Wronskian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave
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1 Journal of Modern Physics, 0,, - Published Online February 0 ( Wronsian Representation of Solutions of NLS Equation, and Seventh Order Rogue Wave Pierre Gaillard Université de Bourgogne, Dion, France Pierre.Gaillard@u-bourgogne.fr Received November, 0; revised December 0, 0; accepted January, 0 ABSTRACT In this paper, we use the representation of the solutions of the focusing nonlinear Schrödinger equation we have constructed recently, in terms of wronsians; when we perform a special passage to the limit, we get quasi-rational solutions expressed as a ratio of two determinants. We have already construct breathers of orders N =, 5, in preceding wors; we give here the breather of order seven. Keywords: Riemann Theta Functions; Fredholm Determinant; Wronsian; NLS Equation; Peregrine Breathers; Ahmediev Solutions. Introduction From fundamental wor of Zaharov and Shabat in [,], a lot of research has been carried out on the nonlinear Schrödinger equation (NLS). The case of periodic and almost periodic algebro-geometric solutions to the focusing NLS equation were first constructed in 7 by Its and Kotlyarov []. The first quasi-rational solutions of NLS equation were construted in by Peregrine []; they are nowadays called worldwide Peregrine breathers. In, Eleonsi, Ahmediev and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [5,]. Other families of higher order were constructed in a series of articles by Ahmediev et al. [7,] using Darboux transformations. In 0, it has been shown in [] that rational solutions of NLs equation can be writen as a quotient of two wronsians. In this paper, we use a result [] giving a new representation of the solutions of the NLS equation in terms of a ratio of two wronsians determinants of even order N composed of elementary functions; the related solutions of NLS are called of order N. When we perform the passage to the limit when some parameter tends to 0, we got families of multi-rogue wave solutions of the focusing NLS equation depending on a certain number of parameters. It allows to recognize the famous Peregrine breather [] and also higher order Peregrine s breathers constructed by Ahmediev [7,]. Recently, another representation of the solutions of the focusing NLS equation, as a ratio of two determinants has been given in [] using generalized Darboux transform. A new approach has been done in [] which gives a determinant representation of solutions of the focusing NLS equation, obtained from Hirota bilinear method, derived by reduction of the Gram determinant representation for Davey-Stewartson system. We have already given breathers of order N = to N = in []. Here, we construct the breather of order N = 7 which shows the efficiency of this method.. Expression of Solutions of NLS Equation in Terms of Wronsian Determinant and Quasi-Rational Limit.. Solutions of NLS Equation in Terms of Wronsian Determinant We briefly recall results obtained in [,]. We consider the focusing NLS equation iv v v v 0. t xx () From [], the solution of the NLS equation can be written in the form I A x t I A x t Ar a, N det, vx, t exp iti. det, In (), the matrix is defined by a expi x txr, The terms,, xr, and (). () are functions of the Copyright 0 SciRes.
2 P. GAILLARD 7 parameters,,,n and satisfying the relations 0, N, N. They are given by the following equations,,,, N, N, N,,, N. The terms xr, r, are defined by i x r N r, ln,. i The coefficients are defined by : 0, N, N N. We consider the following functions y x i tixr, y r sin, N, (5) r ycosx itixr, y, N N. We use the following notations: r, yx itixr, y, N. Wr y W,, N is the wronsian W rydet. y (),,,N by We consider the matrix d D r d,,,n () defined ex i x tx r, N, N. N p, Then we get the following lin between Fredohlm and Wronsian determinants [] Theorem. where N det I D 0 W,, 0, (7) r r r N N expi r y. It can be deduced the following result: Theorem. The function v defined by 0 0 W vx, t exp it i. () W is solution of the NLS Equation () iv v v v 0. t xx.. Quasi-Rational Solutions of NLS Equation In the following, we tae the limit when the parameters for N and for N N. c For simplicity, we denote d the term. We consider the parameter written in the form d, N. When goes to 0, we realize limited expansions at order p, for N, of the terms d d, d d d, d d, xr, i r arctan d d, d d, N d d d, N d d, N xrn, i r arctan d d We have the central result formulated in [] : Theorem. The function v defined by v x t it i W 0 0, exp lim, 0 W is a quasi-rational solution of the NLS Equation () iv v v v 0. t xx. () () Proof: Let r, be the complex number r, x itixr,, r ;, N. We use the following functions: sin,,, cos,,, sin,,, cos,,, () Copyright 0 SciRes.
3 P. GAILLARD for N, and cos, N,, N,, N,, N 5, sin,, sin, () cos, for N N. We define the functions, for N, N in the same way, where the term, in, is replaced by,. Then it is clear that W, 0 det,,n D qx, t: :. () W det D,,,N All the functions, and, and their derivatives depend on and can all be prolonged by continuity when 0. l For simplicity we denote, the term l, 0 x,, t 0 l, l and, the term, x,,0 t. Then we use the expansions N l l l N, xt,,, O, l0 l! N, N, N l l l N N, xt,, N, O, l0 l! N, N. We have the same expansions for the functions N l l l N, xt,,, O, l0 l! N, N, l0 l!. N l l l N N, xt,, N, O, N, N., The components of the columns and N + are C, for 0 respectively equal by definition to, 0 for 0 0 N, for CN of D, and, 0 0 C, 0 N, for C N of D. At the first step of the reduction, we replace the columns C by C C and C N by CN CN for N, for D ; we do the same changes for D. Each component of the column C of D can be rewritten as l! N l l l l, and the column CN replaced by l! N l l l l N, for N. For D, we have the same reductions, each component of the column C of can be rewritten as l! N l l l l, and the column C N replaced by l! N l l l l l N, for N. We can factorize in D and D in each column and N the term for N, and so simp- lify these common terms in numerator and denominator. If we restrict the developments at order in columns and N, we get respectively, 0 for the component of D, N, for the component of CN of D, and, 0 for the component of C, N, for the component of C N of D. This algorithm can be continued until the columns C N, C N of D and C, C of D N N. Then taing the limit when tends to 0, q x, t Q x, t can be replaced by 0 N 0 N,,, N, N 0 N 0 N,,, N, N 0 N 0 N N, N, NN, NN,, : 0 N 0 N Q x t,,, N, N 0 N 0 N,,, N, N 0 N 0 N N, N, N, N N, N () Copyright 0 SciRes.
4 P. GAILLARD Each element of these determinants is a polynomial in x and t. So the solution of the NLS equation taes the form vx, texpiti Q x, t with Q x,t a rational function in x and t, and which ends the proof.. Seventh-Order Breather Solution of NLS Equation.. Fiber-Optics Case To get solutions of NLS equation written in the context of fiber optics iux u tt u u 0, (5) from these of (), we can mae the following changes of variables t X, xt. () Equation (5) plays a fundamental role in optics and is the obect of active research as recent wor [] attests it where the solutions of the two-breathers are studied... Case of the Initial Conditions In the case of order N = 7, we mae an expansion at order. Taing the limit when 0 with d =, N, the solution of NLS Equation (5) taes the form n x, t vx, t exp iti. d x, t Because of the length of the complete analytical expression, we only give it in the appendix. We give here the expression of the solution in the form, ihx, t Qx, t G x t vx, t expiti in the case t = 0: , G x X X X X X X X X X X X X X X X X X X X X X X X X X X X ; Hx,0 0; , Q x X X X X X X X 57000X 7775X X X X X X X X X X X X X X X X X X X X Copyright 0 SciRes.
![50 P. GAILLARD Figure. Solution to the NLS equation, N = 7. Remar. The expressions of G x,0 and Q x,0 can be easily verified from the recursive formulae given in [].](/docs-images/92/109821424/images/5-2.jpg ".. Plot in the (x, t) Coordinates Please see Figure.. Conclusion The method described in the present paper provides a powerful tool to get explicitly solutions of the NLS equation.")
5 50 P. GAILLARD Figure. Solution to the NLS equation, N = 7. Remar. The expressions of G x,0 and Q x,0 can be easily verified from the recursive formulae given in []... Plot in the (x, t) Coordinates Please see Figure.. Conclusion The method described in the present paper provides a powerful tool to get explicitly solutions of the NLS equation. To the best of my nowledge, it is the first time that the breather of order seven solution of the NLS equation is presented. It confirms the conecture about the shape of the rogue wave in the x, t coordinates, the maximum of amplitude equal to N + = 5 and the degree of polynomials in x and t here equal to 5 as already formulated in [7]. This new formulation gives the possibility, by introduction of parameters in the arguments of preceding functions defined in the text, to create an infinite set of non singular solutions of NLS equation. It will be the next step of the wor which will open a large way to future researches in this domain. REFERENCES [] V. E. Zaharov, Stability of Periodic Waves of Finite Amplitude on a Surface of a Deep Fluid, Journal of Applied Mechanics and Technical Physics, Vol., No.,, pp. -. [] V. E. Zaharov and A. B. Shabat, Exact Theory of Two Dimensional Self-Focusing and One Dimensinal Self- Modulation of Waves in Nonlinear Media, Soviet Physics JETP, Vol., 7, pp. -. [] A. R. Its and V. P. Kotlyarov, Explicit Expressions for the Solutions of Nonlinear Schrödinger Equation, Doclady Aademii Nau SSSR, Vol. 5, No., 7, pp. 5-. [] D. Peregrine, Water Waves, Nonlinear Schrödinger Equations and Their Solutions, Journal of the Australian Mathematical Society, Vol. 5,, pp. -. [5] N. Ahmediev, V. Eleonsy and N. Kulagin, Generation of Periodic Trains of Picosecond Pulses in an Optical Fiber: Exact Solutions, Soviet Physics JETP, Vol., 5, pp. -. [] N. Ahmediev, V. Eleonsii and N. Kulagin, Exact First Order Solutions of the Nonlinear Schrödinger Equation, Theoretical and Mathematical Physics, Vol. 7, No.,, pp. 0-. doi:.07/bf075 [7] N. Ahmediev, A. Aniewicz and J. M. Soto-Crespo, Rogue Waves and Rational Solutions of Nonlinear Schrödinger Equation, Physical Review E, Vol. 0, 00, Article ID: 00. [] D. J. Kedziora, A. Aniewicz and N. Ahmediev, Circular Rogue Wave Clusters, Physical Review E, Vol., 0, Article ID: 05. [] P. Dubard, P. Gaillard, C. Klein and V. B. Matveev, On Multi-Rogue Waves Solutions of the NLS Equation and Position Solutions of the KdV Equation, European Physical Journal, Vol. 5, No., 0, pp doi:.0/epst/e0-05- [] P. Gaillard, Families of Quasi-Rational Solutions of the NLS Equation and Multi-Rogue Waves, Journal of Physics A: Mathematical and Theoretical, Vol., No., 0, pp. -5. doi:./75-///50 [] A. Aniewicz, N. Ahmediev and P. A. Clarson, Rogue Waves, Rational Solutions, the Patterns of Their Zeros and Integral Relations, Journal of Physics A: Mathematical and Theoretical, Vol., 0, pp. - [] B. Guo, L. Ling and Q. P. Liu, Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions, Physical Review E, Vol. 5, No., 0, Article ID: 007. doi:./physreve [] Y. Ohta and J. Yang, General High-Order Rogue Waves and Their Dynamics in the Nonlinear Schrödinger Equation, Proceedings of the Royal Society A, Vol., No., 0, pp doi:./rspa.0.00 [] P. Gaillard, Wronsian Representation of Solutions of the NLS Equation and Higher Peregrine Breathers, Journal of Mathematical Sciences: Advances and Applications, Vol., No., 0, pp Copyright 0 SciRes.
6 P. GAILLARD 5 Appendix Rather than to give the analytical expression in the form vx t prefers to give that inspired by Ahmediev et al. in []. The solution of NLS equation taes the form, with N = 7 with n x, t, expit i, to shorten the formulation one d x, t x t e n x, t GN x,t ihn, vn x, t expiti N d x, t QN x,t N G X, T N N N 0 N N 0 g T X H XT, h T X N NN 0 q T X, g 0, g 0, g, g 0, g T, g 0, g T T g 50 g 5700T 5500T T g , 7 0, , 0, g 0T 0500T 50T 0T 75 g 7770T 070T 70T 00T 0 5 7, 0, g T T X, T T T g5 g 0, g 0, T T 70000, Q T T , it i 700T 5000, g0 7700T 500T 55550T 50T 0500T g 0, g T T T 00500T 57700T g7 0, T T 70000T , g 000T 75700T 77000T T T g5 0, T T T T , T T , 0 g 05550T T 050T T T g 0, 0700T T 75000T Copyright 0 SciRes.
7 5 P. GAILLARD g T T T g 0, T T T T 00000T 77500T T T , 0 g T 0T 57570T 00700T 570T T T T T T T T , g 0, 0 g 50700T T T 500T T 0000T T T T T T T T , g7 0, g 700T T 00T T 0 000T T 5000T T T T T T T T , g5 0, 0 g 5000T 70700T 700T T T T T 77000T T T T T T T T , g 0, 0 g 700T 55050T 7000T T T 5000T T T T T T T T T T T , 0 Copyright 0 SciRes.
8 P. GAILLARD 5 g 0, 0 g T 700T 55500T 775T 05000T T T T T T T T T T T T T , g 0, 0 g T T 57000T 500T T T T T T T T T T T T T , g7 0, g 00T 77700T T T 0 00T T T T T T T T T T T T T T T , g5 0, g 50T 0500T 57500T T T T T T T T T T T T T T T T T T , 0 Copyright 0 SciRes.
9 5 P. GAILLARD g 0, g 500T 77700T 50T 00T g 0, T T T T T T T T T T T T T T T T T , 0 g 700T 57T T 50570T T T 0700T T T T T T T T T T T T T T T T , g 0, 0 g 0500T 7000T T 57000T 5000T T T T T T T T T T T T T T T T T T T , g7 0, Copyright 0 SciRes.
10 P. GAILLARD 55 g T T T T g5 0, T T T 00T T T T T T T T T T T T T T T T T , 50 g 50T 5700T 0T 55500T 5700T T T T T T T T T T T T ) T T T T T T T T T , g 0, 5 50 g 00T 555T 0T 500T T T 07000T T T T T T T T T T T T T T T T T T T T , g 0, 0 Copyright 0 SciRes.
11 5 P. GAILLARD g 50T 70T 5T 00050T 000T T T T T T T T T T T T T T T T T T T T T T T , h 0, h 0, h T, h 0, h 75T T, h 0, h50 T T 70 T, h 0, 7 5 h 00T 000T 0500T 5500 T, h7 0, 7 5 h 00T 000T 70T 70000T 5000 T, h5 0, h T T T T T T , h 0, 7 h 0T 00T 50T T 00700T 50700T T, h 0, h0 0T 550T 070T 5500T T 7700T T, h 0, T 7 5 h 0T 500T 00700T T 0000T T T 5000T T, h7 0, 7 5 h 0T 07700T T T 0500T T 57000T 7000T T T, h5 0, 7 5 h 50T 7700T T 57T T T T 7000T T T T, h 0, Copyright 0 SciRes.
12 P. GAILLARD 57 h T T T h 0, T T T T T T T T T, 5 h0 700T 07550T 57770T 7550T T T T T T T T T T, h 0, 7 5 h 500T 75700T T T T T T T T T T T T T, h7 0, 7 5 h 500T 5000T T 5000T T T T T T T T T T T T, h5 0, 7 5 h 700T 55700T 7570T T T T T T T T T T T T T T, h 0, 7 h 500T 500T 070T T 5000T T T T T T T T T T T T T, Copyright 0 SciRes.
13 5 P. GAILLARD h 0, 5 h0 50T 7750T 770T T T T T T T T T T T T T T T T, h 0, 7 5 h 0T 0700T 70000T 50700T T T T T T T T T T T T T T T T, h7 0, 7 5 h 0T T T 75500T 57500T T T T T T T T T T T T T T T T, h5 0, h 0T 75500T 0050T 5550T T 05700T T T T T T T T T T T T T T T T, 5 7 Copyright 0 SciRes.
14 P. GAILLARD 5 h 0, h 0T 5000T 0070T T h 0, T T T T T T T T T T T T T T T T T T, 7 h 00T 5T 50570T T 5770T h 0, T T T T T T T T T T T T T T T T T T, 7 5 h 00T 7000T T T T T T 77000T T T T T T T T T T T T T T T T T, h7 0, 7 7 Copyright 0 SciRes.
15 0 P. GAILLARD h T T T T h5 0, T T T T T T T T T T T T T T T T T T T T, h T 00T 000T 50T T T T T T T T T T T T T T T T T T T T T T T, h 0, h 75T 5T 0T T 0500T T 50700T T T T T T T T T T T T T T T T T T T T T, h 0, T Copyright 0 SciRes.
16 P. GAILLARD h T T T T T T 70000T 05700T T T T T T T T T T T T T T T T T T T T T, q, q 0, q T, q 0, q 7T T q50 7T 7T T 0, q 0, q 075T 7700T 550T q7 0, 55500T 775, 0, q 0, q T T T T T , q5 0, q 770T 070T 70500T 000T 77000T T , q 0, q 00T 0T 00T 7000T 500T T 57000, q 0, 5 00T q0 5T 50000T 50T T T q 0, 57000T 70500T T 7775, q 000T 5500T 000T 57000T 5000T 000T T T T , q7 0, 0 q T 500T 0550T 77000T T 57700T T T 75550T T , q5 0, 5 Copyright 0 SciRes.
17 P. GAILLARD q T T T T q 0, T T T T T , 00T q 0755T 770T 7070T T 70775T 50000T T T T T T T , q 0, 0 0 q0 70T 70T T T 07000T T T T T T T T T , q 0, 0 q 000T 700T 700T 0700T T T 00000T T T T T T T T , q7 0, q 70T 500T 55500T 5000T 05500T T T T T T T T T T , q5 0, 0 0 q 0755T 7770T 7700T T T T 70000T T T T T T T T T T , q 0, T Copyright 0 SciRes.
18 P. GAILLARD q T T T q 0, T T 0T 00000T T T T T T T T T T T , 0 q0 T 0T 0550T T T 77000T T T T T T T T T T T T T , q 0, q 000T T 5500T 750T T T T T T T T T T T T T T T T , q7 0, 0 q 5T 700T 75550T 05700T T T T T T T T T T T T T T T T T , Copyright 0 SciRes.
19 P. GAILLARD q5 0, q 00T 50T 00T T q 0, T T T T T T T T T T T T T T T T T , q 770T 750T 5500T 500T 70700T T 7500T T T T T T T T T T T T T T T T T T , q 0, 0 0 q 0T 70T 7550T 000T T 000T T T T T T T T T T T T T T T T T T , q 0, Copyright 0 SciRes.
20 P. GAILLARD 5 q T T T T q7 0, T T T T T T T T T T T T T T T T T T T T , 50 q 7T 00T 570T 7000T 75700T 0 000T T T T T T T T T T T T T T T T T T T T , q 0, q 7T T 05050T 7000T 7700T T T T T T T T T T T T T T T T T T T T T 57500T , q 0, Copyright 0 SciRes.
21 P. GAILLARD q T 7T 05T T 07000T 77700T T T T T T T T T T T T T T T T T T T T T T , q 0, q T 05T 07T 05T T T T T 70775T T T T T T T T T T T T T T T T T T T T Copyright 0 SciRes.
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