Two-breather solutions for the class I infinitely extended nonlinear Schrodinger equation and their special cases
NNoname manuscript No. (will be inserted by the editor)
Two-breather solutions for the class I infinitely extendednonlinear Schr¨odinger equation and their special cases
M. Crabb and N. Akhmediev
Received: date / Accepted: date
Abstract
We derive the two-breather solution of theclass I infinitely extended nonlinear Schr¨odinger equa-tion. We present a general form of this multi-parametersolution that includes infinitely many free parametersof the equation and free parameters of the two-breathercomponents. Particular cases of this solution includerogue wave triplets, and special cases of ’breather-to-soliton’ and ‘rogue wave-to-soliton’ transformations. Thepresence of many parameters in the solution allows oneto describe wave propagation problems with higher ac-curacy than with the use of the basic NLSE.
Keywords
Infinitely extended NLSE · breathers · rogue waves PACS
The nonlinear Schr¨odinger equation [1,2] (NLSE) hasvarious applications in describing ocean waves [3,4,5],pulses in optical fibres [6,7,8], Bose-Einstein conden-sates [9,10,11,12], waves in the atmosphere [13], plasma[14] and many other physical systems [15,16,17,18,19].Various extensions of the NLSE have been considered[20,21,22] that increase the accuracy of the descriptionof nonlinear wave phenomena in these systems by in-corporating higher-order effects [23,24,25,26]. Higher-order terms in these extensions are responsible for lineardispersion, as well as nonlinear effects such as self-phasemodulation, pulse self-steepening, the Raman effect,
M. CrabbOptical Sciences Group, Research School of Physics and Engi-neering, The Australian National University, Canberra, ACT,2600, Australia. E-mail: [email protected] and so on [7,27]. These higher-order terms are impor-tant in nonlinear optics [28,29], ocean wave dynamics[30,31,32] and especially in modelling high-amplituderogue wave phenomena [33,34,35].Adding higher-order terms generally results in theloss of integrability of the resulting equation. This meansthat exact solutions cannot be written in analyticalform, making the treatment more complicated. How-ever, a special choice of the higher-order operators inthese extensions allows us to keep the integrability. Thepower of using these operators consists in the possi-bility of applying arbitrary real coefficients to each ofthese operators, thus significantly extending the rangeof physical problems that can be solved in exact form.It was found that the NLSE can be extended to arbi-trarily high orders of these operators [36,37,38], andthese operators have been explicitly presented up toeighth order [37]. Using their recurrence relations, theycan be calculated to any order, although the explicitform quickly becomes cumbersome. Nevertheless, thereare no conceptual difficulties in construction of theseequations. Moreover, infinitely many terms can be con-sidered when finding solutions of these equations.Presently, there are two sets of these operators thatcan be used for infinite-order extensions of the NLSE.We call them the class I [36,37,38] and class II [39,40]infinite extensions of the NLSE [41]. The presence oftwo independent extensions enables the more accuratedescription of physical problems with greater flexibil-ity. Here, we deal exclusively with the class I extension.The class II extension is more involved, and will be leftbeyond the scope of the present work.In this paper, we find 2-breather solutions of theclass I infinitely extended NLSE equation. These aremulti-parameter solutions that involve both the free pa-rameters of the equation, and free parameters of the a r X i v : . [ n li n . S I] S e p M. Crabb and N. Akhmediev solution, which together control the features of the twobreather components, such as their localisation, propa-gation, and their relative position and frequencies. Thepresence of an infinite number of free parameters allowsus to consider many particular cases, such as breather-to-soliton conversion, which is exclusive to higher-orderextensions of the basic equation.We also derive several limiting cases, the most im-portant one of which is the general second-order roguewave solution, a particular case of the 2-breather col-lision. However, only a limited number of special casescan be given in the frame of a single manuscript. Weleave others for future work in this direction.
First, we give a brief exposition of the class I infinitelyextended nonlinear Schr¨odinger equation. It is the in-tegrable equation written in general form [36,37] iψ x + F ( ψ, ψ ∗ ) = 0 , (1)where the operator F ( ψ, ψ ∗ ) is defined through F = ∞ (cid:88) n =1 ( α n K n − iα n +1 K n +1 ) , (2)with the operators K n defined recursively by the in-tegrals of the nonlinear Schr¨odinger equation [36], andwhere each coefficient α n is an arbitrary real number;that is, K n ( ψ, ψ ∗ ) = ( − n δδψ ∗ (cid:90) p n +1 dt, where p n is the n -th integral of the basic nonlinearSchr¨odinger equation, and p n +1 can be defined recur-sively as p n +1 = ψ ∂∂t (cid:18) p n ψ (cid:19) + n (cid:88) r =1 p n − r p r , p = | ψ | . The four lowest order operators K n ( n = 2 , , ,
5) de-rived in this way are: K ( ψ, ψ ∗ ) = ψ tt + 2 | ψ | ψ,K ( ψ, ψ ∗ ) = ψ ttt + 6 | ψ | ψ t ,K ( ψ, ψ ∗ ) = ψ tttt + 8 | ψ | ψ tt + 6 | ψ | ψ ++4 ψ | ψ t | + 6 ψ t ψ ∗ + 2 ψ ψ ∗ tt .K ( ψ, ψ ∗ ) = ψ ttttt + 10 | ψ | ψ ttt + 10( ψ | ψ t | ) t ++20 ψ ∗ ψ t ψ tt + 30 | ψ | ψ t . (3)A few others can be found in [38]. The operators K n involve linear terms with derivatives of order n, andnonlinear terms involving t -derivatives of the function ψ and its complex conjugate ψ ∗ .As already mentioned, the numbers α n can take anyvalues whatsoever, and do not need to be viewed as rep-resenting small perturbations for the equation (1) to becompletely integrable. This allows us to find solutionsfor which any order of dispersion can be taken into ac-count without the need for approximation or numeri-cal techniques. This extension substantially widens therange of applicability of the NLSE for solving nonlinearwave evolution problems.When only α (cid:54) = 0 , we have the fundamental, or‘basic’ nonlinear Schr¨odinger equation: iψ x + α K ( ψ, ψ ∗ ) = iψ x + α ( ψ tt + 2 | ψ | ψ ) = 0 , (4)which includes the lowest-order dispersion and self-phasemodulation terms. Further, if only α and α are nonzero,we have the integrable Hirota equation [42]: iψ x + α ( ψ tt + 2 | ψ | ψ ) − iα ( ψ ttt + 6 | ψ | ψ t ) = 0 . (5)Adding the fourth-order operator, K , into Eq.(5), givesthe Lakshmanan-Porsezian-Daniel (LPD) equation [43,44], and so on.Again, the coefficients α n are finite and arbitrary.However, physical applications, in general, require dis-persive effects to decrease rapidly in strength with in-creasing order n . Convergence will thus not be an issuein practice for series involving α n , and we will thereforebe comfortable leaving the operator F for the wholeequation (1), as well as any other associated parame-ters, in the form of an infinite series when necessary.While the operators K n in (3) rapidly become morecomplicated and the resulting differential equation oforder n becomes much harder to solve, exact solutionscan be found explicitly by using already known solu-tions to the NLSE as a guide, and a large class ofbreather and soliton solutions are already known [37,38]. In previous works [37,38], we have seen that theeffect of nonzero odd order operators is to transform t as t (cid:55)→ t + vx with v being a function of all coefficients α n +1 . The effect of the nonzero even order operatorsis to transform x as x (cid:55)→ Bx, with B being a functionof the parameters α n .In this work we extend this approach to a generalfamily of second-order solutions, so we introduce pa-rameters B and B , and v and v , to play an anal-ogous role for the two distinct breather components.This enables us to generalise the 2-breather to the infi-nite extension of the NLSE, and we now proceed to theanalysis of these solutions. wo-breather solutions for the class I infinitely extended nonlinear Schr¨odinger equation and their special cases 3 Higher analogues of the Akhmediev breathers can beobtained through iterations of the Darboux transforma-tion [45,46]. After transforming the plane wave solution e ix with a Darboux transformation, with an eigenvalue λ such that λ (cid:54) = − , and repeating this transformationtwice, we get the 2-breather solution to the basic NLSE.This can then be generalised to the 2-breather solutionof the extended equation. The general 2-breather solu-tion is of the form ψ ( x, t ) = (cid:26) G ( x, t ) + iH ( x, t ) D ( x, t ) (cid:27) e iφx , (6)where G ( x, t ) = − ( κ − κ ) (cid:26) κ δ κ cosh δ B x cos κ t −− κ δ κ cosh δ B x cos κ t −− ( κ − κ ) cosh δ B x cosh δ B x (cid:27) , (7) H ( x, t ) = − κ − κ ) (cid:26) δ δ κ sinh δ B x cos κ t −− δ δ κ sinh δ B x cos κ t −− δ sinh δ B x cosh δ B x ++ δ cosh δ B x sinh δ B x (cid:27) , (8) D ( x, t ) = 2( κ + κ ) δ δ κ κ cos κ t cos κ t + 4 δ δ ×× (sinh δ B x sinh δ B x + sin κ t sin κ t ) −− (2 κ − κ κ + κ ) cosh δ B x cosh δ B x −− κ − κ ) (cid:26) δ κ cosh δ B x cos κ t −− δ κ cosh δ B x cos κ t (cid:27) , (9)Here κ and κ are the modulation parameters, δ m = κ m (cid:112) − κ m is the growth rate of the modulational instability foreach breather component, and the shorthand notation t m indicates t m = t + v m x for m = 1 , . Note thatwhenever t m appears, we have ignored a constant ofintegration, and we have also done the same when-ever δ m B m x appears. The most general solution allowsfor the replacements t m (cid:55)→ t m − T m , and δ m B m x (cid:55)→ δ m B m ( x − X m ) , where T m and X m are real constantswhich determine relative positions along the axes of t and x, respectively, which we might include to incor-porate a time delay in one breather component, for in-stance. For the time being, we set these constants to be both zero without substantial loss, to address theirsignificance later.The phase factor φ is independent of the modula-tion, since this part has no physical effect on the mod-ulation when it is real, and here it takes the same realvalue as it does for the plane wave solution, i.e. φ = ∞ (cid:88) n =1 (cid:18) nn (cid:19) α n . (10)The values B m determine the modulation frequency ofeach component, and the parameters v m , although theycannot be considered velocities in the usual sense, in-troduce a tilt to | ψ | relative to the axes of x and t. Theyare given explicitly by B m = ∞ (cid:88) n =1 (cid:18) nn (cid:19) nF (1 − n, ; κ m ) α n , (11) v m = ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) F ( − n, ; κ m ) α n +1 , (12)with m = 1 , , where F ( a, b ; c ; z ) is the Gaussian hy-pergeometric function. Note that there is a simple re-lationship between v m and B m : the coefficient of α n in B m is twice the coefficient of α n − in v m . The firsttwo terms of B m for the two-breather solution havebeen previously given in [47]. Our new solution extendsthese coefficients to arbitrary orders of dispersion andnonlinearity.Also notice that the parameters are the same func-tions of κ m , for both m = 1 , . This is at least suggestedby symmetry. If two successive Darboux transforma-tions generate a 2-breather solution, then there must betwo independent eigenvalues, corresponding to two in-dependent modulation parameters. Physically, we couldreason that there should be no way of knowing whichbreather component is which, so the order in which eachcomponent was generated by Darboux transformationshould be equally irrelevant. If so, it should then followthat B is the same function of κ as B is of κ , andsimilar for v and v , and this would also imply that B m and v m are the same functions as for the single-breathersolution, which are already known [38]. It is worth con-sidering whether this property extends to the general n -breather solution: i.e. whether, in general, we can find B , . . . , B n and v , . . . , v n which are the same functionsof their respective modulation parameters κ , . . . , κ n , but we do not answer this question here.The growth rate δ m in both components will be realwhen κ m is real, but the eigenvalues of the Darbouxtransformation are free to take any complex value at all,although the transformations are trivial when the eigen-values are real, and thus so are the modulation parame-ters. Real-valued modulation parameters correspond to M. Crabb and N. Akhmediev
Akhmediev breathers, whereas imaginary-valued mod-ulation parameters correspond to Kuznetsov-Ma soli-tons, the functional form of the breathers being other-wise equivalent. An example which shows the differencebetween real and imaginary modulation parameters isgiven in Fig. (2). In Fig. (3) we give an example of theeffects of altering the ratio of the modulation of the twocomponents.Fig. 1:
The 2-breather solution (6) of Eq.(1). The mod-ulation parameters are at a ratio κ : κ = 1 : 2 . Pa-rameters of the equation are: α = , α = , α = ,α = , α = , with all higher α n = 0 . The waveprofile is tilted in the ( x, t ) -plane due to the nonzero v m . Fig. 2:
A collision between an Akhmediev breather andKuznetsov-Ma soliton, with κ = 1 , and κ = i. Here α n = 1 /n ! up to n = 8 , with all higher α n = 0 . Fig. 3:
The 2-breather solution with α n = 1 /n ! up to n = 8 , with all higher α n = 0 , but now with κ = , and κ = 1 . If we choose parameters α n such that B m = 0 , the 2-breather solution may then behave in a way which isunique to the extension of the nonlinear Schr¨odingerequation [47], in the sense that it is only when higherorders of dispersion and nonlinearity are accounted forthat it is possible to take B m = 0 without obtaining atrivial or otherwise degenerate solution.For example, if we choose α such that B = 0 forall κ , then writing B = B, it is easy to show that B must take the value B = ∞ (cid:88) n =1 (cid:18) n + 2 n + 1 (cid:19) ( n + 1) E (cid:0) − n, ; κ | κ (cid:1) α n +2 , where we define the function E as the difference of hy-pergeometric functions: E ( a, b ; c ; z | z ) = F ( a, b ; c ; z ) − F ( a, b ; c ; z ) . We can then simplify the general 2-breather solutionconsiderably. We obtain G ( x, t ) = ( κ − κ ) (cid:26) κ δ κ cosh δ Bx cos κ t −− κ δ κ cos κ t − ( κ − κ ) cosh δ Bx (cid:27) ,H ( x, t ) = 2 δ ( κ − κ ) sinh δ Bx (cid:18) − δ κ cos κ t (cid:19) ,D ( x, t ) = 2( κ + κ ) δ δ κ κ cos κ t cos κ t ++ 4 δ δ sin κ t sin κ t −− (2 κ − κ κ + κ ) cosh δ Bx −− κ − κ ) (cid:18) δ κ cos κ t − wo-breather solutions for the class I infinitely extended nonlinear Schr¨odinger equation and their special cases 5 − δ κ cosh δ Bx cos κ t (cid:19) . (13)An example of this solution is given in Fig. 4. The dif-ference of this solution from the one shown in Fig. 1 isthat the wave profiles at x → ±∞ are not plane waves.The periodic set of tails from each breather maximumextends to infinity, reminiscent of periodically repeat-ing solitons. This is the phenomenon that is known asbreather-to-soliton conversion [47]. Clearly, these ‘soli-tons’ do not have a separate spectral parameter relatedto them.Fig. 4: A wave profile of a ‘breather-to-soliton conver-sion’. We use the same set of parameters as in Fig. 3,except α is now chosen such that B = 0 . This choiceextends to infinity the tails of the breathers that wouldotherwise decay.
When one of the modulation parameters, say κ , tendsto zero, we obtain the semirational limit, i.e. a solutionobtained as a combination of polynomials and circularor hyperbolic functions. Then, writing κ for κ , and δ for δ , the functions G , H , and D become G ( x, t ) = κ { κ (1 + 4 t + 4 B x ) − } cosh δB x ++ κδ cos κt ,H ( x, t ) = 2 κB x ( δ cos κt − κ cosh δB x ) ++ δκ (1 + 4 t + 4 B x ) sinh δB xD ( x, t ) = δκ { − κ (1 + 4 t + 4 B x ) } cos κt ++ 4 δB x sinh δB x + δt sin κt −− { κ (1 + 4 t + 4 B x ) } cosh δB x, (14) and the parameters B and v are reduced to B = ∞ (cid:88) n =1 (cid:18) nn (cid:19) nα n , (15)and v = ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) α n +1 . (16)This semirational 2-breather solution is a superpositionof a Peregrine solution with the Akhmediev breather,since taking the limit κ → The 2-breather solution in the semirational limit.Here the nonzero modulation parameter is κ = 1 , with α n the same as in Fig. (1). It can be considered as asuperposition of the Akhmediev breather with the Pere-grine solution. If both eigenvalues of the Darboux transformation aretaken to be equal, so that both modulation parameters κ m are also equal, we obtain the case of degeneratebreathers. Direct calculations provide no solution. Inthis case, one modulation parameter should instead betaken as a small perturbation from the other, say | κ − κ | = ε. Then, we take the limit as the perturbation ε becomes arbitrarily small, so that the solution remains M. Crabb and N. Akhmediev well-defined at all times. Namely, if we put κ = κ, and κ = κ + ε, we have B = ∞ (cid:88) n =1 (cid:18) nn (cid:19) nF (1 − n, ; ( κ + ε ) ) α n , and v = ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) F ( − n, ; ( κ + ε ) ) α n +1 . Next, recalling the identity ddz F ( a, b ; c ; z ) = abc F ( a + 1 , b + 1; c + 1; z ) , take the Maclaurin series of the G ( x, t ), H ( x, t ), and D ( x, t ) with respect to ε. In the limit as ε → , theratio of these series will be a well-defined solution withequal eigenvalues; it is thus sufficient to consider onlythe lowest-order non-vanishing terms in the series ex-pansion for D ( x, t ) in ε, which in this case happen tobe the coefficients of ε . By this method we obtain thedegenerate 2-breather solution in the form (6) with G ( x, t ) = 2 κ (cid:20) δBx + (cid:26)(cid:18) κB − δ κ B −− δ B (cid:48) (cid:19) x sinh δBx −− κδ (cid:18) − δ κ (cid:19) cosh δBx (cid:27) cos κ ( t + vx ) −− { t + ( v + κv (cid:48) ) x } δ cosh δBx sin κ ( t + vx ) (cid:21) ,H ( x, t ) = 2 κ (cid:20) (cid:26)(cid:18) δ κ − (cid:19) κB + 2 δ B (cid:48) (cid:27) x ++ δ (cid:26) (cid:18) δ κ − (cid:19) Bx − δ κ B (cid:48) (cid:27) x cosh δBx ×× cos κ ( t + vx ) + κδ (cid:18) δ κ − (cid:19) sinh 2 δBx −− δ sinh δBx { cos κ ( t + vx ) ++ κ sin κ ( t + vx ) }{ t + ( v + κv (cid:48) ) x } (cid:21) ,D ( x, t ) = κ δ (cid:20) − κ (cid:18) δ κ (cid:19) − δ κ ( t + vx ) −− δ (cid:18) − δ κ (cid:19) B x −
32 cosh 2 δBx −− δ κ (cid:26)(cid:18) − δ κ (cid:19) B − δ κ B (cid:48) (cid:27) x sinh δBx ×× cos κ ( t + vx ) − δ (cid:26) κ cos κ ( t + vx ) ++ 4 δ κ { t + ( v + κv (cid:48) ) x } sin κ ( t + vx ) (cid:27) cosh δBx + + 16 δ κ (cid:26) κ ( t + vx ) + (cid:18) (cid:18) − δ κ (cid:19) κBB (cid:48) x −− δ B (cid:48) (cid:19) κ x − v (cid:48) { t + ( v + κv (cid:48) ) x } (cid:19)(cid:27)(cid:21) , (17)where B = B , and we use B (cid:48) and v (cid:48) to denote the par-tial derivatives of B and v with respect to ε evaluatedat the point ε = 0 , i.e. when κ → κ. That is, B (cid:48) = κ ∞ (cid:88) n =1 (cid:18) nn (cid:19) n (1 − n ) F (2 − n, ; κ ) α n , and v (cid:48) = − κ ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) nF (1 − n, ; κ ) α n +1 , etc. We drop the subscripts due to the fact that as ε → The degenerate 2-breather solution. We take theset of α n the same as in Fig. 1, and the modulationparameters κ = κ = . The two breathers collidewith the high peak at the origin due to the synchronisedphases.
The degenerate breather solution is a one-parameterfamily of solutions which represents the collision of twobreathers with the same modulation parameter κ , or,equivalently, with equal frequencies. It can be consid-ered a generalisation of the known 2-soliton solutionfor the class I extension of the nonlinear Schr¨odingerequation [50]. When the frequency of the degenerate breather tends tozero, the spacing between the successive peaks in Fig.6 becomes infinitely large, pushing them out to infinity. wo-breather solutions for the class I infinitely extended nonlinear Schr¨odinger equation and their special cases 7
What remains at the origin is the second-order roguewave. In order to derive this solution, we take the limit κ → κ →
0. The derivativesof
G, H, and D with respect to κ at the point κ = 0vanish up to O ( ε ). The resulting functions G , H, and D become polynomials: G ( x, t ) = 12 {− B − BB (cid:48)(cid:48) − vv (cid:48)(cid:48) ) x ++ 80 B x − v (cid:48)(cid:48) xt + 96 B x ( t + vx ) ++ 24( t + vx ) + 16( t + vx ) } , (18) H ( x, t ) = 576 B (cid:48)(cid:48) x + 2304 B (cid:48)(cid:48) x ( t + vx ) − Bx { −− B + 16 B (cid:48)(cid:48) ) Bx − B x ++ 192 v (cid:48)(cid:48) x ( t + vx ) − B x ( t + vx ) ++ 24( t + vx ) − t + vx ) } , (19) D ( x, t ) = − − { B − BB (cid:48)(cid:48) + 64 B (cid:48)(cid:48) −− v − v (cid:48)(cid:48) ) v (cid:48)(cid:48) } x − { B − B v ++ 16(3 v − B ) BB (cid:48)(cid:48) + 16( v − B ) vv (cid:48)(cid:48) } x −− B + 3 B v + 3 v ) B x + 576 v (cid:48)(cid:48) xt −− v (cid:48)(cid:48) xt + 288( B − BB (cid:48)(cid:48) − vv (cid:48)(cid:48) ) x t −− B x t + 576 { ( B − B (cid:48)(cid:48) ) Bv ++ 4( B − v ) v (cid:48)(cid:48) } x t − B vx t −− B + 6 v ) B x t − B ++ 2 v ) B vx t − t + vx ) −− t + vx ) − t + vx ) , (20)where, in the same limit as κ → ,B = ∞ (cid:88) n =1 (cid:18) nn (cid:19) nα n ,v = ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) α n +1 . The first-order derivatives B (cid:48) and v (cid:48) vanish as κ → , but the second-order derivatives still remain, and in thelimit as κ → B (cid:48)(cid:48) = − ∞ (cid:88) n =1 (cid:18) nn (cid:19) n ( n − α n ,v (cid:48)(cid:48) = − ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) nα n +1 . This solution is shown in Fig. 7. It is, naturally, thesecond-order rogue wave, but slanted and rescaled inthe ( x, t )-plane relative to the second-order rogue waveof the NLSE [51,52]. Fig. 7:
The second-order rogue wave, Eqs.(18),(19),(20)obtained from the degenerate 2-breather solution shownin Fig. 6 in the limit κ → , with some stretching dueto higher-order effects. It is well known that the general n -th order rogue wavehas the remarkable property of being able to split into n ( n +1) first-order components [48]. The second-orderrogue wave discussed above is only a particular case ofa more general rogue wave structure, where all threefirst-order components are located at the origin, andhave merged into one single peak. In order to obtain themore general solution where the three components arenot merged together, known as the rogue wave triplet[49], we re-introduce the constants of integration intothe general 2-breather solution, i.e. δ m B m x (cid:55)→ δ m ( B m x − εX m ) ,t + v m x (cid:55)→ t − T m ε + v m x, where X m and T m are arbitrary, and the parameter ε isintroduced to make sure that the Taylor series in the de-generate limit still vanishes up to O ( ε ) . The values of X m and T m determine the location of the componentsof the breather components. They add additional freeparameters to the solution which we have previouslygiven for the restricted case in which X m = T m = 0 . Notice also that we do not make the replacement x (cid:55)→ x − X m ε directly, but, for simplicity’s sake, instead de-fine X m to account for the higher-order terms in B m .In order to further simplify parametrisation, we as-sume that X m and T m are functions of the modulationparameter κ, and are of the order O ( κ ). Then, definingfree parameters ξ and η independent of κ, such that κξ = 48( X − X ) ,κη = 48( T − T ) , M. Crabb and N. Akhmediev we have in the limit as κ → ψ ( x, t ) = (cid:40) G ( x, t ) + i ˆ H ( x, t )ˆ D ( x, t ) (cid:41) e iφx , (21)withˆ G ( x, t ) = G ( x, t ) − ξBx − η ( t + vx ) , (22)ˆ H ( x, t ) = H ( x, t ) + 12 ξ − ξB x −− ηBx ( t + vx ) + 48 ξ ( t + vx ) , (23)ˆ D ( x, t ) = D ( x, t ) − ( ξ + η ) + 12 { ξ (3 B − B (cid:48)(cid:48) ) −− ηv (cid:48)(cid:48) } x + 16 ξB x + 12 η (1 + 4 B x ) ×× ( t + vx ) − ξBx ( t + vx ) − η ( t + vx ) , (24)where ˆ G, ˆ H and ˆ D now contain two new free param-eters, ξ and η, which determine the separation of thefundamental rogue wave components in the triplet [49],and where G, H, and D are as given in Eqs. (18)-(20),for the particular case in which ξ = η = 0. An exampleof the formation of rogue wave triplets, correspondingto nonzero ξ and η, is shown in Fig. 8. When both ξ = 0and η = 0, all three components merge at the origin, asin Fig. 7.Fig. 8: The second-order rogue wave triplet (21), withseparation parameters ξ = − η = 480 , and the extendedequation parameters given by α = , α = , α = , α = , α = , α = . For this choice ofparameters, we have B = , v = , B (cid:48)(cid:48) = − , and v (cid:48)(cid:48) = − . Generally, the coefficient B in Eq. (21) determinesthe degree of localisation along the x -axis. Larger valuesof B will correspond to narrower peaks, whereas smaller values of B will correspond to broader peaks, and B =0 to minimal localisation. A point of interest here isthat it is again possible to choose a parametrisation forwhich B is any fixed constant. If we choose, for instance, α = c − ∞ (cid:88) n =1 (cid:18) n + 2 n + 1 (cid:19) ( n + 1) α n +2 , we end up with B = c, where c is a free parameter.However, B (cid:48)(cid:48) is entirely independent of the choice of c ,since the coefficient of α in B (cid:48)(cid:48) is zero. As the simplestexample, we consider the completely de-localised case, B = 0 , with B (cid:48)(cid:48) remaining arbitrary. The rogue wavesolution then reduces to (21) withˆ G ( x, t ) = G ( x, t ) − η ( t + vx ) , ˆ H ( x, t ) = H ( x, t ) + 12 ξ + 48 ξ ( t + vx ) , ˆ D ( x, t ) = D ( x, t ) − ( ξ + η ) − ξB (cid:48)(cid:48) + ηv (cid:48)(cid:48) ) x ++ 12 η ( t + vx ) − η ( t + vx ) . where G ( x, t ) = 12 {− − v (cid:48)(cid:48) x ( t + vx ) + 24( t + vx ) ++ 16( t + vx ) } ,H ( x, t ) = 576 B (cid:48)(cid:48) x { t + vx ) } ,D ( x, t ) = − − { B (cid:48)(cid:48) + v (cid:48)(cid:48) } x + 576 v (cid:48)(cid:48) x ( t + vx ) −− v (cid:48)(cid:48) x ( t + vx ) − t + vx ) −− t + vx ) − t + vx ) . Here, G , H , and D are as given for the case wherethe components are merged and B = 0, and ˆ G, ˆ H, ˆ D incorporate the shifting of the first-order componentsthrough ξ and η. When B = 0, the second-order rogue wave acquiressoliton-like tails similar to those in Fig. 4. When, ad-ditionally, ξ = η = 0 , rogue waves merge at the originto form a second-order rogue wave with extended tails.This case is shown in Fig. 9. When ξ or η is not zero,the components split, resulting in the disappearance ofthe central peak. This case is shown in Fig. 10. Here,the central peak is absent but the long tails remain,consisting of de-localised first-order components. Conclusions
We have derived the general 2-breather solution for theclass I infinitely extended nonlinear Schr¨odinger equa-tion, and given many limiting cases; namely, breather-to-soliton conversions, the semirational limit, the de-generate 2-breather, and, probably most importantly,the general second-order rogue wave solution. Thesesolutions completely describe a large family of second-order solutions to the class I extension of the NLSE, wo-breather solutions for the class I infinitely extended nonlinear Schr¨odinger equation and their special cases 9
Fig. 9:
The second-order rogue wave solution with‘soliton’-like tails when α chosen such that B = 0 , and ξ = η = 0 . Other parameters are the same as in Fig. 8.
Fig. 10:
The second-order rogue wave solution with‘soliton’-like tails when B = 0 , but now ξ = η = 48 . and exhibit rich behaviour.These results provide a more detailed analysis of theformation of nonlinear wave structures such as breathersand rogue waves when higher-order effects come intoplay, and leaves a large range of future related workwide open. Acknowledgements
The authors gratefully acknowledge thesupport of the Australian Research Council (Discovery ProjectDP150102057).
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