Doubly-Periodic Solutions of the Class I Infinitely Extended Nonlinear Schrodinger Equation
DDoubly-Periodic Solutions of the Class I Infinitely Extended Nonlinear Schr¨odingerEquation
M. Crabb and N. Akhmediev
Optical Sciences Group, Research School of Physics and Engineering,The Australian National University, Canberra, ACT, 2600, Australia
We present doubly-periodic solutions of the infinitely extended nonlinear Schr¨odinger equationwith an arbitrary number of higher-order terms and corresponding free real parameters. Solutionshave one additional free variable parameter that allows to vary periods along the two axes. Thepresence of infinitely many free parameters provides many possibilities in applying the solutions tononlinear wave evolution. Being general, this solution admits several particular cases which are alsogiven in this work.
PACS numbers: 05.45.Yv, 42.65.Tg, 42.81.qb
I. INTRODUCTION
Evolution equations are a powerful tool for describinga great variety of physical effects. These include pulsepropagation in optical fibers [1, 2], nonlinear ocean wavephenomena [3, 4], plasma [5, 6] and atmospheric [7, 8]waves, and the dynamics of Bose-Einstein condensates[9–11] to mention only a few. Using evolution equations,one can explain phenomena that would otherwise be dif-ficult to interpret. Examples of such phenomena includesolitons [12, 13], modulation instability [14–16], super-continuum generation [17], Fermi-Pasta-Ulam recurrence[18], rogue waves [19–23], etc. It is especially helpfulwhen the evolution equations under study are integrable[24]. Unfortunately, this is not always the case. Not allevolution equations are integrable [25–27]. Finding newintegrable equations [28–30], and extending the existingones to allow for incorporating new, physically relevantterms [31–35], is therefore an important direction of re-search in nonlinear dynamics.The nonlinear Schr¨odinger equation (NLSE) is oneof the fundamental examples of a completely integrableequation [36, 37] which finds application in the descrip-tion of water waves [38], pulses in optical fibres [39],amongst other areas of physics. In neither of these fieldsis the NLSE absolutely accurate. Extensions of the NLSEthat have physical relevance are therefore essential, andthese have been considered in a number of works that in-clude both optical applications and water waves [26, 27].In general, these extensions lift the integrability formost particular physical problems. However, in specialcases, we can obtain extensions which remain integrable,and, in addition, we can add infinitely many higher-orderterms with variable coefficients representing the strengthof these effects, adding substantial flexibility to the evo-lution equation.There are two types of extensions of the NLSE [40–44]. For clarity, we call them here the class I and classII extensions. The next higher-order terms in the classI extension correspond to the Hirota equation [40–42],while the next higher-order terms in the class II exten-sion correspond to the Sasa-Satsuma equation [43, 44]. Both of these extensions take into account higher orderdispersive effects, without any restriction on the magni-tude of these effects, i.e. without requiring them to besmall perturbations. In practice, waves are affected bymore than just second-order dispersion, so solutions tothe infinite equations are an important development inthat they allow a generalisation of the fundamental struc-tures which appear in the ‘basic’ nonlinear Schr¨odingerequation to account for these effects. When the num-ber of higher-order terms is limited to the third order,integrability can be achieved with variation of two freeparameters [45]. For infinitely extended equations, thenumber of free parameters is also infinite.The presence of two classes of integrable extensionsthus widens the range of problems that can be solvedanalytically. Remarkably, solutions to both classes canbe found in general form, even for the case of an infinitenumber of terms, and an infinite number of correspond-ing parameters. In order to find these solutions, we canstart with the known solutions of the NLSE and extendthem, recalculating the parameters of the solution. Thiscan be done for soliton, breather and rogue wave solu-tions [41, 42]. In the present work, we further expandthis approach to doubly-periodic solutions. They includeas particular cases solitons and breathers.To be specific, we start with the standard focusingNLSE: i ∂ψ∂x + α (cid:18) ∂ ψ∂t + 2 | ψ | ψ (cid:19) = 0 (1)where ψ = ψ ( x, t ) is the wave envelope, x is the dis-tance along the fibre or along the water surface, while t is the retarded time in the frame moving with the groupvelocity of wave packets. The coefficient α scales thedispersion and nonlinear terms in a way convenient forthe extensions. The infinite extension of Eq. (1) can bewritten in the form i ∂ψ∂x + ∞ (cid:88) n =1 ( α n K n [ ψ ] − iα n +1 K n +1 [ ψ ]) = 0 , (2)where the K n [ ψ ] are n th order differential operators, andthe coefficients α n are arbitrary real numbers. a r X i v : . [ n li n . S I] S e p Here we deal with the class I extension, and exactforms for the class I form of the operators K n [ ψ ] aregiven in [41]. The four lowest order operators K n arepresented below. K [ ψ ] = ψ tt + 2 | ψ | ψ, (3) K [ ψ ] = ψ ttt + 6 | ψ | ψ t ,K [ 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 ttttt + 10 | ψ | ψ ttt + 10( ψ | ψ t | ) t ++20 ψ ∗ ψ t ψ tt + 30 | ψ | ψ t . The coefficients α n determine the strength of the disper-sive effects of order n , as well as higher-order nonlineareffects. The whole infinite equation (2) is integrable forarbitrary values of α n . For example, the equation withthe terms up to the third order is the Hirota equation i ∂ψ∂x + α (cid:18) ∂ ψ∂t + | ψ | ψ (cid:19) − iα (cid:18) ∂ ψ∂t + 6 | ψ | ∂ψ∂t (cid:19) = 0(4)while including up to fourth order terms gives theLakshmanan-Porsezian-Daniel (LPD) equation [31], andso on. Particular solutions of the first-order to the equa-tion (2) have been given in [41, 42]. Solutions of themKdV equation, which is a particular case of (2), areprovided in [46]. Thus, any of the extensions of (2) withonly a few nonzero terms can be considered individually.Among the more general families of solutions tothe nonlinear Schr¨odinger equation (1) are the doubly-periodic solutions [47]. The two periods of this family canbe varied, thus providing particular cases in the form ofsolitons, breathers, cnoidal, dnoidal and Peregrine waveswhen one or two of these periods tend to infinity or zero[47]. Unlike breather solutions, however, these doubly-periodic solutions do not decay in either space or time,and instead have the special property of being periodicin both the x and t variables.In this work, we show that doubly periodic solutionscan be found for the class I equation. This solution in-cludes infinitely many parameters α n in full generality.We also show that particular limiting cases of this familyinclude the Akhmediev breather and soliton solutions. II. DOUBLY PERIODIC SOLUTIONS
There are two types of doubly periodic solutions to thenonlinear Schr¨odinger equation, which can be classifiedas type-A and type-B [48]. Each of them is expressed interms of Jacobi elliptic functions, with the modulus k asthe free parameter of the family. First, we consider thetype-A solutions. A. Type-A Solutions
Type-A solutions of Eq. (2) are of the form ψ ( x, t ) = k sn ( Bx/k, k ) − iC ( t + vx ) dn ( Bx/k, k ) k − kC ( t + vx ) cn ( Bx/k, k ) e iφx (5)where C ( t ) = (cid:114) k k cn (cid:32)(cid:114) k t, (cid:114) − k (cid:33) . The constants B , v and φ in the solution (5) are given interms of the coefficients α n of the equation (2). Takinginto account the lowest order terms, step by step, we find: B = 2 α + 8 α + (cid:18) − k (cid:19) α + (cid:18) − k (cid:19) α + (cid:18) − k + 12 k (cid:19) α + (cid:18) − k + 144 k (cid:19) α + · · · (6) φ = 2 α + (cid:18) − k (cid:19) α + (cid:18) − k (cid:19) α + (cid:18) − k + 6 k (cid:19) α + (cid:18) − k + 60 k (cid:19) α + (cid:18) − k + 432 k − k (cid:19) α + · · · (7) v = 4 α + (cid:18) − k (cid:19) α + (cid:18) − k (cid:19) α + (cid:18) − k + 6 k (cid:19) α + (cid:18) − k + 72 k (cid:19) α + (cid:18) − k + 576 k − k (cid:19) α + · · · (8)An important observation here is that the expressionfor v which is responsible for the ‘tilt’ in ( x, t )-plane dis-cussed below includes only odd-order coefficients α n . Ifthese are zero, v is also zero.In order to determine the general forms, with all α n in-cluded, we note that only one of these sets of polynomialcoefficients is algebraically independent. The coefficientof α n − in v is half the coefficient of α n in B, and isalso the sum of the coefficients of α n in B and φ. Itis therefore sufficient to determine the coefficients of B, since, if we let B = ∞ (cid:88) n =1 B n α n , then v = ∞ (cid:88) n =1 12 B n +1 α n +1 , φ = ∞ (cid:88) n =1 ( B n +1 − B n ) α n . Calculating further the other terms of B n , we find thatthey are the polynomials B n = 2 n − (cid:98) n (cid:99) (cid:88) r =0 ( − r (2 r )!( n − r − r ( r !) ( n − r − k r , where the summation ends at (cid:98) n (cid:99) terms, (cid:98)·(cid:99) being thefloor function, i.e. (cid:98) m (cid:99) is the largest integer which is notgreater than m . Now the full expression for B is givenexplicitly by the series formula B = ∞ (cid:88) n =1 n − (cid:98) n (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − r − r (cid:19) k r α n . (9)It also follows that v = ∞ (cid:88) n =1 n (cid:98)
12 ( n +1) (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − rr (cid:19) k r α n +1 , (10)and the phase factor φ is φ = ∞ (cid:88) n =1 n − (cid:26) (cid:98)
12 ( n +1) (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − rr (cid:19) k r −− (cid:98) n (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − r − r (cid:19) k r (cid:27) α n (11)We plot an example of the type-A solution for Eq. (2)in Fig. 1. For this example, we take the modulus k = 0 . α n = 1 /n ! up to n = 10, restrictingourselves with the case when all terms higher than n = 10are zero. We can see from Fig. 1 that v introduces a tiltFIG. 1: Type-A solution for Eq. (2), with k = 0 . ,α n = 1 /n ! up to n = 10 and all other α n = 0 . Noticethat v is nonzero. to the solutions and appears to operate similarly to avelocity parameter in a boost transformation. However, note that v cannot be interpreted exactly as a velocity, asthere is no function f such that we could write ψ ( x, t ) = f ( t + vx ) as we could do with a travelling wave, exceptin the case that φ = B = 0 . From equation (5), we can see that the parameter
B/k can be associated with a frequency of the modulationalong the x axis when v = 0. On the other hand, the realquarter-period along the t axis is: (cid:114) k K (cid:32)(cid:114) − k (cid:33) , where K ( k ) denotes the complete elliptic integral of thefirst kind with modulus k . However, just as v cannotbe precisely interpreted as a velocity, neither can B/k be thought of as a modulation frequency exactly, exceptwhen v = 0 and the solution is periodic along the x -axis. B. The Akhmediev Breather Limit
In the limit as modulus k → , we havelim k → B = ∞ (cid:88) n =1 (cid:18) nn (cid:19) nF (1 − n, ; ) α n , (12)lim k → φ = ∞ (cid:88) n =1 (cid:18) nn (cid:19) α n (13)lim k → v = ∞ (cid:88) n =1 (cid:18) nn (cid:19) (2 n + 1) F ( − n, ; ) α n +1 (14)where F ( a, b ; c ; z ) is Gauss’ hypergeometric function.The type-A solution then reduces to the Akhmedievbreather with modulation parameter √ k → ψ ( x, t ) = √ Bx − i cos √ t + vx ) √ Bx − cos √ t + vx ) e iφx (15)with B, φ , and v given by (12), (13), and (14), respec-tively. An example of this limiting case is plotted in Fig.2. C. Type-B Solutions
Type-B solutions can be considered as the analytic con-tinuation of the type-A solutions for values of the mod-ulus k >
1. Using the corresponding transformations ofthe elliptic functions [49] with modulus κ = 1 /k , thesesolutions take the form ψ ( x, t ) = κe iφx √ κ sn( Bx, κ ) − iA ( t + vx ) cn( Bx, κ ) √ κ − A ( t + vx ) dn( Bx, κ ) (16)where the function A ( t ) is given by A ( t ) = cd (cid:32) √ κt, (cid:114) − κ κ (cid:33) , FIG. 2:
The limiting case k → of the type-A solutions,with α n = ( n !) / (2 n )! up to n = 12 , all other α n = 0 .This is the Akhmediev breather solution of Eq. (2),periodic in t and with the growth-decay cycle in x . and κ is in the range 0 < κ < . In this case, we find: B = ∞ (cid:88) n =1 n − (cid:98) n (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − r − r (cid:19) κ r α n , (17) φ = ∞ (cid:88) n =1 n − (cid:26) (cid:98)
12 ( n +1) (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − rr (cid:19) κ r −− (cid:98) n (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − r − r (cid:19) κ r (cid:27) α n (18) v = ∞ (cid:88) n =1 n (cid:98)
12 ( n +1) (cid:99) (cid:88) r =0 ( − r r (cid:18) rr (cid:19)(cid:18) n − rr (cid:19) κ r α n +1 . (19)These are just the same polynomials as previously givenfor the type-A solutions, but with reciprocal argument κ = 1 /k .We plot an example of type-B solutions in Fig. 3.The solution is qualitatively different from the type-Asolution as the location of maxima are now different.The limit as κ → k → κ , we can vary the periods ofthe type-B solutions while always keeping the functionsanalytic, so that in the limit κ → κ → ψ ( x, t ) = 2 e iφx sech(2 t + vx ) (20) FIG. 3: The type-B solution (16), where κ = 0 . ,α n = 1 /n ! up to n = 8 , with all other α n = 0 . The peaksof this solution are aligned along lines of constant τ = t + vx. with v = ∞ (cid:88) n =1 n − α n +1 and φ = ∞ (cid:88) n =1 n α n , which is the general soliton solution for the equation (2),up to scaling [41]. III. PHASE PORTRAIT OF SOLUTIONS
The transformation of the two periodic solutions intothe Akhmediev breather (AB) when k → κ → ξ = Bx, τ = t + vx, and define thenew function u ( ξ, τ ) = ψ ( x, t ) e − iφx . (21)FIG. 4: The phase portrait of the periodic dynamicsaround the Akhmediev breather shown by the blackcurves. The type-A solution is shown by the red curvewhile the type-B solution is shown by the blue curve.Here u is defined by (21), and the two saddle points areat u = ± . The trajectories are drawn along the lines ξ = Bx and τ = t + vx . Thus, evolution is in ξ alonglines of constant τ. The counterbalancing exponential factor allows us to stopthe rotation of ψ around the origin in the complex plane.Then it is easy to see that the trajectory correspondingto the AB solution satisfies the equation { Re u ( ξ, T ) } + { Im u ( ξ, T ) − } = 2 , T = nπ √ , (22)where Re u and Im u are the real and imaginary partsof u, respectively, and n is any integer. The trajectoriesdefined by Eq. (22) are circular so long as we trace theevolution in ξ along these lines of constant τ . They areshown as black curves in Fig. 4. Similar precautionsshould be taken for the doubly-periodic orbits.The difference between the type-A and type-B solu-tions can be seen clearly from Fig. 4. Trajectories forthe type-A solutions never cross the real axis, whereastrajectories for the type-B solutions do. Therefore, eachtime they complete one full path, the phase change iseither zero or 2 π. The periodicity of solutions along the x -axis depends on the strength of the dispersion, or thevalues of the coefficients α n , through φ, B , and v . IV. THE CASE WITH ZERO EVEN ORDERTERMS
In the absence of any even order terms in Eq. (2), itbecomes real: ∂ψ∂x − ∞ (cid:88) n =1 α n +1 K n +1 [ ψ ] = 0 , (23)Then we have φ = 0 and B = 0, and the type-B solutiontakes the real-valued form ψ ( x, t ) = u ( τ ) = κA ( τ ) √ κ − A ( τ ) , (24)where τ = t + vx. Note that here v can be interpreted asa velocity since u has the form of a travelling wave.The real quarter-period in τ is equal to the realquarter-period in t of the usual type-B solutions, whichis 1 √ κ K (cid:32)(cid:114) − κ κ (cid:33) . The real quarter-period in x , for fixed t, is √ κK ( κ ) /v since φ = B = 0 . In particular, with the normalisation α = − α n = 0 , this becomes the periodic solution to themKdV equation ψ x + ψ ttt + 6 ψ ψ t = 0 , given by ψ ( x, t ) = κ cd (cid:16) √ κ ( t − x ) , (cid:113) − κ κ (cid:17) √ κ − cd (cid:16) √ κ ( t − x ) , (cid:113) − κ κ (cid:17) . (25)We plot an example of this solution in Fig. 5.FIG. 5: Plot of real-valued mKdV equation solution ψ given by (25), where k = . Here the type-B solution(16) reduces to a periodic solution propagating withspeed v = 4 . The degree of generality of our solutions allows one toconsider many other particular cases. For example, someof the polynomial coefficients have real zeros for certainvalues of n . Taking k = causes the influence of fourth-order dispersion on φ to vanish, as well as the effect of thesixth-order dispersion on the modulation frequency B, and similar for eighth-order dispersion in φ when k = .Considering all these cases can be useful for practicalapplication of these solutions. V. CONCLUSION
We have presented doubly-periodic solutions of type-Aand type-B for the class I infinitely extended nonlinearSchr¨odinger equation. These solutions are expressed interms of Jacobi elliptic functions, and have two variableperiods along the two axes of the system. Being rathergeneral, they include important cases of solutions: amongthem, the Akhmediev breather and the soliton solution are the limiting cases when the modulus of the ellipticfunctions is one or zero. As another particular case, wegive a periodic solution of the mKdV equation.As the equation under consideration has an infinitenumber of free parameters, this can be useful in mod-elling various physical problems of nonlinear wave evo-lution with a large degree of flexibility in choosing theparameters. The integrability of this equation allows oneto write all solutions in explicit form, adding significantlymore power into the analysis.
Acknowledgments
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