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Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters

  • Pierre GaillardEmail author
  • Mickaël Gastineau
Open Access
Research

Abstract

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the (xt) plan in the cases of orders 3 and 4. For this, we use a representation of solutions to NLS equation as a quotient of two determinants by an exponential depending on t. This formulation gives in the case of the order 3 and 4, solutions with, respectively 4 and 6 parameters. With this method, beside Peregrine breathers, we construct all characteristic patterns for the modulus of solutions, like triangular configurations, ring and others.

Keywords

NLS equation Peregrine breathers Rogue waves 

Introduction

The rogue waves phenomenon currently exceed the strict framework of the study of ocean’s waves [1, 2, 3, 4] and play a significant role in other fields; in nonlinear optics [5, 6], Bose–Einstein condensate [7], superfluid helium [8], atmosphere [9], plasmas [10], capillary phenomena [11] and even finance [12]. In the following, we consider the one-dimensional focusing nonlinear equation of Schrödinger (NLS) to describe the phenomena of rogue waves. The first results concerning the NLS equation date from the Seventies. Precisely, in 1972 Zakharov and Shabat solved it using the inverse scattering method [13, 14]. The case of periodic and almost periodic algebro-geometric solutions to the focusing NLS equation was first constructed in 1976 by Its and Kotlyarov [15, 16]. The first quasi rational solutions of NLS equation were constructed in 1983 by Peregrine [17]. In 1986 Akhmediev, Eleonskii and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [18, 19, 20]. Other analogues of the Peregrine breathers of order 3 and 4 were constructed in a series of articles by Akhmediev et al. [21, 22, 23] using Darboux transformations. The present paper presents multi-parametric families of quasi rational solutions of NLS of order N in terms of determinants of order 2N dependent on \(2N-2\) real parameters. The aim of this paper was to try to distinguish among all the possible configurations obtained by different choices of parameters, one those which have a characteristic in order to try to give a classification of these solutions.

Expression of solu-tions of NLS equation in terms of a ratio of two determinants

We consider the focusing NLS equation
$$\begin{aligned} i v_{t}+ v_{xx} + 2|v|^{2}v=0. \end{aligned}$$
(1)
To solve this equation, we need to construct two types of functions \(f_{j,k}\) and \(g_{j,k}\) depending on many parameters. Because of the length of their expressions, one defines the functions \(f_{\nu ,\mu }\) and \(g_{\nu ,\mu }\) of argument \(A_{\nu }\) and \(B_{\nu }\) only in the appendix. We have already constructed solutions of equation NLS in terms of determinants of order 2N which we call solution of order N depending on \(2N-2\) real parameters. It is given in the following result [24, 25, 26, 27]:

Theorem 1

The functions v defined by
$$\begin{aligned} v(x,t) = \frac{\det ((n_{jk})_{j,k\in [1,2N]})}{\det ((d_{jk})_{j,k\in [1,2N]})} e^{2it -i \varphi } \end{aligned}$$
(2)
are quasi-rational solution of the NLS Eq. (1) depending on \(2N-2\) parameters \(\tilde{a}_{j}\), \(\tilde{b}_{j}\), \(1\le j \le N-1,\) where
$$\begin{aligned} n_{j1}&=f_{j,1}(x,t,0), \nonumber \\ n_{jk}&=\frac{\partial ^{2k-2} f_{j,1}}{\partial \epsilon ^{2k-2}}(x,t,0), \, \nonumber \\ n_{jN+1}&=f_{j,N+1}(x,t,0), \, \quad \nonumber \\ n_{jN+k}&=\frac{\partial ^{2k-2}f_{j,N+1}}{\partial \epsilon ^{2k-2}}(x,t,0), \, \nonumber \\ d_{j1}&=g_{j,1}(x,t,0), \, \nonumber \\ d_{jk}&=\frac{\partial ^{2k-2}g_{j,1}}{\partial \epsilon ^{2k-2}}(x,t,0), \nonumber \\ d_{jN+1}&=g_{j,N+1}(x,t,0), \nonumber \\ d_{jN+k}&=\frac{\partial ^{2k-2}g_{j,N+1}}{\partial \epsilon ^{2k-2}}(x,t,0), \nonumber \\ 2\le k &\le N , \, 1\le j \le 2N \end{aligned}$$
(3)
The functions f and g are defined in (9), (10), (11), (12).

Patterns of quasi rational solutions to the NLS equation

The solutions \(v_{N}\) to NLS Eq. (2) of order N depending on \(2N-2\) parameters \(\tilde{a}_{j}\), \(\tilde{b}_{j}\) (for \(1\le j \le N-1\)) have been already explicitly constructed and can be written as
$$\begin{aligned} v_{N}(x,t) = \frac{n(x,t)}{d(x,t)} \exp (2it ) \end{aligned}$$
$$\begin{aligned} = \left(1-\alpha _{N}\frac{G_{N}(2x,4t)+iH_{N}(2x,4t)}{Q_{N}(2x,4t)} \right) e^{2it} \end{aligned}$$
with
$$\begin{aligned} \begin{array}{l} G_N(X,T)=\sum _{k=0}^{N(N+1)}g_k(T)X^k,\\ H_N(X,T)=\sum _{k=0}^{N(N+1}h_k(T)X^k,\\ Q_N(X,T)=\sum _{k=0}^{N(N+1}q_k(T)X^k. \end{array} \end{aligned}$$
For order 3 these expressions can be found in [28]; in the case of order 4, they can be found in [29]. In the following, based on these analytic expressions, we give a classification of these solutions by means of patterns of their modulus in the plane (xt).

Patterns of quasi-rational solutions of order 3 with 4 parameters

\(P_{3}\) breather

If we choose all parameters equal to 0, \(\tilde{a}_{1}=\tilde{b}_{1}= \ldots = \tilde{a}_{N-1}=\tilde{b}_{N-1}=0\), we obtain the classical Peregrine breather given by
Fig. 1

Solution \(P_{3}\) to NLS, N=3, \(\tilde{a}_{1}=\tilde{b}_{1}=\tilde{a}_{2}=\tilde{b}_{2}=0\)

Triangles

To shorten, the following notations are used: for example, the sequence \(1A3+1T3\) means that the structure has one arc of 3 peaks and one triangle of 3 peaks. If we choose \(\tilde{a}_{1}\) or \(\tilde{b}_{1}\) not equal to 0 and all other parameters equal to 0, we obtain triangular configuration with 6 peaks.
Fig. 2

Solution 1T6 to NLS, N=3, \(\tilde{a}_{1}=10^4\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=0\)

Fig. 3

Solution 1T6 to NLS, N=3, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=10^4\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=0\)

Rings

If we choose \(\tilde{a}_{2}\) or \(\tilde{b}_{2}\) not equal to 0, all other parameters equal to 0, we obtain ring configuration with peaks.
Fig. 4

Solution \(1R5+1\) to NLS, N=3, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^{6}\), \(\tilde{b}_{2}=0\)

Fig. 5

Solution \(1R5+1\) to NLS, N=3, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=10^5\)

Arcs

If we choose \(\tilde{a}_{1}\) and \(\tilde{a}_{2}\) not equal to 0 and all other parameters equal to 0 (and vice versa, \(\tilde{b}_{1}\) and \(\tilde{b}_{2}\) not equal to 0 and all other parameters equal to 0), we obtain deformed triangular configuration which we can call as arc structure.
Fig. 6

Solution \(1A3+1T3\) to NLS, N=3, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=10^4\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=5\times 10^6\)

Fig. 7

Solution \(1A3+1T3\) to NLS, N=3, \(\tilde{a}_{1}=10^6\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^{10}\), \(\tilde{b}_{2}=0\)

Patterns of quasi rational solutions of order 4 with 6 parameters

\(P_{4}\) breather

If we choose all parameters equal to 0, \(\tilde{a}_{1}=\tilde{b}_{1}= \ldots = \tilde{a}_{N-1}=\tilde{b}_{N-1}=0\), we obtain the classical Peregrine breather given in the following figure.
Fig. 8

Solution \(P_{4}\) to NLS, N=4, \({\tilde{a}}_{1}={\tilde{b}}_{1}={\tilde{a}}_{2}={\tilde{b}}_{2}={\tilde{a}}_{3}={\tilde{b}}_{3}=0\)

Triangles

To shorten, we use the notations defined in the previous section. If we choose \(\tilde{a}_{1}\) or \(\tilde{b}_{1}\) not equal to 0 and all other parameters equal to 0, we obtain triangular configuration with 10 peaks.
Fig. 9

Solution 1T10 to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=0\), \(\tilde{b}_{3}=0\)

Rings

If we choose \(\tilde{a}_{2}\) or \(\tilde{a}_{3}\) not equal to 0 and all other parameters equal to 0 (or vice versa \(\tilde{b}_{2}\) or \(\tilde{b}_{3}\) not equal to 0 and all other parameters equal to 0), we obtain ring configuration with 10 peaks.
Fig. 10

Solution 2R5 / 5 to NLS, N=4, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^5\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=0\), \(\tilde{b}_{3}=0\)

Fig. 11

Solution \(1R7+P_{2}\) to NLS, N=4, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=10^8\), \(\tilde{b}_{3}=0\)

Arcs

If we choose two parameters non equal to 0, \(\tilde{a}_{1}\) and \(\tilde{a}_{2}\), or \(\tilde{a}_{1}\) and \(\tilde{a}_{3}\) not equal to 0, or \(\tilde{a}_{2}\) and \(\tilde{a}_{3}\) and all other parameters equal to 0 (or vice versa for parameters b), we obtain arc configuration with 10 peaks.1
Fig. 12

Solution \(2A3/4I+T3\) to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^6\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=0\), \(\tilde{b}_{3}=0\)

Fig. 13

Solution \(2A3/4I+T3\) to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^6\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=0\), \(\tilde{b}_{3}=0\), sight top

Fig. 14

Solution \(2A4/3+1T3\) to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=5 \times 10^7\), \(\tilde{b}_{3}=0\)

Fig. 15

Solution \(2A4/3+1T3\) to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=0\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=5 \times 10^7\), \(\tilde{b}_{3}=0\), sight top

Fig. 16

Solution \(2A3/4+1T3\) to NLS, N=4, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^6\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=3 \times 10^8\), \(\tilde{b}_{3}=0\)

Fig. 17

Solution \(2A3/4+1T3\) to NLS, N=4, \(\tilde{a}_{1}=0\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^6\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=3 \times 10^8\), \(\tilde{b}_{3}=0\), sight top

Triangles inside rings

If we choose three parameters non equal to 0, \(\tilde{a}_{1}\), \(\tilde{a}_{2}\) and \(\tilde{a}_{3}\) and all other parameters equal to 0 (or vice versa for parameters b), we obtain ring with inside triangle.
Fig. 18

Solution \(1A7+1T3\) to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^3\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=10^9\), \(\tilde{b}_{3}=0\)

Fig. 19

Solution \(1A7+1T3\) to NLS, N=4, \(\tilde{a}_{1}=10^3\), \(\tilde{b}_{1}=0\), \(\tilde{a}_{2}=10^3\), \(\tilde{b}_{2}=0\), \(\tilde{a}_{3}=10^9\), \(\tilde{b}_{3}=0\), sight top

Conclusion

We have presented here patterns of modulus of solutions to the NLS focusing equation in the (xt) plane. This study can be useful at the same time for hydrodynamics as well for nonlinear optics; many applications in these fields have been realized, as it can be seen in recent works of Chabchoub et al. [30] or Kibler et al. [31]. This study tries to bring all possible types of patterns of quasi-rational solutions to the NLS equation. We see that we can obtain \(2^{N-1}\) different structures at the order N. Parameters a or b give the same type of structure. For \(a_{1}\ne 0\) (and other parameters equal to 0), we obtain triangular rogue wave; for \(a_{j}\ne 0\) (\(j\ne 1\) and other parameters equal to 0) we get ring rogue wave; in the other choices of parameters, we get in particular arc structures (or claw structure). This type of study has been realized in preceding works. Akhmediev et al, study the order \(N=2\) in [32], \(N=3\) in [33]; the case \(N=4\) was studied in particular (\(N=5,\, 6\) were also studied) in [34, 35] showing triangle and arc patterns; only one type of ring was presented. The extrapolation was done until the order \(N=9\) in [36]. Ohta and Yang [37] presented the study of the case cas \(N=3\) with rings and triangles. Recently, Ling and Zhao [38] presented the cases \(N=2,\, 3, \,4\) with rings, triangle and also claw structures.

In the present study, one sees appearing richer structures, in particular the appearance of a triangle of 3 peaks inside a ring of 7 peaks in the case of order \(N=4\); to the best of my knowledge, it is the first time that this configuration for order 4 is presented. In this way, we try to bring a better understanding to the hierarchy of NLS rogue wave solutions. It will be relevant to go on this study with higher orders.

Footnotes

  1. 1.

    In the following notations 2A4 / 3I, I meaning Reversed.

References

  1. 1.
    Kharif, C., Pelinovsky, E., Slunyaev, A.: Rogue waves in the Ocean. Springer (2009)Google Scholar
  2. 2.
    Akhmediev, N., Pelinovsky, E.: Discussion and debate: rogue waves - towards a unifying concept? Eur. Phys. Jour. Spec. Top. V 185 (2010)Google Scholar
  3. 3.
    Kharif, Ch., Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon. Eur. Jour. Mech. B Fluid 22(6), 603–634 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Slunyaev, A., Didenkulova, I., Pelinovsky, E.: Rogue waters. Cont. Phys. 52(6), 571–590 (2003)ADSCrossRefGoogle Scholar
  5. 5.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)ADSCrossRefGoogle Scholar
  6. 6.
    Dudley, J.M., Genty, G., Eggleton, B.J.: Optical rogue waves. Opt. Express 16, 3644 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    Bludov, Y.V., Konotop, V.V., Akhmediev, N.: Matter rogue waves. Phys. Rev. A 80(033610), 1–5 (2009)Google Scholar
  8. 8.
    Ganshin, A.N., Efimov, V.B., Kolmakov, G.V., Mezhov-Deglin, L.P., McClintok, P.V.E.: Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium. Phys. Rev. Lett. 101, 065303 (2008)ADSCrossRefGoogle Scholar
  9. 9.
    Stenflo, L., Marklund, M.: Rogue waves in the atmosphere. J. Plasma Phys. 76(3–4), 293–295 (2010)ADSCrossRefGoogle Scholar
  10. 10.
    Ruderman, M.S.: Freak waves in laboratory and space plasmas. Eur. Phys. J. Spec. Top. 185, 57–66 (2010)CrossRefGoogle Scholar
  11. 11.
    Shats, M., Punzman, H., Xia, H.: Matter rogue waves. Phys. Rev. Lett. 104(104503), 1–5 (2010)Google Scholar
  12. 12.
    Yan, Z.Y.: Financial rogue waves. Commun. Theor. Phys. V 54, 5, 947, 1-4, (2010)Google Scholar
  13. 13.
    Zakharov, V.E.: Stability of periodic waves of finite amplitude on a surface of a deep fluid. J. Appl. Tech. Phys 9, 86–94 (1968)Google Scholar
  14. 14.
    Zakharov, V.E., Shabat, A.B.: Exact theory of two dimensional self focusing and one dimensinal self modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)ADSMathSciNetGoogle Scholar
  15. 15.
    Its, A.R., Kotlyarov, V.P.: Explicit expressions for the solutions of nonlinear Schrödinger equation. Dockl. Akad. Nauk. SSSR S A V 965, 11 (1976)zbMATHGoogle Scholar
  16. 16.
    Its, A.R., Rybin, A.V., Salle, M.A.: Exact integration of nonlinear Schrödinger equation. Teore. i Mat. Fiz. 74(1), 29–45 (1988)Google Scholar
  17. 17.
    Peregrine, D.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B 25, 16–43 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Akhmediev, N., Eleonsky, V., Kulagin, N.: Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions. Sov. Phys. JETP V 62, 894–899 (1985)Google Scholar
  19. 19.
    Eleonskii, V., Krichever, I., Kulagin, N.: Rational multisoliton solutions to the NLS equation, Soviet Doklady 1986 sect. Math. Phys. 287, 606–610 (1986)MathSciNetGoogle Scholar
  20. 20.
    Akhmediev, N., Eleonskii, V., Kulagin, N.: Exact first order solutions of the nonlinear Schrödinger equation. Theor. Math. Phys. 72(2), 183–196 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of nonlinear Schrödinger equation. Phys. Rev. E 80(026601), 1–9 (2009)zbMATHGoogle Scholar
  22. 22.
    Akhmediev, N., Ankiewicz, A., Clarkson, P.A.: Rogue waves, rational solutions, the patterns of their zeros and integral relations. J. Phys. A Math. Theor. 43(122002), 1–9 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Chabchoub, A., Hoffmann, H., Onorato, M., Akhmediev, N.: Super rogue waves: observation of a higher-order breather in water waves. Phys. Rev. X 2(011015), 1–6 (2012)Google Scholar
  24. 24.
    Gaillard, P.: Families of quasi-rational solutions of the NLS equation and multi-rogue waves. J. Phys. A Meth. Theor. 44, 1–15 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Gaillard, P.: Wronskian representation of solutions of the NLS equation and higher Peregrine breathers. J. Math. Sci. Adv. Appl. 13(2), 71–153 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Gaillard, P.: Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves. J. Math. Phys. V 54, 013504-1-32 (2013)Google Scholar
  27. 27.
    Gaillard, P.: Multi-parametric deformations of the Peregrine breather of order N solutions to the NLS equation and multi-rogue waves. Adv. Res. 4, 346–364 (2015)CrossRefGoogle Scholar
  28. 28.
    Gaillard, P.: Deformations of third order Peregrine breather solutions of the NLS equation with four parameters. Phys. Rev. E V 88, 042903-1-9 (2013)Google Scholar
  29. 29.
    Gaillard, P.: Six-parameters deformations of fourth order Peregrine breather solutions of the NLS equation. J. Math. Phys. V 54, 073519-1-22 (2013)Google Scholar
  30. 30.
    Chabchoub, A., Hoffmann, N.P., Akhmediev, N.: Rogue wave observation in a water wave tank, Phys. Rev. Lett. V 106, 204502-1-4 (2011)Google Scholar
  31. 31.
    Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790–795 (2010)CrossRefGoogle Scholar
  32. 32.
    Ankiewicz, A., Kedziora, D.J., Akhmediev, N.: Rogue wave triplets. Phys Lett. A 375, 2782–2785 (2011)ADSCrossRefzbMATHGoogle Scholar
  33. 33.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: The phase patterns of higher-order rogue waves. J. Opt. V 15, 064011-1-9 (2013)Google Scholar
  34. 34.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Circular rogue wave clusters. Phys. Rev. E 84(056611), 1–7 (2011)zbMATHGoogle Scholar
  35. 35.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Triangular rogue wave cascades. Phys. Rev. E 86(056602), 1–9 (2012)zbMATHGoogle Scholar
  36. 36.
    Kedziora, D.J., Ankiewicz, A., Akhmediev, N.: Classifying the hierarchy of the nonlinear Schrödinger equation rogue waves solutions. Phys. Rev. E V 88, 013207-1-12 (2013)Google Scholar
  37. 37.
    Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468(2142), 1716–1740 (2012)ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Ling, L., Zhao, L.C.: Simple determinant representation for rogue waves of the nonlinear Shrodinger equation. Phys. Rev. E V 88, 043201-1-9 (2013)Google Scholar

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Authors and Affiliations

  1. 1.Université de BourgogneDijonFrance
  2. 2.IMCCE, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités UPMC UnivParisFrance

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