Higher order Peregrine breathers and multi-rogue waves solutions of the NLS equation

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Higher order Peregrine breathers and multi-rogue waves solutions of the NLS equation

Pierre Gaillard

To cite this version:

Pierre Gaillard. Higher order Peregrine breathers and multi-rogue waves solutions of the NLS equation.

2012. �hal-00589556v3�

Higher order Peregrine breathers and multi-rogue waves solutions of

the NLS equation.

+Pierre Gaillard, ⁺ Universit´e de Bourgogne, Dijon, France : e-mail: [email protected],

April 25, 2011

Abstract

This work is a continuation of a recent paper in which we have constructed a multi-parametric family of solutions of the focusing NLS equation given in terms of Wronskians determinants of order 2N com- posed of elementary trigonometric functions. When we perform a special passage to the limit when all the periods tend to infinity, we get a family of quasi-rational solutions. Here we construct Peregrine breathers of ordersN = 4,5,and the multi-rogue waves corresponding the 10 or 15 peak formation in frame of the NLS model first explained by Matveev et al. in 2010. In the cases N = 4,5 we get comfortable formulas to study the deformation of higher Peregrine breather of or- der 4 to the 10 rogue-waves or order 5 to the 15 rogue-waves solution via variation of the free parameters of our construction.

1 Introduction

The nonlinear Schr¨odinger equation (NLS) was first derived by Zakharov [16]

in 1968. The solutions were first given by Zakharov and Schabat. The case of periodic and almost periodic algebro-geometric solutions to the focusing NLS

Akhmediev and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Pere- grine breather[3]. Other families of higher order were constructed in a series of articles by Akhmediev et al. [1, 2] using Darboux transformations. In 2010 it has been shown in [6] that rational solutions of NLS equation can be written as a quotient of two wronskians and it has been recovered as partic- ular case, Akhmediev’s quasi-rational solutions of NLS equation.

In this paper, we extend a result [10] giving a new representation of the solutions of the NLS equation in terms of a ratio of two wronskians deter- minants of even order 2N 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’s breather [14] and also higher order Peregrine’s breathers constructed by Akhmediev [1, 4].

We treat in the following the cases of order N = 4 and N = 5. We get for an arbitrary choice of the parameters the shape of Akhmediev’s breathers;

we can also get, for particular parameters, the apparition of peaks of similar amplitude for the modulus of the solution v in the (x;y) coordinates.

2 Expression of solutions of NLS equation in terms of Wronskian determinant and quasi- rational limit

2.1 Solutions of NLS equation in terms of Wronskian determinant

We briefly recall results obtained in [10]. We consider the focusing NLS equation

ivt+vxx+ 2|v|²v = 0. (1)

From [10], the solution of the NLS equation can be written in the form

In (2), the matrix Ar = (aνµ)1≤ν,µ≤2N is defined by aνµ = (−1)^ǫν Y

λ6=µ

γ_λ+γ_ν γλ −γµ

exp(iκνx−2δνt+xr,ν +eν). (3) The terms ǫν are defined by :

ǫν = 0, 1≤ν ≤N

ǫ_ν = 1, N + 1≤ν ≤2N. (4)

We consider the following functions

φ^r_ν(y) = sin(κνx/2 +iδνt−ixr,ν/2 +γνy−ieν), 1≤ν ≤N,

φ^r_ν(y) = cos(κνx/2 +iδνt−ixr,ν/2 +γνy−ieν), N + 1 ≤ν≤2N. (5) We use the following notations :

Θν =κνx/2 +iδνt−ixr,ν/2 +γνy−ieν, 1≤ν ≤2N.

Wr(y) =W(φ1, . . . , φ2N) is the wronskian

Wr(y) = det[(∂_y^µ−1φν)ν, µ∈[1,...,2N]]. (6) We consider the matrix Dr = (dνµ)ν, µ∈[1,...,2N] defined by

dνµ= (−1)^ǫνQ

λ6=µ

γ_λ+γν

γ_λ−γµ

exp(iκνx−2δνt+xr,ν−ieν), 1≤ν ≤2N, 1≤µ≤2N,

with

xr,ν = (r−1) lnγν −i γ_ν +i.

Then we get the following link between Fredohlm and Wronskian determi- nants [9]

Theorem 2.1

det(I+D_r) =k_r(0)×W_r(φ₁, . . . , φ_2N)(0), (7)

It can be deduced the following result : Theorem 2.2 The function v defined by

v(x, t) = W3(0)

W1(0)exp(2it−iϕ). (8)

is solution of the NLS equation (1)

ivt+vxx+ 2|v|²v = 0.

2.2 Quasi-rational solutions of NLS equation

In the following, we take the limit when the parametersλj →1 for 1≤j ≤N and λ_j → −1 for N+ 1 ≤j ≤2N.

For simplicity, we denote dj the term ^√cj₂.

We consider the parameter λj written in the form

λj = 1−2ǫ²d²_j, 1≤j ≤N. (9) When ǫ goes to 0, we realize limited expansions at order p, for 1≤ j ≤ N, of the terms

κj = 4djǫ(1−ǫ²d²_j)^1/2,δj = 4djǫ(1−2ǫ²d²_j)(1−ǫ²d²_j)^1/2, γj =djǫ(1−ǫ²d²_j)^−1/2, xr,j = (r−1) ln^1+iǫd_1−iǫd^{j(1−ǫ2d2j)−1/2}

j(1−ǫ²d²_j)^−1/2,

κN+j = 4djǫ(1−ǫ²d²_j)^1/2, δN+j =−4djǫ(1−2ǫ²d²_j)(1−ǫ²d²_j)^1/2, γN+j = 1/(djǫ)(1−ǫ²d²_j)^1/2, xr,N+j = (r−1) ln¹_1+iǫd^{−iǫdj(1−ǫ2d2j)−1/2}

j(1−ǫ²d²_j)^−1/2. The parameters aj and bj, for 1≤N are chosen in the form

a_j = ˜a_jǫ^M−1, b_j = ˜b_jǫ^M−1, 1≤j ≤N, M = 2N. (10) We have the central result given in [10] :

Theorem 2.3 With the parameters λj defined by (9), aj and bj chosen as in (10), for 1≤j ≤N, the functionv defined by

v(x, t) = exp(2it−iϕ) lim

ǫ→0

W₃(0)

W1(0), (11)

is a quasi-rational solution of the NLS equation (1) iv_t+v_xx+ 2|v|²v = 0,

3 Quasi-rational solutions of order N

To get solutions of NLS equation written in the context of fiber optics iux+1

2utt+u|u|² = 0, (12)

from these of (1), we can make the following changes of variables t→X/2

x→T. (13)

In the following, because of the complexity of the expression, we give all the solutions in the case where the parameters d_j are equal to j and all the parameters aj and bj are equal to 0.

The solution of NLS equation can be written in the form vN(x, t) = n(x, t)

d(x, t) exp(2it−iϕ) = (1−αN

GN(2x,4t) +iHN(2x,4t) Q_N(2x,4t) )e^2it with

G_N(X, T) = PN(N+1)

k=0 g_k(T)X^k HN(X, T) = PN(N+1)

k=0 h_k(T)X^k QN(X, T) = PN(N+1)

k=0 q_k(T)X^k

3.1 Case N=4

In the case of order N = 4, we make an expansion at order 7. Taking the limit when ǫ→0, the solution of NLS equation (12) takes the form

v(x, t) = n(x, t)

d(x, t)exp(2it−iϕ).

α4= 4, g20= 0, g19= 0, g18= 10, g17= 0, g16= 270T²+ 270, g15= 0, g14= 1800T⁴

−3600T²+ 9000, g13= 0, g12= 5880T⁶−54600T⁴−12600T²+ 189000, g11= 0, g10= 11340T⁸−176400T⁶+ 189000T⁴−378000T²−1077300, g9= 0,

g8= 13860T¹⁰−207900T⁸+ 2356200T⁶+ 1701000T⁴−56983500T²−4819500, g7= 0, g6= 10920T¹²−18480T¹⁰+ 6967800T⁸+ 56095200T⁶−342657000T⁴ +198450000T²−11907000, g5= 0 g4= 5400T¹⁴+ 163800T¹²+ 9034200T¹⁰ +107919000T⁸−615195000T⁶+ 178605000T⁴+ 654885000T²+ 178605000, g3= 0, g2= 1530T¹⁶+ 133200T¹⁴+ 5506200T¹²−116802000T¹⁰−1731334500T⁸

+2532222000T⁶−893025000T⁴+ 4643730000)T²+ 223256250, g1= 0, g0= 190T¹⁸+ 33150T¹⁶+ 1294200T¹⁴+ 3288600T¹²+ 48629700T¹⁰

−2015401500T⁸−1845585000T⁶+ 14586075000)T⁴+ 2098608750T²−44651250,

h20= 0, h19= 0, h18= 10T, h17= 0, h16= 90T³−270T, h15= 0, h14= 360T⁵

−6000T³−5400T, h13= 0, h12= 840T⁷−29400T⁵+ 12600T³−138600T, h11= 0, h10= 1260T⁹−65520T⁷+ 259560T⁵−529200T³−1984500T, h9= 0,

h8= 1260T¹¹−77700T⁹+ 718200T⁷−5329800T⁵−6142500T³+ 29767500T, h7= 0, h6= 840T¹³−48720T¹¹+ 718200T⁹+ 2973600T⁷−72765000T⁵ +436590000T³+ 146853000T,h5= 0, h4= 360T¹⁵−12600T¹³+ 138600T¹¹

−5859000T⁹−328293000T⁷+ 1075599000T⁵+ 773955000T³+ 535815000T, h3= 0, h2= 90T¹⁷+ 1200T¹⁵−189000T¹³−40143600T¹¹

−307786500T⁹+ 2085426000T⁷−4465125000T⁵+ 4405590000T³−1205583750T, h1= 0, h0= 10T¹⁹+ 930T¹⁷−86040T¹⁵−7018200T¹³−48100500T¹¹−542902500T⁹

+6039117000T⁷+ 12942909000T⁵+ 937676250T³, q20= 1, q19= 0, q18= 10T²+ 10, q17= 0, q16= 45T⁴−270T²+ 405, q15= 0,

q14= 120T⁶−1800T⁴+ 1800T²+ 16200, q13= 0, q12= 210T⁸−4200T⁶+ 6300T⁴ +113400T²+ 425250, q11= 0, q10= 252T¹⁰−3780T⁸+ 63000T⁶

+718200T⁴+ 3005100T²+ 1644300, q9= 0, q8= 210T¹²+ 1260T¹⁰ +255150T⁸−567000T⁶+ 23388750T⁴−31468500T²+ 17435250, q7= 0, q6= 120T¹⁴+ 5880T¹²+ 476280T¹⁰+ 16443000T⁸+ 162729000T⁶

−154791000T⁴+ 130977000T²+ 130977000, q5= 0, q4= 45T¹⁶+ 5400T¹⁴ +459900T¹²+ 19845000T¹⁰+ 153798750T⁸+ 702513000T⁶−89302500T⁴ +1250235000T²+ 111628125,q3= 0, q2= 10T¹⁸

+2250T¹⁶+ 225000T¹⁴+ 4422600T¹²−99508500T¹⁰−224248500T⁸ +9704205000T⁶+ 15181425000T⁴−1920003750T²+ 223256250, q1= 0, q0=T²⁰+ 370T¹⁸+ 44325T¹⁶+ 2208600T¹⁴+ 62795250T¹²+ 693384300T¹⁰ +6641129250T⁸+ 4346055000T⁶+ 14042818125)T⁴+ 2902331250T²+ 22325625

The plot of the modulus of v in the (x, t) coordinates gives :

Figure 1: Solution of NLS, N=4a.

If we choose a1 = 1, a2 = 1, a3 = 1, a4 = 1, b1 = 1, b2 = 1, b3 = 1, b4 = 1, we have the same type of analytical expression for the modulus ofv;

we don’t reproduce here.

We only give shape of the modulus of v in the (x, t) coordinates :

Figure 2: Solution of NLS, N=4b.

If we choosea1 = 0,a2 = 0, a3 = 0, a4 = 0, b1 = 1000000, b2 = 1000000, b3 = 1000000, b4 = 1000000, the shape of the modulus of v in the (x, t) coordinates is given by :

Figure 3: Solution of NLS, N=4c.

In these three plots, we can see the deformation of the initial Akhmediev’s breather to multi-rogue waves, by choosing different types of the parameters among the 8 given by our formulation.

3.2 Case N=5

In the case N = 5, we realize an expansion at order 9 in ǫ. From (11), we get solutions of NLS equation. Taking the limit when ǫ→0, the solution of NLS equation takes the form

v(x, t) = n(x, t)

d(x, t)exp(2it−iϕ).

α5= 60, g30= 0, g29= 0, g28= 1, g27= 0, g26= 42T²+ 42, g25= 0, g24= 455T⁴

−1050T²+ 2415, g23= 0, g22= 2548T⁶−30660T⁴−13860T²+ 119700, g21= 0, g20= 9009T⁸−226380T⁶+ 171990T⁴−343980T²+ 3221505, g19= 0, g18= 22022T¹⁰

−838530T⁸+ 4142460T⁶−44100T⁴−36713250T²−40153050, g17= 0,

g16= 39039T¹²−1844766T¹⁰+ 22431465T⁸−9075780T⁶−259473375T⁴−2703484350T²

−370010025, g15= 0, g14= 51480T¹⁴−2522520T¹²+ 61319160T¹⁰

+39803400T⁸−773955000T⁶−21896973000T⁴+ 33756345000T²−2893401000, g13= 0, g12= 51051T¹⁶−2023560T¹⁴+ 104367060T¹²+ 629483400T¹⁰

+6114046050T⁸−132697164600T⁶+ 554979316500T⁴+ 319310019000T²+ 30787036875, g11= 0, g10= 38038T¹⁸−589050T¹⁶+ 124369560T¹⁴+ 1700266680T¹²

+37748127060T¹⁰−446713728300T⁸+ 2431707075000T⁶+ 1380509487000T⁴ +4238859255750T²+ 1299806817750, g9= 0, g8= 21021T²⁰+ 570570T¹⁸ +112372785T¹⁶+ 1735587000T¹⁴−43189665750T¹²−2318934687300T¹⁰

+10714665764250T⁸−20464596621000T⁶+ 35015175365625T⁴+ 40381027706250T² +5540260123125, g7= 0, g6= 8372T²²+ 769692T²⁰+ 78618540T¹⁸

+570662820T¹⁶−223349124600T¹⁴−2950615722600T¹²+ 16520555280600T¹⁰

−11401393059000T⁸+ 147193042090500T⁶+ 422927620447500T⁴−99598095922500T² +17840228332500, g5= 0, g4= 2275T²⁴+ 415380T²²+ 39897270T²⁰

−30649500T¹⁸−148598863875T¹⁶−1555875783000T¹⁴−2135859799500T¹²

+94593530241000T¹⁰−98463038821875T⁸+ 2611250197762500T⁶−159203362106250T⁴

−83293781287500T²−21709549378125, g3= 0, g2= 378T²⁶+ 114450T²⁴ +12621420T²²+ 89037900T²⁰−283320450T¹⁸+ 1545272004150T¹⁶

+12633981885000T¹⁴−118201467699000T¹²+ 1380551057313750T¹⁰ +7814079083238750T⁸+ 3521850108367500T⁶+ 4776100863187500T⁴

−1406247137268750T²−13291560843750, g1= 0, g0= 29T²⁸+ 13230T²⁶ +1814295T²⁴+ 74845260T²²−764250795T²⁰−204794909550T¹⁸

−3849793565625T¹⁶−34193820087000T¹⁴+ 942733356807375T¹² +1889980437035250T¹⁰+ 13147594251868125T⁸+ 3164572952887500T⁶

−3369410673890625T⁴−124940671931250T²+ 3987468253125,

h30= 0, h29= 0, h28=T, h27= 0, h26= 14T³−42T, h25= 0, h24= 91T⁵−1610T³

−1365T, h23= 0, h22= 364T⁷−14700T⁵+ 13860T³−64260T, h21= 0, h20= 1001T⁹

−67452T⁷+ 411894T⁵−97020T³−2546775T, h19= 0, h18= 2002T¹¹−190190T⁹ +2572500T⁷−3342780T⁵−6769350T³−39756150T, h17= 0, h16= 3003T¹³

−358974T¹¹+ 7821765T⁹−39225060T⁷−73327275T⁵+ 439963650T³+ 2114980875T, h15= 0, h14= 3432T¹⁵−471240T¹³+ 13736520T¹¹−135513000T⁹−793686600T⁷ +2779093800T⁵+ 51116751000T³+ 24754653000T, h13= 0, h12= 3003T¹⁷

−434280T¹⁵+ 14403060T¹³−248776920T¹¹−793072350T⁹−2707651800T⁷+ 375945664500T⁵

−81098577000T³+ 297337138875T, h11= 0, h10= 2002T¹⁹−275814T¹⁷

+8148168T¹⁵−362872440T¹³−114704100T¹¹−90682521300T⁹+ 1534457471400T⁷

−1772183107800T⁵+ 1117897625250T³+ 1493718266250)T, h9= 0, h8= 1001T²¹

−113190T¹⁹+ 836325T¹⁷−501931080T¹⁵−15705928350T¹³−400107348900T¹¹ +4976480045250T⁹−11450902365000T⁷+ 30510953731125T⁵−5820156483750T³

−21110374254375T, h7= 0, h6= 364T²³−24332T²¹−2084460T¹⁹

−528432660T¹⁷−31926371400T¹⁵+ 150244907400T¹³+ 11823972489000T¹¹

−3962494809000T⁹+ 7158970633500T⁷+ 132364254802500T⁵−455536249717500T³

−63681344842500T, h5= 0, h4= 91T²⁵+ 420T²³−1450890T²¹−337761900T¹⁹

−18543465675T¹⁷+ 274020553800T¹⁵+ 5724951088500T¹³+ 48513868893000T¹¹

+171111381643125T⁹+ 1334157649492500T⁷−1694171881946250T⁵−515712560737500T³

−131586452353125)T, h3= 0, h2= 14T²⁷+ 1470T²⁵−409500T²³

−111637260T²¹−3311799750T¹⁹+ 88973271450T¹⁷−3045655809000T¹⁵

−34947318861000T¹³+ 1002802178873250T¹¹+ 1999468016831250T⁹−6800738923597500T⁷ +4269249343012500T⁵−1666761729806250T³+ 204690036993750T, h1= 0,

h0=T²⁹+ 238T²⁷−43701T²⁵−14070420T²³−1034990775T²¹−32505382350T¹⁹ +259820563275T¹⁷+ 13855420996200T¹⁵+ 406907765530875T¹³+ 497730743291250T¹¹ +1983581436965625T⁹−10570073675332500T⁷−7864084888813125T⁵−224184326231250T³ +73103584640625)T

q30= 1, q29= 0, q28= 15T²+ 15, q27= 0, q26= 105T⁴−630T²+ 945, q25= 0, q24= 455T⁶−7875T⁴+ 4725T²+ 64575, q23= 0, q22= 1365T⁸−39900T⁶

+103950T⁴+ 548100T²+ 3709125, q21= 0, q20= 3003T¹⁰−114345T⁸+ 859950T⁶+ 4035150T⁴ +34827975T²+ 133656075, q19= 0, q18= 5005T¹²−200970T¹⁰+ 3649275T⁸+ 220500T⁶

+277333875T⁴+ 959505750T²+ 1115785125, q17= 0, q16= 6435T¹⁴−204435T¹²+ 10174815T¹⁰ +42170625T⁸+ 2030639625T⁶+ 7693410375T⁴−27357820875T²+ 24214372875, q15= 0,

q14= 6435T¹⁶−59400T¹⁴+ 21035700T¹²+ 451672200T¹⁰+ 2902331250T⁸+ 79622109000T⁶

−319613647500T⁴+ 191285955000T²+ 463546951875, q13= 0, q12= 5005T¹⁸ +155925T¹⁶+ 33585300T¹⁴+ 1481098500T¹²+ 42118035750T¹⁰+ 639849435750T⁸

−1190848837500T⁶+ 1787210932500T⁴+ 4850130403125T²+ 5581517878125, q11= 0 q10= 3003T²⁰+ 279510T¹⁸+ 40951575T¹⁶+ 2550025800T¹⁴+ 112585249350T¹² +1486454400900T¹⁰+ 2935114197750T⁸+ 10430710605000T⁶+ 58973741229375T⁴ +49590883833750T²+ 14657286301875, q9= 0, q8= 1365T²²+ 246015T²⁰ +36850275T¹⁸+ 2719544625T¹⁶+ 98273999250T¹⁴−830307854250T¹²

−8598553724250T¹⁰+ 211739487041250T⁸+ 162726680615625T⁶ +731900852746875T⁴−370328202365625T²+ 93610564228125, q7= 0, q6= 455T²⁴+ 134820T²²+ 23403870T²⁰+ 1942384500T¹⁸+ 36981653625T¹⁶

−1371507795000T¹⁴+ 3080287318500T¹²+ 299367020421000T¹⁰+ 3135310421315625T⁸ +10570433743012500T⁶−3151872128081250T⁴+ 1114718902762500T²

+412481438184375, q5= 0, q4= 105T²⁶+ 46725T²⁴+ 9856350T²² +950244750T²⁰+ 28094731875T¹⁸+ 11015463375T¹⁶−7594552507500T¹⁴ +191792925292500T¹²+ 6041183185209375T¹⁰+ 13451797993921875T⁸ +29454394197768750T⁶+ 1342447645218750T⁴+ 5894807234203125T²

+118075580755453125T⁴−5861578332093750T²+ 299060118984375, q1= 0, q0=T³⁰+ 855T²⁸+ 275625T²⁶+ 44441775T²⁴+ 4060783125T²² +207533751075T²⁰+ 5923312282125T¹⁸+ 77461769896875T¹⁶

+1691986493491875T¹⁴+ 21127132873153125T¹²+ 60580010182426875T¹⁰ +225021251512378125T⁸+ 50098108080234375T⁶+ 67806897644390625T⁴ +5881515673359375T²+ 19937341265625

Here we obtain the Akhmediev’s breather of order 5. Again, we can note the presence of N(N−1)−1 local maximums; the global maximum of |v|is equal to 11. We represent the modulus of v in the (x, t) coordinates and we get :

Figure 4: Solution of NLS, N=5a.

In the following cases, we only give the plots for the modulus ofv in the (x, t) coordinates.

If we choose a₁ = 1, a₂ = 1, a₃ = 1, a₄ = 1, b₁ = 1, b₂ = 1, b₃ = 1, b₄ = 1, we get :

Figure 5: Solution of NLS, N=5b.

If we choose a1 = 0, a2 = 0, a3 = 0, a4 = −0, a5 = 0, b1 = 1000000, b2 = 1000000,b3 = 1000000,b4 = 1000000, we get The shape of the modulus of v in the (x, t) coordinates is given by :

Figure 6: Solution of NLS, N=5c.

3.3 Case N=6

In the case N = 6, we realize an expansion at order 9 in ǫ. From (11), we get solutions of NLS equation. Taking the limit when ǫ→0, the solution of NLS equation takes the form

v(x, t) = n(x, t)

exp(2it−iϕ).