The Fifth Order Peregrine Breather and Its Eight-Parameter Deformations Solutions of the NLS Equation

Commun. Theor. Phys. 61 (2014) 365 369 Vol. 61, No. 3, March 1, 2014 The Fifth Order Peregrine Breather and Its Eight-Parameter Deformations Solutions of the NLS Equation Pierre Gaillard Unversité de Bourgogne, Campus de Mirande, 9 Avenue Alain Savary, 21000 Dijon, France (Received July 12, 2013; revised manuscript received September 22, 2013) Abstract We construct here explicitly new deformations of the Peregrine breather of order 5 with 8 real parameters. This gives new families of quasi-rational solutions of the NLS equation and thus one can describe in a more precise way the phenomena of appearance of multi rogue waves. With this method, we construct new patterns of different types of rogue waves. We get at the same time, the triangular configurations as well as rings isolated. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings. PACS numbers: 47.10.A-, 47.35.Fg, 47.54.Bd Key words: NLS equation, Peregrine breathers, Ahmediev s solutions 1 Introduction One can use different approaches to modeling the evolution of deep water waves. In this study, we use of the nonlinear Schrödinger equation (NLS). [1 2] The story of the nonlinear Schrödinger equation has begun in 1972 where Zaharov and Shabat first solved it using the inverse scattering method. [3 4] Its and Kotlyarov constructed in 1976 periodic and almost periodic solutions of the focusing NLS equation. [5] It is in 1979 that Ma found the first breather type solution of the NLS equation. [6] Then Peregrine constructed in 1983 the first quasi-rational solutions of NLS equation. [7] Eleonsi, Ahmediev, and Kulagin obtained the first higher order analogue of the Peregrine breather [8 9] in 1986. Ahmediev et al. in a series of articles, [10 11] constructed other analogues of Peregrine breathers, using Darboux transformations. The notion of rogue waves first appeared in studies of ocean waves. This term of rogue wave (or frea wave) was first introduced in the scientific community by Draper in 1964. [12] We recall that the common criteria for rogue waves in the ocean, is that the vertical distance from trough to crest is two or more times greater than the average wave height among one third of the highest waves in a time series (10 to 30 min). It was in 1995 that the first recorded by scientific measurement in North Sea was realized on the oil platform of Draupner located between Norway and Scotland. Then it moved to other domains of physics; in nonlinear optics, [13] Bose Einstein condensate, [14] atmosphere, [15] and even finance. [16] A lot of experiments about solutions of NLS equation have been realized these last years. In particular, the basic Peregrine soliton has been studied very recently in [17 18]; furthermore, the NLS equation accurately describes physical rogue waves of relatively high order according to the wor Ref. [19]. Actually, there is growing interest in studying higher order rational solutions. In 2010, rational solutions of the NLS equation have been written as a quotient of two Wronsians in [20]. In 2011, an other representation of the solutions of the NLS equation has been constructed in [21], also in terms of a ratio of two Wronsians determinants of order 2N. In 2012, an other representation of the solutions of the focusing NLS equation, as a ratio of two determinants has been given by by Guo, Ling, and Liu, in [22] using generalized Darboux transform. Ohta and Yang [23] have given a new approach where solutions of the focusing NLS equation are given by means of a determinant representation, obtained from Hirota bilinear method. At the beginning of the year 2012, one obtained a representation in terms of determinants, which does not involve limits. [24] These first two formulations given in [21, 24] did depend in fact only on two parameters; this was first pointed out by Matveev in 2012 for the order 3 and 4. Then we found for the order N (for determinants of order 2N), solutions depending on 2N 2 real parameters. With this new method, we construct news deformations at order 5 with 8 real parameters. The aim of this paper is to present new solutions depending this time on strictly more than two parameters, to get all the possible patterns for the solutions of NLS equation. As it will be shown in the following, we construct solutions depending on 8 parameters, which give the (analogue) Peregrine breather as particular case when all the parameters are equal to 0: for this reason, we will call these solutions, 8 parameters deformations of the Peregrine of order 5. The structure of the paper is as the following. E-mail: Pierre.Gaillard@u-bourgogne.fr c 2013 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

366 Communications in Theoretical Physics Vol. 61 We give first the new result concerning the new solutions of the NLS equation and the notations used. We construct new quasi rational solutions depending on 2N 2 parameters at the order N. After, one builds various drawings to illustrate the evolution of the solutions according to the parameters. One obtains at the same time triangular configurations and ring structures with a maximum of 15 peas. The complete analytical expression of the solutions depending on 8 parameters is found; it taes more than 14 000 pages; we can not publish it in this text. These deformations give by new patterns a best understanding of the NLS equation. 2 Degenerate Determinant Representation of Solutions of NLS Equation We consider the focusing NLS equation iv t + v xx + 2 v 2 v = 0. (1) We consider 2N real parameters λ ν, ν = 1,...,2N satisfying the relations 0 < λ j < 1, λ N+j = λ j, 1 j N. (2) We define the terms κ ν, δ ν, γ ν as functions of parameters of λ ν ; they are given by κ ν = 2 1 λ 2 ν, δ ν = κ ν λ ν, 1 λν γ ν =, (3) 1 + λ ν κ N+j = κ j, δ N+j = δ j, γ N+j = 1/γ j, j = 1,...,N. (4) The terms x r,ν (r = 1, 3) are defined by x r,ν = (r 1)ln γ ν i γ ν + i, 1 ν 2N. (5) The parameters e ν are defined by e j = ia j b j, e N+j = ia j + b j, 1 j N, (6) where a j and b j, for 1 j N are arbitrary real numbers. We use the following notations : A ν = κ ν x/2 + iδ ν t ix 3,ν /2 ie ν /2, B ν = κ ν x/2 + iδ ν t ix 1,ν /2 ie ν /2, 1 ν 2N, with κ ν, δ ν, x r,ν defined in (3), (4), and (5). The parameters e ν are defined by (6). The parameters a j and b j, for 1 N are chosen in the form a j = b j = N 1 =1 N 1 =1 ã ǫ 2+1 j 2+1, b ǫ 2+1 j 2+1, 1 j N. (7) Below we define the following functions : f 4j+1, = γ 4j 1 sin A, f 4j+2, = γ 4j cosa, f 4j+3, = γ 4j+1 sin A, f 4j+4, = γ 4j+2 cosa, (8) for 1 N, and f 4j+1, = γ 2N 4j 2 cosa, f 4j+2, = γ 2N 4j 3 sin A, f 4j+3, = γ 2N 4j 4 cosa, f 4j+4, = γ 2N 4j 5 sina, (9) for N + 1 2N. We define the functions g j, for 1 j 2N, 1 2N in the same way, we replace only the term A by B. g 4j+1, = γ 4j 1 sin B, g 4j+2, = γ 4j cosb, g 4j+3, = γ 4j+1 sin B, g 4j+4, = γ 4j+2 cosb, (10) for 1 N, and g 4j+1, = γ 2N 4j 2 cosb, g 4j+2, = γ 2N 4j 3 sin B, g 4j+3, = γ 2N 4j 4 cosb, g 4j+4, = γ 2N 4j 5 sinb, (11) for N + 1 2N. Then we get the following result: Theorem 1 The function v defined by v(x, t) = exp(2it iϕ) det((n j) j, [1,2N] ) det((d j)j, [1,2N] ) is a quasi-rational solution of the NLS equation (1) iv t + v xx + 2 v 2 v = 0, (12) depending on 2N 2 parameters ã j, bj, 1 j N 1, where n j1 = f j,1 (x, t, 0), n j = 2 2 f j,1 (x, t, 0), ǫ2 2 n jn+1 = f j,n+1 (x, t, 0), n jn+ = 2 2 f j,n+1 ǫ 2 2 (x, t, 0), d j1 = g j,1 (x, t, 0), d j = 2 2 g j,1 (x, t, 0), ǫ2 2 d jn+1 = g j,n+1 (x, t, 0), d jn+ = 2 2 g j,n+1 ǫ 2 2 (x, t, 0), 2 N, 1 j 2N. (13) The functions f and g are defined in (8), (9), (10), (11).

No. 3 Communications in Theoretical Physics 367 Setch of the proof We postpone to give the complete proof to an other publication in order not to weigh down this text. We use the ideas exposed in [24]. We use the expression of the solutions of the NLS equation given in terms of quotient of determinants of functions defined in Ref. [24]; it is the formula (21) with the functions f j and g j defined by (17), (18), (19), (20) of [24]. We replace the coefficients a j and b j of [24] by these defined in (7) of this present paper. We realize expansions of these functions in ǫ at order 2N. Then we combine columns of the determinants of the preceding formula (21) of [24] in order to eliminate ǫ 0 in column 2,..., ǫ 2(N 1) in column N, ǫ 0 in column N + 1,..., ǫ 2(N 1) in column 2N. Then we tae the limit when ǫ tends to 0 and we get the result given by (12). 3 Quasi-Rational Solutions of Order 5 with Eight Parameters We have already constructed in [21] solutions for the cases N = 1, 2, 3, and in [24 26] for N = 4, 5, 6, 7, with two parameters. Because of the length of the expression v of the solution of NLS equation with eight parameters, we can not give here. We only construct figures to show deformations of the fifth Peregrine breathers. Conversely to the study with two parameters given in preceding wors, [21,24] we get other type of symmetries in the plots in the (x, t) plane. We give some examples of this fact in the following. 3.1 Peregrine Breather of Order 5 If we choose ã 1 = b 1 = ã 2 = b 2 = ã 3 = b 3 = ã 4 = b 4 = 0, we obtain the classical Peregrine breather of order 5 (Fig. 1). Fig. 1 Solution of NLS, N = 5, ã 1 = b 1 = ã 2 = b 2 = ã 3 = b3 = ã 4 = b 4 = 0; Peregrine breather P 5. Fig. 2 Solution of NLS, N = 5, ã 1 = 10 4 ; triangle with 15 peas. Fig. 3 Solution of NLS, N = 5, b 1 = 10 4 ; triangle with 15 peas. Fig. 4 Solution of NLS, N = 5, ã 2 = 10 5 ; 3 rings of 5 peas without central pea. With other choices of parameters, we obtain different types of configurations: triangles, ring structures, and configurations with multiple concentric rings with a maximum of 15 peas.

368 Communications in Theoretical Physics Vol. 61 3.2 Variation of Parameters In the case of the variation of one parameter, we obtain different types of configurations with a maximum of 15 peas. Fig. 5 Solution of NLS, N = 5, b 2 = 10 6 ; 3 rings of 5 peas without central pea. Fig. 6 Solution of NLS, N = 5, ã 3 = 10 6 ; 2 concentric rings of 7 peas with one pea in the center. Fig. 7 Solution of NLS, N = 5, b 3 = 10 10 ; 2 concentric rings of 7 peas with one pea in the center. Fig. 8 Solution of NLS, N = 5, ã 4 = 10 10 ; a ring with 9 peas and in the center the Peregrine of order 3. a maximum of 15 peas (Figs. 2 and 3). For a 2 0 or b 2 0, we have 3 concentric rings with 5 peas on each of them without central pea (Figs. 4 and 5). For a 3 0 or b 3 0, we obtain 2 concentric rings with 7 peas on each of them with a central pea (Figs. 6 and 7). At last, for a 4 0 or b 4 0, we have 1 ring with 9 peas with inside (for large values of parameters) the apparition of the Peregrine breather of order 3 with 6 peas (Figs. 8 and 9). Fig. 9 Solution of NLS, N = 5, b 4 = 10 10 ; a ring with 9 peas and in the center the Peregrine of order 3. In the cases a 1 0 or b 1 0 we obtain triangles with 4 Conclusion In the present paper we construct for the first time to my nowledge, explicitly solutions of the NLS equation of order N = 5 with 2N 2 = 8 real parameters. These solutions expressed in terms of polynomials of degree N(N + 1) in x and t tae more than 14 000 pages; it can not be published in this text. It is given in [27]. By different choices of these parameters, we obtained new patterns in the (x; t) plane; we recognized ring struc-

No. 3 Communications in Theoretical Physics 369 ture as already observed in the case of deformations depending on two parameters. [21,24,28] We obtain triangular configurations; it was already reported in [23, 29]. We get new concentric rings configurations. All the previous results were written in the hydrodynamic case; it can be rewritten in the optic case by the following transformations: t X/2, x T. (14) So these results can be useful at the same time for hydrodynamics as well for nonlinear optics. Many applications have been realized recently in hydrodynamics or in non-linear optics, as recent wors of Chabchoub et al. [17] or Kibler et al. [30] attest it in particular. It would be important to continue this study for the higher orders in order to give a better understanding of the phenomenon of rogue waves. References [1] A.R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, Elsevier, New Yor (2010). [2] H.C. Yuen and B.M. Lae, Phys. Fluids 18 (1975) 956. [3] V.E. Zaharov, J. Appl. Tech. Phys. 9 (1968) 86. [4] V.E. Zaharov and A.B. Shabat, Sov. Phys. JETP 34 (1972) 62. [5] A.R. Its and V.P. Kotlyarov, Docl. Aad. Nau. SSSR 965 (1976). [6] Y.C. Ma, Stud. Appl. Math. 60 (1979) 43. [7] D. Peregrine, J. Austral. Math. Soc. Ser. B 25 (1983) 16. [8] N. Ahmediev, V. Eleonsy, and N. Kulagin, Sov. Phys. JETP 62 (1985) 894. [9] N. Ahmediev, V. Eleonsii, and N. Kulagin, Th. Math. Phys. 72 (1987) 183. [10] N. Ahmediev, A. Aniewicz, and J.M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601. [11] N. Ahmediev, A. Aniewicz, and P.A. Clarson, J. Phys. A: Math. Theor. 43 (2010) 122002. [12] L. Draper, Oceanus 10 (1964) 13. [13] D.R. Solli, C. Ropers, P. Koonath, and B. Jalali, Nature (London) A 450 (2007) 1054. [14] Y.V. Bludov, V.V. Konotop, and N. Ahmediev, Phys. Rev. A 80 (2009) 033610. [15] L. Stenflo and M. Marlund, J. Phys. Plasma 76 (2010) 293. [16] Z.Y. Yan, Commun. Theor. Phys. 54 (2010) 947. [17] A. Chabchoub, N.P. Hoffmann and N. Ahmediev, Phys. Rev. Lett. 106 (2011) 204502. [18] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Ahmediev, and J.M. Dudley, Nat. Phys. 6 (2010) 790. [19] A. Chabchoub, H. Hoffmann, M. Onorato, and N. Ahmediev, Phys. Rev. X 2 (2012) 011015. [20] P. Dubard, P. Gaillard, C. Klein, and V.B. Matveev, Eur. Phys. J. Special Topics 185 (2010) 247. [21] P. Gaillard, J. Phys. A: Meth. Theor. 44 (2011) 435204. [22] B. Guo, L. Ling, and Q.P. Liu, Phys. Rev. E 85 (2012) 026607. [23] Y. Ohta and J. Yang, Proc. R. Soc. A 468 (2012) 1716. [24] P. Gaillard, J. Math. Phys. 54 (2013) 013504. [25] P. Gaillard, J. Math. Sci. Adv. Appl. 13 (2012) 71. [26] P. Gaillard, J. Mod. Phys. 4 (2013) 246. [27] P. Gaillard, hal-00819359 (2013). [28] P. Dubard and V.B. Matveev, Nonlinearity 26 (2013) 93. [29] D.J. Kedziora, A. Aniewicz, and N. Ahmediev, Phys. Rev. E 86 (2012) 056602. [30] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Ahmediev, and J.M. Dudley, Nat. Phys. 6 (2010) 790.