The height of an n th-order fundamental rogue wave for the nonlinear Schrödinger equation
aa r X i v : . [ n li n . S I] M a r THE HEIGHT OF AN n TH-ORDER FUNDAMENTAL ROGUE WAVE FORTHE NONLINEAR SCHR ¨ODINGER EQUATION
LIHONG WANG , , CHENGHAO YANG , JI WANG , JINGSONG HE ∗ School of Mechanical Engineering & Mechanics, Ningbo University, Ningbo, P. R. China State Key Laboratory of Satellite Ocean Environment Dynamics (Second Institute of Oceanography, SOA),P. R. China Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P. R. China
Abstract.
The height of an n th-order fundamental rogue wave q [ n ]rw for the nonlinear Schr¨odingerequation, namely (2 n +1) c , is proved directly by a series of row operations on matrices appearedin the n -fold Darboux transformation. Here the positive constant c denotes the height of theasymptotical plane of the rogue wave. Keywords : Rogue wave, Nonlinear Schr¨odinger equation, Darboux transformation
PACS numbers: 02.30.Ik, 42.65.Tg, 42.81.Dp, 05.45.Yv1.
Introduction
The high-intensity light is of growing importance in the creation of few-cycle pulses at at-tosecond scale [1–3], which in general implies the vast boost of the high bit-rate optical fibercommunication and the birth of the so called “attosecond physics” [4]. Parallel to above ex-tensive researches on this featured light from the view of the optics, another kind of largeamplitude optical wave, namely optical rogue wave (RW) for the nonlinear Schr¨odinger(NLS)equation, has been paid much attention since the experimental observation at year 2011 [5],almost 30 years later of the discovery of the solution [6]. This wave is described vividly as whichappears from nowhere and disappears without a trace [7]. The first-order RW, which is alsocalled Peregrine soliton [6], has a simple profile: two hollows allocated in the two sides of thecenter peak on a non-vanishing asymptotic plane, and maximum amplitude of the peak is threetimes of the height of the plane. However, higher-order RWs have several different interestingpatterns [8–16]. For example, a fundamental pattern is consisted of a central main peak andseveral gradually decreasing peaks allocated in two sides on a non-vanishing asymptotical plane.There exists a conjecture [17–19]: the height of a n th-order RW under the fundamental patternis (2 n + 1) times of the height of the asymptotical plane, which has been confirmed for severalRWs in both theory up to twelfth-order [20] and experiment results up to fifth-order [21]. Thus,higher-order RWs have larger amplitude (larger power), and hence can be more destructive insome disastrous events and be more useful to generate large-intensity short optical pulses. Thehigher-order RWs provide higher possibility for observation because of their large amplitudes,so that we can use or avoid them with more conveniences in real physical systems [22–25].Therefore, it is physically important to pay more attention on the height of the higher-orderRWs.The first mathematical proof of the above expression for the maximal amplitude was givenin [19]. Namely, the recurrence relations (7) and (8) of this work lead directly to the explicitformulae (2 n + 1) for the expression of interest. Another work that gave the proof for the same ∗ Corresponding author: [email protected], [email protected]. xpression for the maximal amplitude is [26]. The problem was also addressed in [27]. Thepurpose of this paper is to provide a direct proof of the above conjecture, which differs fromabove two works given in [19, 26]. The proof of the conjecture is a highly non-trivial workbecause the formula of the higher-order RW for the NLS is extremely cumbersome such thata fifth-order RW with eight parameters takes more than fourteen thousands pages [28]. Thenonlinear Schr¨odinger equation [29, 30] is the form of iq t + q xx + 2 | q | q = 0 . (1)Here q = q ( x, t ) represents the envelop of electric field, t is a normalized spatial variable and x is a normalized time variable. In optics, the squared modulus | q | usually denotes a measurablequantity optical power (or intensity). The NLS equation is a widely applicable integrable system[31] in physics, which is solved by several methods such as the inverse scattering method [31],the Hirota method [32] and the Darboux transformation(DT) [33]. Recently, the height ofmulti-breather of the NLS has been given in references [27, 34], but which can not imply theheight of the RWs because of the appearing of an indeterminate form under the degenerationof eigenvalues, namely a i → in reference [27] or ν i → n th-order breather,a new formula of the n th-order RW q [ n ]rw ( x, t ) is given by using a newly introduced function Cramer in section II. In section III, we provide a direct proof of the conjecture on the heightof the q [ n ]rw ( x, t ) by a series of row operations on the corresponding matrix appeared in the n -foldDT .2. The n th-order Rogue Waves of the NLS generated by the n -fold DT We recall briefly to construct higher-order RWs from higher-order breathers of the NLS bythe DT [13]. In order to construct the DT, it is necessary to introduce a proper “seed” asfollows: q [0] = ce iρ , (2)with ρ = ax + (2 c − a ) t, a ∈ R , c > i = −
1. Corresponding to this “seed” solution, theeigenfunction of the Lax equation associated with λ is expressed by φ ( λ ) = (cid:18) e i ρ e − i ρ (cid:19) ϕ ( λ ) , (3) ϕ ( λ ) = (cid:18) ϕ ( λ ) ϕ ( λ ) (cid:19) = (cid:18) ce id ( λ ) + i (cid:0) λ + a + h ( λ ) (cid:1) e − id ( λ ) ce − id ( λ ) + i (cid:0) λ + a + h ( λ ) (cid:1) e id ( λ ) (cid:19) , (4)in which h ( λ ) = q(cid:0) λ + a (cid:1) + c , d ( λ ) = ( x + (2 λ − a ) t ) h ( λ ). In order to construct n th-foldDT of the NLS, introduce following 2 n eigenfunctions f k − , φ ( λ k − ) = (cid:18) f k − , f k − , (cid:19) for λ k − , (5)and f k , (cid:18) − f ∗ k − , f ∗ k − , (cid:19) = (cid:18) f k, f k, (cid:19) for λ k = λ ∗ k − , (6) nd k = 1 , , , · · · , n . Here the asterisk denotes the complex conjugation. Using above “seed”solution and the eigenfunctions, the n -fold DT generates an n th-order breather of the NLS [13] q [ n ] = q [0] − i | ∆ [ n ]1 || ∆ [ n ]2 | . (7)Here, | · | denotes the determinant of a matrix, and two matrices in q [ n ] are∆ [ n ]1 = ( α , α , · · · , α n − , α n +1 ) , ∆ [ n ]2 = ( α , α , · · · , α n − , α n ) , (8)with α k − = λ k − f , λ k − f , ... λ k − n f n, = λ k − ϕ , λ k − ϕ , ... λ k − n ϕ n, e i ρ , ( k = 1 , , · · · , n + 1) , (9)and α k = λ k − f , λ k − f , ... λ k − n f n, = λ k − ϕ , λ k − ϕ , ... λ k − n ϕ n, e − i ρ , ( k = 1 , , · · · , n ) . (10)It can be seen that the n th-order breather q [ n ] has two variables x and t , two real parameters a and c , and n unique complex spectrum parameters λ k corresponding eigenfunctions f k ( k =1 , , , · · · , n − n th-order breather, introduce Cramer ( · ) function, Cramer (∆) = | ∆ || ∆ | . (11)Here, ∆ = ( ~a , ~a , · · · , ~a n | ~a n +1 ) is an augmented matrix of 2 n -dimensional column vectors ~a j ( j = 1 , , , · · · , n + 1), ∆ = ( ~a , ~a , · · · , ~a n − , ~a n +1 ) and ∆ = ( ~a , ~a , · · · , ~a n − , ~a n ) aretwo sub-matrices of ∆. Setting∆ [ n ] = ( α , α , · · · , α n − , α n | α n +1 ) = ∆ [ n ] ( f j , f j ; λ j ) , (12)in which the second equality just means that ∆ [ n ] is generated by functions f j ( j = 1 , , , · · · , n − f k and λ k ( k = 1 , , , · · · , n ) are given by (6). Using newly introduced Cramer function, then n th-order breather (7) can be re-written as q [ n ] = q [0] − i Cramer (cid:0) ∆ [ n ] (cid:1) , (13)which will be used to study the height of the n th-order RWs.Simplify determinants in numerator and denominator of Cramer (cid:0) ∆ [ n ] (cid:1) simultaneously byusing (9) and (10), then Cramer (cid:0) ∆ [ n ] ( f j , f j ; λ j ) (cid:1) = Cramer (cid:0) ∆ [ n ] ( ϕ j , ϕ j ; λ j ) (cid:1) · e iρ . (14)Equation (14) shows that Cramer (cid:0) ∆ [ n ] (cid:1) is generated equivalently by functions ϕ j ( j = 1 , , ,. . . , n −
1) except of a phase ρ . Moreover, the Cramer function has an important property. emma 1 Set P is an invertible 2 n × n matrix, then Cramer ( P · ∆) = Cramer (∆) . (15) Proof: Cramer ( P · ∆) = | P ∆ || P ∆ | = | P || ∆ || P || ∆ | = | ∆ || ∆ | = Cramer (∆) . (cid:3) This lemma shows that the ratio of the two determinants in
Cramer ( P · ∆) is invariant underelementary row operations, and will be used repeatedly to do row operations on the matricesappeared in the n th-order RW.It is known that an n th-order RW q [ n ]rw of the NLS is obtained from an n th-order breather q [ n ] (13) by higher-order Taylor expansion of an indeterminate form which is appeared from thedouble degeneration λ j → λ → λ = − a + ic ( j = 1 , , , · · · , n −
1) [13]. According to (13)and (14), the n th-order RW of the NLS becomes q [ n ]rw = q [ n ]rw ( x, t ) = q [0] − i Cramer (cid:0) ∆ [ n ]rw (cid:1) · e iρ . (16)Here, matrix ∆ [ n ]rw is defined by following elements (cid:0) ∆ [ n ]rw (cid:1) j,k = 1[ j +12 ]! ∂ [ j +12 ] ∂ √ ǫ (∆ [ n ] ( ϕ , ϕ ; λ + ǫ )) j,k | ǫ =0 , (17)because of1[ j +12 ]! ∂ [ j +12 ] ∂ √ ǫ (∆ [ n ] ( ϕ j , ϕ j ; λ + ǫ )) j,k | ǫ =0 = 1[ j +12 ]! ∂ [ j +12 ] ∂ √ ǫ (∆ [ n ] ( ϕ , ϕ ; λ + ǫ )) j,k | ǫ =0 , (18)under the double degeneration of eigenvalues λ j → λ = λ + ǫ . It can be seen clearly that∆ [ n ]rw is just generated by one eigenfunction f , or equivalently by two components ϕ and ϕ of ϕ . By the simplification of the formula of ∆ [ n ]rw , then∆ [ n ]rw = ∆ [ n ]rw ( ϕ , ϕ ) , (19) (cid:0) ∆ [ n ]rw ( ϕ , ϕ ) (cid:1) j,k = j +12 )! ∂ j +12 ∂ √ ǫ (( λ + ǫ ) k − ϕ ) | ǫ =0 , j ∈ Odd , k ∈ Odd , j +12 )! ∂ j +12 ∂ √ ǫ (( λ + ǫ ) k − ϕ ) | ǫ =0 , j ∈ Odd , k ∈ Even , j )! ∂ j ∂ √ ǫ ( − ( λ + ǫ ) k − ϕ ) ∗ | ǫ =0 , j ∈ Even , k ∈ Odd , j )! ∂ j ∂ √ ǫ (( λ + ǫ ) k − ϕ ) ∗ | ǫ =0 , j ∈ Even , k ∈ Even . (20)It it clear that the coefficient of ǫ l in above formula has zero contribution when l is an integer.Formula (16) of the n th-order RW q [ n ]rw ( x, t ) is crucial to study the height in this paper, becauseit is convenient to introduce row operations in order to simplify determinants in numerator anddenominator. 3. The Height of an n th-order fundamental RW The q [ n ]rw ( x, t ) in (16) is an n th-order fundamental rogue wave of the NLS [13], and the heightof its asymptotical plane is c . Because one can always move the central peak to origin of thecoordinate on the ( x, t )-plane, we shall set x = 0 and t = 0 in q [ n ]rw in following theorem to studythe height without the loss of the generality. heorem 1. The height of an n th-order fundamental RW is (cid:12)(cid:12)(cid:12) q [ n ]rw (cid:12)(cid:12)(cid:12) height = (2 n + 1) c . Here n isa positive integer and c is the height of the asymptotical plane. Proof:
We shall prove the theorem by four steps from q [ n ]rw ( x, t ) in (16). The main idea of thecalculation of Cramer (cid:16) ∆ [ n ]rw (cid:17) is to utilize row operations according to Lemma 1 , such thatthe matrix ∆ [ n ]rw becomes a strict upper triangular matrix. Step 1: Simplify the formula of q [ n ]rw (0 , . It is known by double degeneration of eigenvaluesthat there is only one eigenfunction ϕ in q [ n ]rw ( x, t ) associated with eigenvalue λ = λ + ǫ , then h = h ( λ ) = p ( ic + ǫ ) + c = √ icǫ + ǫ = h √ ǫ √ ic, (21)here, h = r ǫ ic = ∞ X k =0 c k ǫ k = c + c ǫ + c ǫ + · · · + c k ǫ k + · · · , (22)with Taylor expansion coefficients c = 1 , c = 12 · (2 ic ) , c k = ( − k − · · · · · · (2 k − k · (2 ic ) k · k ! , ( k = 2 , , · · · ) . (23)Setting x = 0 and t = 0 in q [ n ]rw (16), yields (cid:26) ϕ = i ( ǫ + h ) ,ϕ = i ( ǫ + h ) , (24)and then q [ n ]rw (0 ,
0) = c − i Cramer (cid:0) ∆ [ n ]rw ( i ( ǫ + h ) , i ( ǫ + h )) (cid:1) . (25)Because coefficient of ǫ l in the calculation of ∆ [ n ]rw through (20) has zero contribution when l isan integer, the formula of the n th-order RW is further simplified as q [ n ]rw (0 ,
0) = c − i Cramer (cid:0) ∆ [ n ]rw ( ih, ih )) (cid:1) . (26) Step 2: Remove the common factors in each row of ∆ [ n ]rw . According to (15) in
Lemma1 , the elementary row operations P for ∆ [ n ]rw ( ih, ih )) remove a nonzero common factor i √ ic inodd rows while − i √− ic in even rows, and the transformed matrix ∆ [ n ]rw ( h √ ǫ, h √ ǫ ) is givenin (37). This implies Cramer (cid:0) ∆ [ n ]rw ( ih, ih ) (cid:1) = Cramer (cid:0) P ∆ [ n ]rw ( ih, ih ) (cid:1) = Cramer (cid:0) ∆ [ n ]rw ( h √ ǫ, h √ ǫ ) (cid:1) , (27)and the second equality can be seen from (21). Here P = diag ( r , r , r , r , · · · , r , r ) ,r = ( i √ ic ) − and r = ( − i √− ic ) − . Therefore the n th-order RW becomes q [ n ]rw (0 ,
0) = c − i Cramer (cid:0) ∆ [ n ]rw ( h √ ǫ, h √ ǫ ) (cid:1) . (28) tep 3: Transform ∆ [ n ]rw to be a block upper triangular matrix. We further constructa series of row operation matrices P , P , · · · , P n − (see appendix) acting on ∆ [ n ]rw ( h √ ǫ, h √ ǫ )such that the transformed matrix∆ [ n ]rw ( √ ǫ, √ ǫ ) = P n − · · · P P ∆ [ n ]rw ( h √ ǫ, h √ ǫ )is given by (38). According to Lemma 1 , Cramer (cid:0) ∆ [ n ]rw ( h √ ǫ, h √ ǫ ) (cid:1) = Cramer (cid:0) ∆ [ n ]rw ( √ ǫ, √ ǫ ) (cid:1) , which leads to a new formula of the n th-order RW q [ n ]rw (0 ,
0) = c − i Cramer (cid:0) ∆ [ n ]rw ( √ ǫ, √ ǫ ) (cid:1) . (29) Step 4: Transform ∆ [ n ]rw to be a strict upper triangular matrix. We are now in a positionto do final row operations on the matrix in
Cramer , i.e. P n ∆ [ n ]rw ( √ ǫ, √ ǫ ), such that matrix givenby (38) becomes a strict upper triangular matrix, and thus the Cramer of the transformedmatrix is nic . Here the 2 n × n block diagonal matrix P n is given by P n = diag ( J, J, · · · , J ) , (30)and J = (cid:18) (cid:19) . (31)According to Lemma 1 , Cramer (cid:16) ∆ [ n ]rw ( √ ǫ, √ ǫ ) (cid:17) = nic . Substitute it back into formula (29),then q [ n ]rw (0 ,
0) = c − i ( nic ) = (2 n + 1) c. (32)Therefore, the height of an n th-order fundamental RW of the NLS is (cid:12)(cid:12) q [ n ]rw (cid:12)(cid:12) height = (cid:12)(cid:12) q [ n ]rw (0 , (cid:12)(cid:12) = (2 n + 1) c. (33)This is the end of the proof. (cid:3) Acknowledgments
Acknowledgments.
This work is supported by the NSF of China underGrant No.11671219, and the K.C. Wong Magna Fund in Ningbo University. This study is also sup-ported by the open Fund of the State Key Laboratory of Satellite Ocean Environment Dynamics,Second Institute of Oceanography(No. SOED1708).
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Step 3: P = · · · · · · − c · · · − c ∗ · · · − c · · · − c ∗ · · · − c m − · · · − c ∗ m − · · · (34) P = · · · · · · · · · · · · − c · · · − c ∗ · · · − c · · · − c ∗ · · · − c m − · · · − c ∗ m − · · · (35) · · · P n − = · · · · · · · · · · · · · · · · · · · · · − c · · · − c ∗ (36) he transformed matrix after Step 2 : λ λ · · · λ n − λ n − λ n − − λ ∗ λ ∗ · · · − λ ∗ n − λ ∗ n − − λ ∗ n c c λ c + 1 λ c + 1 · · · λ n − c + (cid:0) n − (cid:1) λ n − λ n − c + (cid:0) n − (cid:1) λ n − λ n c + (cid:0) n (cid:1) λ n − − c ∗ c ∗ − λ ∗ c ∗ − λ ∗ c ∗ + 1 · · · − λ ∗ n − c ∗ − (cid:0) n − (cid:1) λ ∗ n − λ ∗ n − c ∗ + (cid:0) n − (cid:1) λ ∗ n − − λ ∗ n c ∗ − (cid:0) n (cid:1) λ ∗ n − ... ... ... ... . . . ... ... ... c n − c n − λ c n − + c n − λ c n − + c n − · · · P n − j =0 (cid:0) n − j (cid:1) λ n − j − c n − j − P n − j =0 (cid:0) n − j (cid:1) λ n − j − c n − j − P n − j =0 (cid:0) nj (cid:1) λ n − j c n − j − − c ∗ n − c ∗ n − − λ ∗ c ∗ n − − c ∗ n − λ ∗ c ∗ n − + c ∗ n − · · · − P n − j =0 (cid:0) n − j (cid:1) λ ∗ n − j − c ∗ n − j − P n − j =0 (cid:0) n − j (cid:1) λ ∗ n − j − c ∗ n − j − − P n − j =0 (cid:0) nj (cid:1) λ ∗ n − j c ∗ n − j − (37) The transformed matrix after
Step 3 : λ λ λ λ · · · λ n − λ n − λ n − − λ ∗ λ ∗ − λ ∗ λ ∗ · · · − λ ∗ n − λ ∗ n − − λ ∗ n λ λ · · · (cid:0) n − n − (cid:1) λ n − (cid:0) n − n − (cid:1) λ n − (cid:0) nn − (cid:1) λ n − − − λ ∗ λ ∗ · · · − (cid:0) n − n − (cid:1) λ ∗ n − (cid:0) n − n − (cid:1) λ ∗ n − − (cid:0) nn − (cid:1) λ ∗ n − · · · (cid:0) n − n − (cid:1) λ n − (cid:0) n − n − (cid:1) λ n − (cid:0) nn − (cid:1) λ n − − · · · − (cid:0) n − n − (cid:1) λ ∗ n − (cid:0) n − n − (cid:1) λ ∗ n − − (cid:0) nn − (cid:1) λ ∗ n − ... ... ... ... ... ... . . . ... ... ...0 0 0 0 0 0 · · · (cid:0) n (cid:1) λ · · · − − (cid:0) n (cid:1) λ ∗ (38)(38)