Multi-parametric solutions to the NLS equation

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1 Multi-parametric solutions to the NLS equation Pierre Gaillard To cite this version: Pierre Gaillard. Multi-parametric solutions to the NLS equation <hal > HAL Id: hal Submitted on 26 Mar 2015 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Multi-parametric solutions to the NLS equation. + Pierre Gaillard, + Université de Bourgogne, Dijon, France : Pierre.Gaillard@u-bourgogne.fr, March 26, 2015 Abstract The structure of the solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) for the order N in terms of quasi rational functions is given here. We first give the proof that the solutions can be expressed as a ratio of two wronsians of order 2N and then two determinants by an exponential depending on t with 2N 2 parameters. It also is proved that for the order N, the solutions can be written as the product of an exponential depending on t by a quotient of two polynomials of degree N(N + 1) in x and t. The solutions depend on 2N 2 parameters and give when all these parameters are equal to 0, the analogue of the famous Peregrine breather P N. It is fundamental to note that in this representation at order N, all these solutions can be seen as deformations with 2N 2 parameters of the famous Peregrine breather P N. With this method, we already built Peregrine breathers until order N = 10, and their deformations depending on 2N 2 parameters. 1 Introduction The term of rogue wave was introduced in the scientific community by Draper in 1964 [1]. The usual 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). The first rogue wave recorded by scientific measurement in North Sea was made on the oil platform of Draupner in 1995, located between Norway and Scotland. Rogue waves in the ocean have led to many marine catastrophes; it is one of the reasons why these rogue waves turn out to be so important for the scientific community. It becomes a challenge to get a better understanding of their mechanisms of formation. The rogue waves phenomenon currently exceed the strict framewor of the study of ocean s waves and play a significant role in other fields; in nonlinear optics [2], Bose-Einstein condensate [3], atmosphere [4] and even finance [5]. Here, we consider the one dimensional focusing nonlinear Schrödinger equation 1

3 (NLS) to describe the phenomena of rogue waves. The first results concerning the NLS equation date bac the wors of Zaharov and Shabat in 1968 who solved it using the inverse scattering method [6, 7]. The case of periodic and almost periodic algebro-geometric solutions to the focusing NLS equation were first constructed in 1976 by Its and Kotlyarov [8, 9]. In 1977 Kuznetsov found the first breather type solution of the NLS equation [10]; a simular result was given by Ma [11] in The first quasi rational solutions to NLS equation were constructed in 1983 by Peregrine [12]. In 1986 Ahmediev, Eleonsi and Kulagin obtained the two-phase almost periodic solution to the NLS equation and obtained the first higher order analogue of the Peregrine breather [13]. Other analogues of Peregrine breathers of order 3 were constructed and initial data corresponding to orders 4 and 5 were described in a series of articles by Ahmediev et al., in particular in [14, 15] using Darboux transformations. Quite recently, many wors about NLS equation have been published using different methods. In 2010, rational solutions to the NLS equation were written as a quotient of two wronsians [16]. In 2011, the present author constructed in [17] another representation of the solutions to the NLS equation in terms of a ratio of two wronsians of even order 2N composed of elementary functions using truncated Riemann theta functions depending on two parameters; rational solutions were obtained when some parameter tended to 0. In 2012, Guo, Ling and Liu found another representation of the solutions as a ratio of two determinants [19] using generalized Darboux transform; a new approach was proposed by Ohta and Yang in [20] using Hirota bilinear method; finally, the present author has obtained rational solutions in terms of determinants which do not involve limits in [21] depending on two parameters. With this extended method, we present multi-parametric families of quasi rational solutions to the focusing NLS equation of order N in terms of determinants (determinants of order 2N) dependent on 2N 2 real parameters. With this representation, at the same time the well-nown ring structure, but also the triangular shapes also found by Ohta and Yang [20], Ahmediev et al. [25] are given. The aim of this paper is to prove the representation of the solutions to the focusing NLS equation depending this time on 2N 2 parameters; the proof presented in this paper with 2N 2 parameters has been never published. This is the first tas of the paper; then we deduce its particular degenerate representations in terms of a ratio of two determinants of order 2N. The second tas of the paper is to give the proof of the structure of the solution at the order N as the ratio of two polynomials of order N(N + 1) in x and t by an exponential depending on t. This representation maes possible to get all the possible patterns for the solutions to the NLS equation. It is important to stress that contrary to other methods, these solutions depending on 2N 2 parameters give the Peregrine breather as particular case when all the parameters are equal to 0 : for this reason, these solutions will be called 2N 2 parameters deformations of the Peregrine of order N. The paper is organized as follows. First of all, we express the solutions of the NLS equation using Fredholm determinants from these expressed in terms of 2

4 truncated functions theta of Riemann first obtained by Its, Rybin and Salle [9]; the representation given in theorem 2.1 is different from those given in [9]. From that, we prove the representation of the solutions of the NLS equation in terms of wronsians depending on 2N 2 parameters. We deduce a degenerate representation of solutions to the NLS equation depending a priori on 2N 2 parameters at the order N. Then we prove a theorem which states the structure of the quasi-rational solutions to the NLS equation. It was only conjectured in preceding wors [17, 21]. Families depending on 2N 2 parameters for the N-th order as a ratio of two polynomials of x and t multiplied by an exponential depending on t are obtained; it is proved that each of these polynomials have a degree equal to N(N +1). 2 Expression of solutions to the NLS equation in terms of wronsians 2.1 Solutions to the NLS equation in terms of θ functions For r = 1,3, we define θ r (x,t) = { {0;1} exp 2N ( ) 2µ 2N µ>ν,µ,ν=1 ln γν γ µ γ ν+γ µ ν ( 2N + ν=1 iκ νx 2δ ν t+x r,ν + ) } 2n µ=1,µ ν ln γ ν+γ µ +πiǫν γ ν γ µ +e ν ν, Inthisformula,thesymbol {0;1} 2N denotessummationoverall2n-dimensional vectors whose coordinates ν are either 0 or 1. The terms κ ν,δ ν,γ ν and x r,ν are functions of the parameters λ ν, 1 ν 2N; they are defined by the formulas : 1 λ ν 1+λ ν,; κ ν = 2 1 λ 2 ν, δ ν = κ ν λ ν, γ ν = x r,ν = (r 1)ln γν i γ, r = 1,3. (2) ν+i The parameters 1 < λ ν < 1, ν = 1,...,2N, are real numbers such that 1 < λ N+1 < λ N+2 <... < λ 2N < 0 < λ N < λ <... < λ 1 < 1 λ N+j = λ j, j = 1,...,N. The condition (3) implies that κ j+n = κ j, δ j+n = δ j+n, γ j+n = γ 1 j, x r,j+n = x r,j, j = 1,...,N. (4) Complex numbers e ν 1 ν 2N are defined in the following way : e j = ia j b j, e N+j = ia j +b j, 1 j N, a,b R. (5) ǫ ν {0;1}, ϕ, ν = 1...2N are arbitrary real numbers. With these notations, the solution of the NLS equation (1) (3) iv t +v xx +2 v 2 v = 0, (6) 3

5 can be expressed as ([9]) v(x,t) = θ 3(x,t) exp(2it iϕ), (7) θ 1 (x,t) 2.2 From θ functions to Fredholm determinants To get Fredholm determinants, we have to express the functions θ r defined in (1) in terms of subsets of [1,..,2N] ( ) θ r (x,t) = ( 1) ǫν γ ν +γ µ γ ν γ µ exp iκ ν x 2δ ν t+x r,ν +e ν. (8) J {1,..,2N} ν J ν J, µ/ J In (8), the symbol J {1,..,2N} denotes summation over all subsets J of indices of the set {1,..,2N}. Let I be the unit matrix and C r = (c j ) 1 j, 2N the matrix defined by : c νµ = ( 1) ǫν η µ γ ν +γ η η ν γ ν γ η exp(iκ νx 2δ ν t+x r,ν +e ν ), (9) ν J ǫ j = j 1 j N, ǫ j = N +j, N +1 j 2N. (10) Then det(i +C r ) has the following form det(i +C r ) = J {1,...,2N} ν J ( 1) ǫν ν J µ/ J γ ν +γ µ γ ν γ µ exp(iκ νx 2δ ν t+x r,ν +e ν ).(11) Comparing this last expression (11) with the formula (8) at the beginning of this section, we have clearly the identity θ r = det(i +C r ). (12) We can give another representation of the solutions to NLS equation. To do this, let s consider the matrix D r = (d j ) 1 j, 2N defined by : d νµ = ( 1) ǫν γ η +γ ν γ η γ µ exp(iκ νx 2δ ν t+x r,ν +e ν ). (13) η µ We have the equality det(i + D r ) = det(i + C r ), and so the solution of NLS equation taes the form v(x,t) = det(i +D 3(x,t)) exp(2it iϕ). (14) det(i +D 1 (x,t)) 4

6 Theorem 2.1 The function v defined by v(x,t) = det(i +D 3(x,t)) exp(2it iϕ). (15) det(i +D 1 (x,t)) is a solution of the focusing NLS equation with the matrix D r = (d j ) 1 j, 2N defined by d νµ = ( 1) ǫν γ η +γ ν γ η γ µ exp(iκ νx 2δ ν t+x r,ν +e ν ). η µ where κ ν, δ ν, x r,ν, γ ν, e ν being defined in(2), (3) and (5). 2.3 From Fredholm determinants to wronsians We want to express solutions to NLS equation in terms of wronsian determinants. For this, we need the following notations : φ r,ν = sinθ r,ν, 1 ν N, φ r,ν = cosθ r,ν, N +1 ν 2N, r = 1,3, (16) with the arguments Θ r,ν = κ ν x/2+iδ ν t ix r,ν /2+γ ν y ie ν /2, 1 ν 2N. (17) We denote W r (y) the wronsian of the functions φ r,1,...,φ r,2n defined by W r (y) = det[( µ 1 y φ r,ν ) ν,µ [1,...,2N] ]. (18) We consider the matrix D r = (d νµ ) ν,µ [1,...,2N] defined in (13). Then we have the following statement Theorem 2.2 det(i +D r ) = r (0) W r (φ r,1,...,φ r,2n )(0), (19) where ν=1 Θ r,ν) 2N ν 1 ν=2 r (y) = 22N exp(i 2N µ=1 (γ ν γ µ ). Proof : We start to remove the factor (2i) 1 e iθr,ν in each row ν in the wronsian W r (y) for 1 ν 2N. Then W r = 2N ν=1 e iθr,ν (2i) N (2) N W r, (20) 5

7 with W r = (1 e 2iΘr,1 ) iγ 1 (1+e 2iΘr,1 )... (iγ 1 ) 2 (1+( 1) 2N e 2iΘr,1 ) (1 e 2iΘr,2 ) iγ 2 (1+e 2iΘr,2 )... (iγ 2 ) 2 (1+( 1) 2N e 2iΘr,2 ).... (1 e 2iθr,2N ) iγ 2N (1+e 2iΘr,2N )... (iγ 2N ) 2 (1+( 1) 2n e 2iΘr,2N ) The determinant W r can be written as W r = det(α j e j +β j ), where α j = ( 1) (iγ j ) 1, e j = e 2iΘr,j, and β j = (iγ j ) 1, 1 j N, 1 2N, α j = ( 1) 1 (iγ j ) 1, e j = e 2iΘr,j, and β j = (iγ j ) 1, N + 1 j 2N, 1 2N. We want to calculate W r. To do this, we use the following Lemma Lemma 2.1 Let A = (a ij ) i,j [1,...,N], B = (b ij ) i,j [1,...,N], (H ij ) i,j [1,...,N], the matrix formed by replacing in A the jth row of A by the ith row of B Then det(a ij x i +b ij ) = det(a ij ) det(δ ij x i + det(h ij) det(a ij ) ) (21) Proof : We use the classical notations : Ã = (ã ji ) i,j [1,...,N] the transposed matrix in cofactors of A. We have the well nown formula A Ã = deta I. So it is clear that det(ã) = (det(a)). Thegeneraltermoftheproduct(c ij ) i,j [1,..,N] = (a ij x i +b ij ) i,j [1,..,N] (ã ji ) i,j [1,..,N] can be written as c ij = N s=1 (a isx i +b is ) ã js = x i n s=1 a isã js + n s=1 b isã js = δ ij det(a)x i +det(h ij ). We get det(c ij ) = det(a ij x i +b ij ) (det(a)) = (det(a)) N det(δ ij x i + det(hij) det(a) ). Thus det(a ij x i +b ij ) = det(a) det(δ ij x i + det(hij) det(a) ).. We denote U = (α ij ) i,j [1,...,2N], V = (β ij ) i,j [1,...,2N]. By applying the previous lemma, one obtains : W r = det(α ij e i +β ij ) = det(α ij ) det(δ ij e i + det(hij) det(α ij) ) = det(u) det(δ ije i + det(hij) det(u) ), (22) where (H ij ) i,j [1,...,N] is the matrix formed by replacing in U the jth row of U by the ith row of V defined previously. The determinant of U of Vandermonde type is clearly equal to det(u) = i N(2) (γ l γ m ). (23) 2N l>m 1 6

8 To calculate determinant W r, we must compute now det(h ij ). To do that, two cases must be studied : 1. For 1 j N. The matrix H ij is clearly of the VanderMonde type where the j-th row of U in U is replaced by the i-th row of V. Clearly, we have : det(h ij ) = ( 1) N(2N+1)+ (i) N(2) M, (24) where M = M(m 1,...,m 2N ) is the Vandermonde determinant defined by m = γ for j and m j = γ i. Thus we have : det(h ij ) = (i) N(2) 2N l> 1, (m l m ) = (i) N(2) 2N l>m 1,l j,m j (γ l γ m ) l<j ( γ i γ l ) l>j (γ l +γ i ), (25) = ( 1) j (i) N(2) 2N l>m 1,l j,m j (γ l γ m ) l j (γ l +γ i ). To evaluate W r, we must simplify the quotient q ij := det(hij) det(u) : q ij = ( 1)j (i) N(2) 2N l>m 1,l j,m j (γ l γ m) l j (γ l+γ i) i N(2) 2N l>m 1 (γ l γ m) = ( 1) j l j (γ l+γ i) l<j (γj γ l) l>j (γ l γ = ( 1) j l j (γ l+γ i) j) ( 1) j 1 l j (γ l γ j) = l j(γ l+γ i) l j (γ l γ. j) (26) We can replaceq ij by r ij defined by l j(γ l+γ i) l i (γ l γ, because det(δ i) ijx i + det(qij) det(a) ) = det(δ ij x i + det(rij) det(a) ) (similar matrices). We express r ij in terms of absolute value; as j [1;N] and 0 < γ 1 <... < γ N < 1 < γ 2N <... < γ N+1, we have : l i (γ l γ i ) = ( 1) i 1 l i γ l γ i, So the term r ij can be written as r ij = ( 1) i l j γ l+γ i l i γ l γ = i ( 1)ǫ(i) l j γ l+γ i with respect to the notations given in (10) and (13). l j (γ l +γ i ) = l j γ l +γ i. (27) l i γ l γ i = c ije 2iΘr,i(0), (28) 2. The same estimations for N +1 j 2N are made; deth ij is first det(h ij ) = ( 1) N(2N+1)+ (i) N(2) M, (29) with M = M(m 1,...,m 2N ) the Vandermonde determinant defined by m = γ for j and m j = γ i. Thus we have : det(h ij ) = (i) N(2) 2N l> 1, (m l m ) = (i) N(2) 2N l>m 1,l j,m j (γ l γ m ) l<j ( γ i γ l ) l>j (γ l +γ i ), (30) = ( 1) j 1 (i) N(2) 2N l>m 1,l j,m j (γ l γ m ) l j (γ l +γ i ). 7

9 The quotient q ij := det(hij) det(u) equals : q ij = ( 1)j 1 (i) N(2) 2N l>m 1,l j,m j (γ l γ m) l j (γ l+γ i) i N(2) 2N l>m 1 (γ l γ m) = ( 1)j 1 l j (γ l+γ i) l<j (γj γ l) l>j (γ l γ = ( 1) j 1 l j (γ l+γ i) j) ( 1) j 1 l j (γ l γ = l j(γ l+γ i) j) l j (γ l γ. j) We replace q ij by r ij defined by l j(γ l+γ i) l i (γ l γ i) (31), for the same reason as previously exposed. r ij is expressed in terms of absolute value; as j [N + 1;2N] and 0 < γ 1 <... < γ N < 1 < γ 2N <... < γ N+1, we have : l i (γ l γ i ) = ( 1) 2N i+n l i γ l γ i, So the term r ij can be written as r ij = ( 1) N+i l j γ l+γ i l i γ l γ = i ( 1)ǫ(i) l j γ l+γ i with respect to the notations given in (10) and (13). Replacing e i by e 2iΘr,i, det W r can be expressed as l j (γ l +γ i ) = l j γ l +γ i. (32) l i γ l γ i = c ije 2iΘr,i(0), (33) det W r = det(u) det(δ ij e i + det(hij) det(u) ) = det(u) det(δ ije i +r ij ) = det(u) 2N i=1 det(δ e 2iΘi ij +( 1) ǫ(i) l i γ l+γ i e 2iΘ r,i γ l γ i ). (34) We estimate the two members of the last relation (34) in y = 0, and using (23) we obtain the following result det W r (0) = i N(2) 2N l>m 1 (γ l γ m ) 2N i=1 e 2iΘr,i(0) det(δ ij +( 1) ǫ(i) l i γ l+γ i e 2iΘ r,i(0) γ l γ i ) = i N(2) 2N j 1 j=2 i=1 (γ j γ i )e 2i 2N i=1 Θr,i(0) det(δ ij +c ij ) = i N(2) 2N j 1 j=2 i=1 (γ j γ i )e 2i 2N i=1 Θr,i(0) det(i +C r ) = i N(2) 2N j 1 j=2 i=1 (γ j γ i )e 2i 2N i=1 Θr,i(0) det(i +D r ). Therefore, the wronsian W r given by (20 ) can be written as W r (φ r,1,...,φ r,2n )(0) = 2N j=1 eiθr,j(0) (2) 2N (i) N Wr (35) = 2N j=1 eiθr,j(0) (2) 2N (i) N i N(2) 2N j 1 j=2 i=1 (γ j γ i )e 2i 2N i=1 Θr,i(0) det(i +D r ) (36) = (2) 2N 2N j 1 j=2 i=1 (γ j γ i )e i 2N i=1 Θr,i(0) det(i +D r ). As a consequence det(i +D r ) = r (0)W r (φ 1,...,φ 2N )(0). (37) 8

10 2.4 Wronsian representation of solutions to the NLS equation From the initial formulation (15) we have v(x,t) = det(i +D 3(x,t)) det(i +D 1 (x,t)) exp(2it iϕ). Using (19), the following relation between Fredholm determinants and wronsians is obtained and det(i +D 3 ) = 3 (0) W 3 (φ r,1,...,φ r,2n )(0) det(i +D 3 ) = 3 (0) W 3 (φ r,1,...,φ r,2n )(0). As Θ 3,j (0) contains N terms x 3,j 1 j N and N terms x 3,j 1 j N, we have the equality 3 (0) = 1 (0), and we get the following result : Theorem 2.3 The function v defined by v(x,t) = W 3(φ 3,1,...,φ 3,2N )(0) W 1 (φ 1,1,...,φ 1,2N )(0) exp(2it iϕ). is a solution of the focusing NLS equation depending on two real parameters a and b with φ r ν defined in (16) φ r,ν = sin(κ ν x/2+iδ ν t ix r,ν /2+γ ν y ie ν /2), 1 ν N, φ r,ν = cos(κ ν x/2+iδ ν t ix r,ν /2+γ ν y ie ν /2), N +1 ν 2N, r = 1,3, κ ν, δ ν, x r,ν, γ ν, e ν being defined in(2), (3) and (5). 3 Families of multi-parametric solutions to the NLS equation in terms of a ratio of two determinants Solutions to the NLS equation as a quotient of two determinants are constructed. Similar functions defined in a preceding wor [21] are used, but modified as explained in the following. The following notations are needed : X ν = κ ν x/2+iδ ν t ix 3,ν /2 ie ν /2, Y ν = κ ν x/2+iδ ν t ix 1,ν /2 ie ν /2, for 1 ν 2N, with κ ν, δ ν, x r,ν defined in (2). Parameters e ν are defined by (5). Here, is the crucial point : we choose the parameters a j and b j in the form a j = =1 ã j 2+1 ǫ 2+1, b j = =1 9 b j 2+1 ǫ 2+1, 1 j N. (38)

11 Below the following functions are used : for 1 N, and ϕ 4j+1, = γ 4j 1 sinx, ϕ 4j+2, = γ 4j cosx, ϕ 4j+3, = γ 4j+1 sinx, ϕ 4j+4, = γ 4j+2 cosx, (39) ϕ 4j+1,N+ = γ 2N 4j 2 cosx N+, ϕ 4j+2,N+ = γ 2N 4j 3 sinx N+, ϕ 4j+3,N+ = γ 2N 4j 4 cosx N+, ϕ 4j+4,N+ = γ 2N 4j 5 sinx N+, (40) for 1 N. We define the functions ψ j, for 1 j 2N, 1 2N in the same way, the term X is only replaced by Y. for 1 N, and ψ 4j+1, = γ 4j 1 siny, ψ 4j+2, = γ 4j cosy, ψ 4j+3, = γ 4j+1 siny, ψ 4j+4, = γ 4j+2 cosy, (41) ψ 4j+1,N+ = γ 2N 4j 2 cosy N+, ψ 4j+2,N+ = γ 2N 4j 3 siny N+, ψ 4j+3,N+ = γ 2N 4j 4 cosy N+, ψ 4j+4,N+ = γ 2N 4j 5 siny N+, for 1 N. Then it is clear that can be written as q(x,t) := W 3(0) W 1 (0) (42) q(x,t) = 3 1 = det(ϕ j,) j, [1,2N] det(ψ j, ) j, [1,2N]. (43) Werecallthatλ j = 1 2jǫ 2. All thefunctions ϕ j, and ψ j, andtheirderivatives depend on ǫ and can all be prolonged by continuity when ǫ = 0. Then the following expansions are used ϕ j, (x,t,ǫ) = 1 (2l)! ϕ j,1[l] 2l ǫ 2l +O(ǫ 2N ), ϕ j,1 [l] = 2l ϕ j,1 ǫ 2l (x,t,0), ϕ j,1 [0] = ϕ j,1 (x,t,0), 1 j 2N, 1 N, 1 l N 1, ϕ j,n+ (x,t,ǫ) = 1 (2l)! ϕ j,n+1[l] 2l ǫ 2l +O(ǫ 2N ), ϕ j,n+1 [l] = 2l ϕ j,n+1 ǫ 2l (x,t,0), ϕ j,n+1 [0] = ϕ j,n+1 (x,t,0), 1 j 2N, 1 N, 1 l N 1. We have the same expansions for the functions ψ j,. ψ j, (x,t,ǫ) = 1 (2l)! ψ j,1[l] 2l ǫ 2l +O(ǫ 2N ), ψ j,1 [l] = 2l ψ j,1 ǫ 2l (x,t,0), 10

12 ψ j,1 [0] = ψ j,1 (x,t,0), 1 j 2N, 1 N, 1 l N 1, ψ j,n+ (x,t,ǫ) = 1 (2l)! ψ j,n+1[l] 2l ǫ 2l +O(ǫ 2N ), ψ j,n+1 [l] = 2l ψ j,n+1 ǫ 2l (x,t,0), ψ j,n+1 [0] = ψ j,n+1 (x,t,0), 1 j 2N, 1 N, N +1 2N.. Then we get the following result : Theorem 3.1 The function v defined by v(x,t) = exp(2it iϕ) det((n j) j, [1,2N] ) det((d j)j, [1,2N] ) (44) is a quasi-rational solution of the NLS equation (6) where iv t +v xx +2 v 2 v = 0, n j1 = ϕ j,1 (x,t,0), 1 j 2N n j = 2 2 ϕ j,1 ǫ 2 2 (x,t,0), n jn+1 = ϕ j,n+1 (x,t,0), 1 j 2N n jn+ = 2 2 ϕ j,n+1 ǫ 2 2 d j1 = ψ j,1 (x,t,0), 1 j 2N d j = 2 2 ψ j,1 (x,t,0), ǫ 2 2 d jn+1 = ψ j,n+1 (x,t,0), 1 j 2N d jn+ = 2 2 ψ j,n+1 2 N, 1 j 2N ǫ 2 2 The functions ϕ and ψ are defined in (39),(40), (41), (42). (x,t,0), (x,t,0), Proof : The columns of the determinants appearing in q(x, t) are combined successively to eliminate in each column (and N +) of them the powers of ǫ strictly inferior to 2( 1); then each common term in numerator and denominator is factorized and simplified; finally we tae the limit when ǫ goes to 0. Precisely, first of all, the components j of the columns 1 and N +1 are respectively equal by definition to ϕ j1 [0] +0(ǫ) for C 1, ϕ jn+1 [0]+0(ǫ) for C N+1 of 3, and ψ j1 [0]+0(ǫ) for C 1, ψ jn+1[0]+0(ǫ) for C N+1 of 1. At the first step of the reduction, we replace the columns C by C C 1 and C N+ by C N+ C N+1 for 2 N, for 3 ; the same changes for 1 are done. Each component j of the column C of 3 can be rewritten as 1 (2l)! ϕ j,1[l]( 2l 1)ǫ 2l andthecolumnc N+ replacedby 1)ǫ 2l for 2 N. For 1, we have the same reductions, each component j of the column C can be rewritten as C N+ replaced by 1 (2l)! ϕ j,n+1[l]( 2l 1 (2l)! ψ j,1[l]( 2l 1)ǫ 2l and the column 1 (2l)! ψ j,n+1[l]( 2l 1)ǫ 2l for 2 N. The term ǫ 2 for 2 N can factorized in 3 and 1 in each column and N +, and so these common terms can be simplified in numerator and denominator. 11

13 If we restrict the developments at order 1 in columns 2 and N +2, we get respectively ϕ j1 [1]+0(ǫ) for component j of C 2, ϕ jn+1 [1]+0(ǫ) for component j of C N+2 of 3, and ψ j1 [1]+0(ǫ) for component j of C 2, ψ jn+1[1]+0(ǫ) for componentj ofc N+2 of 1. This algorithmcan be continued up to the columns C N, C 2N of 3 and C N, C 2N of 1. Then taing the limit when ǫ tends to 0, q(x,t) can be replaced by Q(x,t) defined by : Q(x,t) := ϕ 1,1 [0]... ϕ 1,1 [N 1] ϕ 1,N+1 [0]... ϕ 1,N+1 [N 1] ϕ 2,1 [0]... ϕ 2,1 [N 1] ϕ 2,N+1 [0]... ϕ 2,N+1 [N 1] ϕ 2N,1 [0]... ϕ 2N,1 [N 1] ϕ 2N,N+1 [0]... ϕ 2N,N+1 [N 1] ψ 1,1 [0]... ψ 1,1 [N 1] ψ 1,N+1 [0]... ψ 1,N+1 [N 1] ψ 2,1 [0]... ψ 2,1 [N 1] ψ 2,N+1 [0]... ψ 2,N+1 [N 1] ψ 2N,1 [0]... ψ 2N,1 [N 1] ψ 2N,N+1 [0]... ψ 2N,N+1 [N 1] So the solution of the NLS equation taes the form : So we get the result given in (44). v(x,t) = exp(2it iϕ) Q(x,t) (45) 4 Families of quasi-rational solutions of order N depending on 2N 2 parameters Here a theorem which states the structure of the quasi-rational solutions to the NLS equation is given. It was only conjectured in preceding wors [17, 21]. Moreover we obtain here families depending on 2N 2 parameters for the Nthorder Peregrine breather including families with 2 parameters constructed in preceding wors and so we get other symmetries in these deformations than those were expected. In this section we use the notations defined in the previous sections. The functions ϕ and ψ are defined in (39), (40), (41), (42). Theorem 4.1 The function v defined by v(x,t) = exp(2it iϕ) det((n j) j, [1,2N] ) det((d j)j, [1,2N] ) (46) is a quasi-rational solution of the NLS equation (6) quotient of two polynomials N(x,t) and D(x,t) depending on 2N 2 real parameters ã j and b j, 1 j N 1. N and D are polynomials of degrees N(N +1) in x and t. 12

14 Proof : From the previous result (45), we need to analyze functions ϕ,1, ψ,1 and ϕ,n+1, ψ,n+1. Functions ϕ,j and ψ,j differ only by the term of the argument x 3,, so only the study of functions ϕ,j will be carried out. Then the study of functions ψ,j can be easily deduced from the analysis of ϕ,j. The expansions of these functions in ǫ are studied. We denote (l j ),j [1,2N] the matrix defined by l j = 2j 2 ǫ 2j 2ϕ 1, l,j+n = 2j 2 ǫ 2j 2ϕ,1+N, 1 j N, 1 2N, 0 x 0 ϕ meaningϕ. Eachcoefficientofthematrix(l j ),j [1,2N] mustbeevaluated, the power of x and t in the coefficient of ǫ 2(m 1) for the column m [1,2N]. We remar that with these notations, the matrix (l j ),j [1,2N] evaluated in ǫ = 0 is exactly (n j ),j [1,2N] defined in (45). Four cases must be studied depending on the parity of. 1. We study l 1 for odd, = 2s+1. l 1 = ( 1) s sin(2ǫ(1 ǫ 2 ) 1 2 x+4iǫ(1 ǫ 2 ) 1 2 (1 2ǫ 2 )t iln 1+iǫ(1 ǫ2 ) 1 2 e 1 ) ǫ 2 (1 ǫ 2 ) iǫ(1 ǫ 2 ) 1 2 p p p = ( 1) s sinǫ( c 2l ǫ 2l x+2i c 2l ǫ 2l (1 2ǫ 2 )t+2 ( 1) l ǫ 2l(1 ǫ2 ) 2l+1 2 (2l+1) ã l ǫ 2l +i bl ǫ 2l +O(ǫ p+1 )) ǫ 2 ( r g 2l ǫ 2l +O(ǫ r+1 )) p r = ( 1) s sinǫ( (c 2l x+d 2l t+f 2l +O(ǫ p+1 ))ǫ 2l ) ǫ 2 ( g 2l ǫ 2l +O(ǫ r+1 )) = q ( 1) l+s ǫ 2l (2l+1)! = q p r ( (c 2n x+d 2n t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l+1 ǫ 1 ( g 2l ǫ 2l +O(ǫ r+1 )) n=0 ( 1) l+s ǫ 2l (2l+1)! ( p r P n (x,t)ǫ 2n ) 2l+1 ǫ 1 g 2l ǫ 2l +O(ǫ t ) n=0 where P n (x,t) is a polynomial of order 1 in x and t. l,1 = = q q ǫ 2l ǫ 2l α α p=2l+1 α α p=2l+1 r β α0,...,α p P 0 (x,t) α0...p p (x,t) αp ǫ 2(α1+2α2+pαp) ǫ 2s g 2l ǫ 2l +O(ǫ t ) r Q α0,...,α p (x,t)ǫ 2(α1+2α2+pαp) ǫ 2s g 2l ǫ 2l +O(ǫ t ), where Q α0,...,α p (x,t) is a polynomial of order 2l+1 in x and t. The terms in ǫ 0 are obtained for l = 0 in the two summations with α 0 = 1. 13

15 For column m, we search the terms in ǫ 2m 2 with the maximal power in x and t. It is obtained for 2l+ 1 = 2m 2, which gives l = m s 1. The notations given in (44) are used. We get the following result Proposition 4.1 deg(n 2s+1,m ) = 2(m s) 1 for s m 1, n 2s+1,m = 0 for s m. (47) 2. We study l 1 for even, = 2s. l 1 = ( 1) s+1 cos(2ǫ(1 ǫ 2 ) 1 2 x+4iǫ(1 ǫ 2 ) 1 2 (1 2ǫ 2 )t iln 1+iǫ(1 ǫ2 ) 1 2 e 1 ) ǫ 2 (1 ǫ 2 ) iǫ(1 ǫ 2 ) 1 2 p p p = ( 1) s+1 cosǫ( c 2l ǫ 2l x+2i c 2l ǫ 2l (1 2ǫ 2 )t+2 ( 1) l ǫ 2l(1 ǫ2 ) 2l+1 2 (2l+1) ã l ǫ 2l +i bl ǫ 2l +O(ǫ p+1 )) ǫ 2 ( r g 2l ǫ 2l +O(ǫ r+1 )) p r = ( 1) s+1 cosǫ( (c 2l x+d 2l t+f 2l +O(ǫ p+1 ))ǫ 2l ) ǫ 2 ( g 2l ǫ 2l +O(ǫ r+1 )) q ( 1) l+d+1 ǫ 2l p r = ( (c 2n x+d 2n t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l ǫ 2 ( g 2l ǫ 2l +O(ǫ r+1 )) (2l)! = n=0 q ( 1) l+s+1 ǫ 2l p r ( P n (x,t)ǫ 2n ) 2l ǫ 2s 2 g 2l ǫ 2l +O(ǫ t ) (2l)! n=0 where P n (x,t) is a polynomial of order 1 in x and t. l,1 = = q q ǫ 2l ǫ 2l α α p=2l α α p=2l r β α0,...,α p P 0 (x,t) α0...p p (x,t) αp ǫ 2(α1+2α2+pαp) ǫ 2s 2 g 2l ǫ 2l +O(ǫ t ) r Q α0,...,α p (x,t)ǫ 2(α1+2α2+pαp) ǫ 2s 2 g 2l ǫ 2l +O(ǫ t ), where Q α0,...,α p (x,t) is a polynomial of order 2l in x and t. The terms in ǫ 0 are obtained for l = 0 in the two summations with α 0 = 1. For column m, we search the terms in ǫ 2m 2 with the maximal power in x and t. It is obtained for 2l+ 2 = 2m 2, which gives l = m s. With the notations given in (44), we have Proposition 4.2 deg(n 2s,m ) = 2(m s) for s m, n 2s,m = 0 for s > m. (48) 14

16 3. We study l M 2 +1 for odd, = 2s+1. l M 2 +1 = ( 1)s cos(2ǫ(1 ǫ 2 ) 1 2 x 4iǫ(1 ǫ 2 ) 1 2 (1 2ǫ 2 )t+iln 1+iǫ(1 ǫ2 ) iǫ(1 ǫ 2 ) 1 2 ǫ M 1 (1 ǫ 2 ) M 1 2 p p p = ( 1) s (cosǫ( c 2l ǫ 2l x 2i c 2l ǫ 2l (1 2ǫ 2 )t 2 ( 1) l ǫ 2l(1 ǫ2 ) 2l+1 2 (2l+1) ã l ǫ 2l +i bl ǫ 2l +O(ǫ p+1 )) ǫ M 1 ( r g 2l ǫ 2l +O(ǫ r+1 )) e M 2 +1) p r = ( 1) s (cosǫ( (c 2l x+d 2l t+f 2l )ǫ 2l +O(ǫ p+1 )) ǫ M 1 ( g 2l ǫ 2l +O(ǫ r+1 )) q ( 1) l+s ǫ 2l p r = ( (c 2n x+d 2n t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l ǫ M 1 ( g 2l ǫ 2l +O(ǫ r+1 )) (2l)! n=0 q ( 1) l+s ǫ 2l p r = ( P n (x,t)ǫ 2n +O(ǫ p+1 )) 2l ǫ M 2s 2 ( g 2l ǫ 2l +O(ǫ r+1 )) (2l)! n=0 where P n (x,t) is a polynomial of order 1 in x and t. = q ǫ 2l l, M 2 +1 = q ǫ 2l α α p=2l β α0,...,α p P 0 (x,t) α0 r...p p (x,t) αp ǫ 2(α1+2α2+pαp) ǫ M 2s 2 g 2l ǫ 2l +O(ǫ t ) α α p=2l r Q α0,...,α p (x,t)ǫ 2(α1+2α2+pαp) ǫ M 2s 2 g 2l ǫ 2l +O(ǫ t ), where Q α0,...,α p (x,t) is a polynomial of order 2l in x and t. The terms in ǫ 0 (column M 2 +1) are obtained for l = 0 in the two summations with α 0 = 1. For column M 2 +m, we search the terms in ǫ2m 2 with the maximal power in x and t. It is obtained for 2l+2(N s 1) = 2m 2, which gives l = m+s N. Then we get the following result Proposition 4.3 deg(n 2s+1,m+ M) = 2m+2s M for s M 2 2 m, n 2s+1,m = 0 for s < M 2 m.(49) 15

17 4. We study l,1+ M 2 l M for even, = 2s = ( 1)s sin(2ǫ(1 ǫ 2 ) 1 2 x 4iǫ(1 ǫ 2 ) 1 2 (1 2ǫ 2 )t+iln 1+iǫ(1 ǫ2 ) iǫ(1 ǫ 2 ) 1 2 ǫ M 1 (1 ǫ 2 ) M 1 2 p p p = ( 1) s sinǫ( c 2l ǫ 2l x 2i c 2l ǫ 2l (1 2ǫ 2 )t 2 ( 1) l ǫ 2l(1 ǫ2 ) 2l+1 2 (2l+1) ã l ǫ 2l +i bl ǫ 2l +O(ǫ p+1 )) ǫ M 1 ( r g 2l ǫ 2l +O(ǫ r+1 )) e M 2 +1) p r = ( 1) s sinǫ( (c 2l x+d 2l t+f 2l )ǫ 2l +O(ǫ p+1 )) ǫ M 1 ( g 2l ǫ 2l +O(ǫ r+1 )) = = q q ( 1) l+s ǫ 2l (2l+1)! ( 1) l+s ǫ 2l (2l+1)! p r ( (c 2n x+d 2n t+f 2n +O(ǫ p+1 ))ǫ 2n ) 2l+1 ǫ M ( g 2l ǫ 2l +O(ǫ r+1 )) n=0 p r ( P n (x,t)ǫ 2n +O(ǫ p+1 )) 2l+1 ǫ M 2s ( g 2l ǫ 2l +O(ǫ r+1 )) n=0 where P n (x,t) is a polynomial of order 1 in x and t. = q ǫ 2l l,1 = q ǫ 2l α α p=2l+1 β α0,...,α p P 0 (x,t) α0 r...p p (x,t) αp ǫ 2(α1+2α2+pαp) ǫ M 2s g 2l ǫ 2l +O(ǫ t ) α α p=2l+1 r Q α0,...,α p (x,t)ǫ 2(α1+2α2+pαp) ǫ M 2s g 2l ǫ 2l +O(ǫ t ), where Q α0,...,α p (x,t) is a polynomial of order 2l+1 in x and t. The terms in ǫ 0 are obtained for l = 0 in the two summations with α 0 = 1. For column M 2 +m, we search the terms in ǫ2m 2 with the maximal power in x and t. It is obtained for 2l+M = 2m 2, which gives l = m+s N 1. Using the notations given in (44), we get the following result Proposition 4.4 deg(n 2s,m+ M 2 n 2s,m+ M 2 ) = 2m+2s M 1 for s M 2 +1 M, = 0 for s < M 2 +1 m. (50) These results can be rewritten in the following way 16

18 Proposition 4.5 deg(n j, ) = 2 j for j 2, n j, = 0 for j > 2, deg(n j, ) = 2 +j 2M 1 for j 2M +1 2, n j, = 0 for j < 2M (51) The degree of the determinant of the matrix (n j ),j [1,2N] can now be evaluated. From the previous analysis, we see that x and t have necessarily the same power in eachn j. The maximalpowerin xand t, is successivelytaenin eachcolumn. It is realized by the following product N N n j,j n N+j,2N+1 j. j=1 j=1 Applying the result given in (51) we get N N deg(det(n j ),j [1,2N] ) = deg(n j,j )+ deg(n N+j,2N+1 j ) j=1 j=1 j=1 N N = 2j j + 2(M +1 j) 2M 1+ M 2 +j j=1 j=1 N N = j + N +1 j = N(N +1). j=1 Itisthesamefordeterminantdet(d j ),j [1,2N], wehavedeg(det(d j ),j [1,2N] ) = N(N +1). Thus the quotient det((n j)j, [1,2N] ) det((d j)j, [1,2N] ) defines a quotient of two polynomials, each of them of degree N(N + 1), and this proves the result. Parameters a 1 = =1 ãǫ and a 1 = =1 ãǫ must be chosen in the following way. The term ǫ must be a power of ǫ to get a nontrivial solution; ǫ must be a strictly positive number a in order to have a finite limit when ǫ goes to 0. If the power of ǫ is superior to 2N 2, the derivations going up to 2N 2, then this coefficient becomes 0 when the limit is taen when ǫ goes to 0 and so has no relevance in the expression of the limit. 17

19 5 Conclusion Here we proved the structure of quasi-rational solutions to the one dimensional focusing NLS equation at order N. They can be expressed as a product of an exponential depending on t by a ratio of two polynomials of degree N(N +1) in x and t. If we choose ã i = b i = 0 for 1 i, we obtain the classical (analogue) Peregrine breather. Thus these solutions appear as 2N 2-parameters deformations of the Peregrine breather of order N. The solutions for orders 3 and 4 first found by Matveev have also been explicitly found by the present author [26, 27]. We have also explicitly found the solutions at order 5 with 8 parameters [28]: these expressions are too extensive to be presented : it taes pages! For other orders 6, 7, 8, the solutions are also explicitly found but are too long to be published in any review. In the relative wors [29, 30, 32, 33, 34] only the analysis has been done and figures of deformations of the Peregrine breathers has been realized. The solutions for order 9 with 16 parameters[33] and respectively for order 10 with 18 parameters are also completely found [34]. We still insist on the fact that quasi rational solutions of NLS equation can be expressed as a quotient of two polynomials of degree N(N +1) in x and t dependent on 2N 2 real parameters by an exponential depending on time. Among these aforementioned solutions of order N, there is one which has the largest module : it is the solution obtained in this representation when all the parameters are equal to 0; one obtains the Peregrine breather order N. His importance is due to the fact that among the solutions of order N, its module is largest, equal to 2N +1. This result first formulated by Ahmediev has just been proved recently [35]. In the recent studies proposed by the author, the solutions of order N can be represented by their module in the plane (x; t). With the representation given in this article, one obtains at ordern, the configurationscontaining N(N+1)/2 peas, except the special case of Peregrine breather. These configurations can be classified according to the values of the parameters a i or b i for i varying between 1 and N 1. It is important to note that the role played by a i or b i for a given index i is the same one, in obtaining the configurations. The study refers to the analysis of the solutions when only one of the parameters is not equal to 0. Among these solutions, one distinguishes two types of configurations; for a 1 or b 1 not equal to 0, one observes triangular configurations with N(N + 1)/2 peas. For a i or b i not equal to 0 and 2 i N 1, one observes concentric rings. The simplest structure is obtained for a or b not equal to 0 : one obtains only one ring of 2N 1 peas with in his center Peregrine breather of order N 2; this fact was also first formulated by Ahmediev. The detailed study of the other structures is being analyzed. We hope to be able to give results soon. We can conclude that the method described in the present paper providesavery efficient and powerful tool to get explicit solutions to the NLS equation and to understand the behavior of rogue waves. There are currently many applications in different fields as recent wors by 18

20 Ahmediev et al. [36] or Kibler et al. [37] attest it in particular. This study leads to a better understanding of the phenomenon of rogue waves, and it would be relevant to go on with higher orders. References [1] Draper, L Frea ocean waves Oceanus V. 10, [2] Solli, D.R. Ropers, C. Koonath, P. Jalali, B Optical rogue waves Nature V. 450, [3] Bludov, Y.V. Konotop, V.V. Ahmediev, N Matter rogue waves, Phys. Rev. A V [4] Stenflo, L. Marlund, M Rogue waves in the atmosphere Jour. Of Plasma Phys. V. 76, N. 3-4, [5] Yan, Z.Y 2010 Financial rogue waves Commun. Theor. Phys. V. 54 N [6] Zaharov, V. E Stability of periodic waves of finite amplitude on a surface of a deep fluid J. Appl. Tech. Phys V [7] Zaharov, V. E. Shabat, A.B. 1972, Exact theory of two dimensional self focusing and one dimensinal self modulation of waves in nonlinear media Sov. Phys. JETP V [8] Its, A.R. Kotlyarov, V.P Explicit expressions for the solutions of nonlinear Schrödinger equation Docl. Aad. Nau. SSSR S. A V. 965, N. 11. [9] Its, A.R. Rybin, A.V. Salle, M.A Exact integration of nonlinear Schrödinger equation Teore. i Mat. Fiz. V. 74 N [10] Kuznetsov, E Solitons in a parametrically unstable plasma Sov. Phys. Dol. V [11] Ma, Y.C The perturbed plane-wave solutions of the cubic nonlinear Schrödinger equation Stud. Appl. Math. V [12] Peregrine, H Water waves, nonlinear Schrödinger equations and their solutions J. Austral. Math. Soc. Ser. B V [13] Ahmediev, N. Eleonsii, V. Kulagin, N Generation of periodic trains of picosecond pulses in an optical fiber : exact solutions Sov. Phys. J.E.T.P. V

21 [14] Ahmediev, N. Aniewicz, N.Soto-Crespo, J.M Rogue waves and rational solutions of nonlinear Schrödinger equation Physical Review E V. 80 N [15] Ahmediev, N.Aniewicz, A.Clarson, P.A Rogue waves, rational solutions, the patterns of their zeros and integral relations J. Phys. A : Math. Theor. V [16] Dubard, P. Gaillard, P. Klein, C. Matveev, V.B On multi-rogue waves solutions of the NLS equation and positon solutions of the KdV equation Eur. Phys. J. Special Topics, V [17] Gaillard, P Families of quasi-rational solutions of the NLS equation and multi-rogue waves J. Phys. A : Meth. Theor. V [18] Gaillard, P Wronsian representation of solutions of the NLS equation and higher Peregrine breathers Jour. Of Math. Sciences : Advances and Applications V [19] Guo, B. Ling, L. Liu, Q.P Nonlinear Schrödinger equation: Generalized Darboux transformation and rogue wave solutions Phys. Rev. E V [20] Ohta, Y. Yang, J General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation Pro. R. Soc. A V [21] Gaillard, P Degenerate determinant representation of solution of the NLS equation, higher Peregrine breathers and multi-rogue waves Jour. Of Math. Phys. V [22] Gaillard, P Wronsian representation of solutions of NLS equation and seventh order rogue waves Jour. Of Mod. Phys. V [23] Gaillard, P. 2013Two parameters deformations of ninth Peregrine breather solution of the NLS equation and multi rogue waves Jour. Of Math. V [24] Gaillard, P Two-parameters determinant representation of seventh order rogue waves solutions of the NLS equation Jour. Of Theor. And Appl. Phys. V [25] Kedziora, D.J.Aniewicz, A.Ahmediev, N Triangular rogue wave cascades Phys. Rev. E V [26] Gaillard, P Deformations of third order Peregrine breather solutions of the NLS equation with four parameters Phys. Rev. E V [27] Gaillard, P Six-parameters deformations of fourth order Peregrine breather solutions of the NLS equation J. Math. Phys. V

22 [28] Gaillard, P The fifth order Peregrine breather and its eight-parameters deformations solutions of the NLS equation Commun. Theor. Phys. V [29] Gaillard, P Ten parameters deformations of the sixth order Peregrine breather solutions of the NLS equation Physica Scripta V [30] Gaillard, P Higher order Peregrine breathers, their deformations and multi-rogue waves Jour. Of Phys. : conferences Series V [31] Gaillard, P. Two parameters wronsian representation of solutions of nonlinear Schrödinger equation, eight Peregrine breather and multi-rogue waves Jour. Of Math. Phys. V [32] Gaillard, P Two parameters wronsian representation of solutions of nonlinear Schrödinger equation, eight Peregrine breather and multi-rogue waves Jour. Of Math. Phys., V [33] Gaillard P, Gastineau M 2015 The Peregrine breather of order nine and its deformations with sixteen parameters solutions of the NLS equation accepted in P.L.A.. [34] Gaillard, P. Gastineau, M Eighteen parameter deformations of the Peregrine breather of order ten solutions of the NLS equation Int. Jour. of Mod. Phys. C V [35] Gaillard, P Other 2N-2 parameters solutions of the NLS equation and 2N+1 highest amplitude of the modulus of the N-th order AP breather accepted in Jour. Of Phys. A [36] Chabchoub, A. Hoffmann, N.P. Ahmediev, N Rogue Wave Observation in a Water Wave Tan Phys. Rev. Lett. V [37] Kibler, B.Fatome, J.Finot, C. Millot, G. Dias, F. Genty, G. Ahmediev, N. Dudley, J.M The Peregrine soliton in nonlinear fibre optics Nature Physics V