesfrom 0 to 0 .
5. Now, substituting the resultant expressions coming out from (11)and considering three admissible pairs of eigenvalues given above in (4), we canset up the double-periodic wave pattern as background for the Hirota Eq. (1).6o generate RWs on this double periodic background, we construct a sec-ond linearly independent solution for the same Lax pair (2) and (3) with thesame eigenvalues. The second linearly independent solution which we intend toconstruct should be non-periodic and grow linearly in x and t . We consider thesecond linearly independent solution ϕ = ( ˆ f , ˆ g ) T for the Lax pair Eqs. (2)and (3) in the formˆ f = f η − g | f | + | g | , ˆ g = g η + 2 ¯ f | f | + | g | , (12)with η is the unknown function which is to be determined. The Wronskianbetween the two solutions (4) and (12) is nonzero, f ˆ g − ˆ f g = f (cid:18) g η + 2 ¯ f | f | + | g | (cid:19) − g (cid:18) f η − g | f | + | g | (cid:19) = 2 , (13)which in turn proves that the second solution is linearly independent from thefirst one.Substituting (12) into (2) and utilizing (5) for ϕ = ( ˆ f , ˆ g ) T , we obtain thefollowing first-order equation for η , that is ∂η ∂x = W := − λ + ¯ λ ) ¯ f ¯ g (cid:16) | ˆ f | + | ˆ g | (cid:17) . (14)Similarly, inserting (12) into (3) and utilizing (3) for ϕ = ( ˆ f , ˆ g ) T , we obtainanother first-order equation for η , namely ∂η ∂t = W := 2( λ + ¯ λ ) (cid:0) r x + (2 λ − λ + 3 i ) r (cid:1) ¯ f | f | + | g | ) + 2( λ + ¯ λ ) (cid:0) − ¯ r x + (2 λ − λ + 3 i )¯ r (cid:1) ¯ g | f | + | g | ) − λ + ¯ λ ) (cid:0) r ¯ r + 2 λ − (2¯ λ − i )( λ − ¯ λ ) (cid:1) ¯ f ¯ g | f | + | g | ) . (15)The two first-order equations (14) and (15) is compatible with each other( W t = W x ) because both the expressions are derived from the compatibleLax equations (2) and (3). Upon integrating the above two expressions, (14)and (15), we obtain η ( x, t ) = Z xx W ( x ′ , t ) dx ′ + Z tt W ( x , t ′ ) dt ′ , (16)where ( x , t ) is arbitrarily fixed. It is very difficult to integrate the Eq. (16)analytically. So we do not present the explicit form of η here. We integrate(16) by Newton-Raphson method. 7ubstituting the double periodic solutions r ( x, t ) given in (5) and (6) andthe eigenfunctions of the second solution ϕ = ( ˆ f , ˆ g ) T of the linear equations(2)-(3) with λ = λ in the one-fold Darboux transformation formula (4)ˆ r ( x, t ) = r ( x, t ) + 2( λ + ¯ λ ) ˆ f ¯ˆ g | ˆ f | + | ˆ g | , (17)where ˆ f and ˆ g are given in (12), we obtain RW solution on the double periodicbackground for the Hirota equation (1). The second solution differs from theone reported for the NLS equation. Figure 2:
RWs on the background of the double-periodic solution (5) with k = 0 . λ = √ τ , λ = √ τ and λ = √ τ . In Fig. 2(a)-(c) we show the surface plots of | ˆ r | of RWs on the double periodicbackground using the solution (5) with k = 0 . λ = √ τ , λ = √ τ and λ = √ τ where τ = 0 . τ = 1 and τ = τ − τ .In this figure, RWs attain their maximum amplitude at their origin, that is( x , t ) = (0 ,
0) in all the cases. Specifically, in Fig. 2(c) we notice that theamplitude of RWs is higher than that of the other two cases at ( x , t ) = (0 , k = 0 .
9, the surface plots of | ˆ r | of RWs on the double periodic backgroundusing the solution (5) with three real eigenvalues, namely λ = √ τ , λ = √ τ and λ = √ τ where τ = 0 . τ = 1 and τ = τ − τ are displayed in8 igure 3: RWs on the background of the double-periodic solution (5) with k = 0 . λ = √ τ , λ = √ τ and λ = √ τ . Fig. 3(a)-(c). We also observe that the amplitude of RWs is higher than thatof the other two cases at ( x , t ) = (0 ,
0) which is demonstrated in Fig. 3(c).
Figure 4:
RWs on the background of the double-periodic solution (6) with k = 0 . λ = √ τ and λ = √ β + iγ . Figure 4 represents the surface plots of | ˆ r | for RWs on the double periodicbackground using the solution (6) with k = 0 . λ = √ τ , τ = 2 β and(b) λ = √ β + iγ , β = 0 .
45 and γ = 0 . igure 5: RWs on the background of the double-periodic solution (6) with k = 0 .
95 forcomplex conjugate eigenvalues λ = √ τ and λ = √ β + iγ . given in Fig. 4 with k = 0 .
4. Summary
In this letter, we have constructed RW solutions on the double-periodic back-ground for the Hirota equation in an algebriac way. One of the efficient toolsto derive RW solution for the given nonlinear partial differential equation is theDarboux transformation method through which one can construct the desiredsolution by appropriately choosing the seed solution. However, in our case,it is very difficult to integrate the underlying equations arising from Lax pairwith the double periodic solution as seed solution. To circumvent this difficultywe brought another procedure through which the necessary expressions for thesquared eigenfunctions and the eigenvalues that appear in the one fold Darbouxtransformation formula can be identified. The procedure which we brought-into identify the eigenfunctions and eigenvalues is the method of nonlineariza-tion of Lax pairs introduced long ago in a different context. With the help ofthis method and the spectral theory of Lax pairs we have captured eigenvaluesand the squared eigenfunctions which appear in the one-fold Darboux transfor-mation formula. With the obtained expressions we create the double periodicwaves as background. We then proceed to generate the RW on the top of thisbackground. For this purpose, we choose a second independent solution forthe Lax system with the same eigenfunctions and eigenvalues with an unknownfunction. Substituting this solution in the Lax pair equations and integratingthem we determine this unknown function. By evaluating the one-fold Darbouxtransformation formula with this second independent solution we create the RWsolution on the top of the double periodic wave. The explicit form of the RWsolution on the double periodic back ground is not explicitly presented since theunknown function which appear in the second independent solution cannot befound analytically. We have presented the localized structures for two differentsets of eigenvalues. The application of this approach to few other evolutionaryequations is under progress. 10 cknowledgments
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 sponsoredby National Board for Higher Mathematics, Government of India under theGrant No. 02011/20/2018NBHM(R.P)/R&D 24II/15064. The authors thankProf. Dmitry E. Pelinovsky for helping to evaluate the numerical part of thiswork.
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