Rogue wave patterns in the nonlinear Schrödinger equation
RRogue wave patterns in the nonlinear Schr¨odinger equation
Bo Yang and Jianke Yang
Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05405, USA
Rogue wave patterns in the nonlinear Schr¨odinger equation are analytically studied. It is shownthat when an internal parameter in the rogue waves (which controls the shape of initial weak pertur-bations to the uniform background) is large, these waves would exhibit clear geometric structures,which are formed by Peregrine waves in shapes such as triangle, pentagon, heptagon and nonagon,with a possible lower-order rogue wave at its center. These rogue patterns are analytically deter-mined by the root structures of the Yablonskii-Vorob’ev polynomial hierarchy, and their orientationsare controlled by the phase of the large parameter. It is also shown that when multiple internalparameters in the rogue waves are large but satisfy certain constraints, similar rogue patterns wouldstill hold. Comparison between true rogue patterns and our analytical predictions shows excellentagreement.
1. INTRODUCTION
The name of rogue waves first appeared in oceanography, where it referred to large spontaneous and unexpectedwater wave excitations that are a threat even to big ships [1, 2]. Later, their counterparts in optics were also reported[3, 4]. Due to their physical importance, rogue waves have received intensive theoretical and experimental studies inthe past decade. On the theoretical front, analytical expressions of rogue waves have been derived in a wide variety ofintegrable physical models, such as the nonlinear Schr¨odinger (NLS) equation for wave-packet propagation in the oceanand optical systems [5–11], the derivative NLS equations for circularly polarized nonlinear Alfv´en waves in plasmasand short-pulse propagation in a frequency-doubling crystal [12–17], the Manakov equations for light transmission inrandomly birefringent fibers [18–23], and the three-wave resonant interaction equations [24–29]. On the experimentalfront, various rogue wave solutions in the NLS equation and defocusing Manakov equations have been observed inwater tanks and optical fibers [30–34]. In these experiments, intimate knowledge of theoretical rogue wave solutionsin the underlying nonlinear wave equations was utilized, which highlights the importance of theoretical developmentson rogue waves for practical rogue wave verifications and predictions.The study of rogue wave patterns is important as this information allows for the prediction of later rogue waveevents from earlier wave forms. Although graphs of low-order rogue waves have been plotted for many integrableequations, and simple patterns such as triangles and rings have been reported, richer patterns arising from high-orderrogue wave solutions have received little attention. For the NLS equation, preliminary investigations on rogue wavepatterns were made in [9, 35, 36] through Darboux transformation and numerical simulations. It was observed in [9]that if a N -th order rogue wave exhibits a single-shell ring structure, then the center of the ring is a ( N − a r X i v : . [ n li n . S I] J a n generalize our analytical results to cases where multiple internal parameters in the rogue waves are large but meetcertain constraints. Sec. 7 concludes the paper.
2. PRELIMINARIES
The nonlinear Schr¨odinger (NLS) equation i u t + 12 u xx + | u | u = 0 (1)arises in numerous physical situations such as water waves and optics [5, 37, 38]. In this article, we consider its roguewave solutions, which are rational solutions which approach a constant-amplitude continuous-wave background as x, t → ±∞ . Since this equation admits Galilean and scaling invariances, we can set the boundary conditions of theserogue waves as u ( x, t ) → e i t , x, t → ∞ , (2)without any loss of generality. Analytical expressions for general rogue waves in the NLS equation have been derived in [8, 10, 11] by variousmethods. However, those expressions are not the best for our solution analysis. Here, we present a simpler expressionfor these solutions, which can be derived by incorporating a new parameterization [17] into bilinear rogue waves ofRef. [11]. These simpler expressions of rogue waves are given by the following theorem.
Theorem 1.
The general NLS rogue waves under boundary conditions (2) are u N ( x, t ) = σ σ e i t , (3)where the positive integer N represents the order of the rogue wave, σ n is a N × N Gram determinant σ n = det ≤ i,j ≤ N (cid:16) φ ( n )2 i − , j − (cid:17) , (4)the matrix elements in σ n are defined by φ ( n ) i,j = min( i,j ) (cid:88) ν =0 ν S i − ν ( x + ( n ) + ν s ) S j − ν ( x − ( n ) + ν s ) , (5)vectors x ± ( n ) = (cid:0) x ± , x ± , · · · (cid:1) are defined by x ± = x ± i t ± n, x ± k = 0 , x +2 k +1 = x + 2 k (i t )(2 k + 1)! + a k +1 , x − k +1 = x − k (i t )(2 k + 1)! + a ∗ k +1 , (6)with k ≥ s = (0 , s , , s , · · · ) are coefficients fromthe expansion ∞ (cid:88) j =1 s j λ j = ln (cid:20) λ tanh (cid:18) λ (cid:19)(cid:21) , (7)the Schur polynomials S k ( x ), with x = ( x , x , . . . ), are defined by ∞ (cid:88) k =0 S k ( x ) (cid:15) k = exp (cid:32) ∞ (cid:88) k =1 x k (cid:15) k (cid:33) , (8)or more explicitly, S k ( x ) = (cid:88) l +2 l + ··· + ml m = k m (cid:89) j =1 x l j j l j ! , (9)and a k +1 ( k = 1 , , · · · , N −
1) are free irreducible complex constants.This theorem will be proved in Appendix A. Since these rogue waves approach a uniform background as t → −∞ , theinternal parameters { a k +1 } in these waves physically control the shape of initial small perturbations to this uniformbackground, which in turn decide the subsequent time evolution and the resulting pattern of rogue waves.As we will show, these rogue wave solutions will exhibit clear and recognizable patterns when some of these N − a , a , · · · , a N − ) get large. It turns out that the resulting rogue patterns are determined bythe root structures of the Yablonskii-Vorob’ev polynomial hierarchy, and this polynomial hierarchy and their rootstructures will be introduced next. Yablonskii-Vorob’ev polynomials arose in rational solutions of the second Painlev´e equation (P II ) [39, 40] w (cid:48)(cid:48) = 2 w + zw + α, (10)where α is an arbitrary constant. It has been shown that this P II equation admits rational solutions if and only if α = N is an integer. In this case, the rational solution is unique and is given by w ( z ; N ) = ddz ln Q N − ( z ) Q N ( z ) , N ≥ , (11) w ( z ; 0) = 0 , w ( z ; − N ) = − w ( z ; N ) , (12)and the polynomials Q N ( z ), now called the Yablonskii-Vorob’ev polynomials, are constructed by the following recur-rence relation Q N +1 Q N − = zQ N − (cid:2) Q N Q (cid:48)(cid:48) N − ( Q (cid:48) N ) (cid:3) , (13)with Q ( z ) = 1, Q ( z ) = z , and the prime denoting the derivative. Later, a determinant expression for thesepolynomials was found in [41]. Let p k ( z ) be the special Schur polynomial defined by ∞ (cid:88) k =0 p k ( z ) (cid:15) k = exp (cid:18) z(cid:15) − (cid:15) (cid:19) . (14)Then, Yablonskii-Vorob’ev polynomials Q N ( z ) are given by the N × N determinant [41] Q N ( z ) = c N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ( z ) p ( z ) · · · p − N ( z ) p ( z ) p ( z ) · · · p − N ( z )... ... ... ... p N − ( z ) p N − ( z ) · · · p N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (15)where c N = (cid:81) Nj =1 (2 j − p k ( z ) = 0 if k <
0. These polynomials are monic polynomials with integer coefficients[42]. The first few Yablonskii-Vorob’ev polynomials are Q ( z ) = z + 4 ,Q ( z ) = z + 20 z − ,Q ( z ) = z ( z + 60 z + 11200) . To define the Yablonskii-Vorob’ev polynomial hierarchy, we let p [ m ] k ( z ) be the generalized Schur polynomial definedby ∞ (cid:88) k =0 p [ m ] k ( z ) (cid:15) k = exp (cid:18) z(cid:15) − m m + 1 (cid:15) m +1 (cid:19) , (16)where m is a positive integer. Then, the Yablonskii-Vorob’ev hierarchy Q [ m ] N ( z ) are given by the N × N determinant[42] Q [ m ] N ( z ) = c N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p [ m ]1 ( z ) p [ m ]0 ( z ) · · · p [ m ]2 − N ( z ) p [ m ]3 ( z ) p [ m ]2 ( z ) · · · p [ m ]4 − N ( z )... ... ... ... p [ m ]2 N − ( z ) p [ m ]2 N − ( z ) · · · p [ m ] N ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)where p [ m ] k ( z ) = 0 if k <
0. When m = 1, Q [1] N ( z ) are the original Yablonskii-Vorob’ev polynomials Q N ( z ). When m > Q [ m ] N ( z ) give higher members of this polynomial hierarchy. All these Q [ m ] N ( z ) polynomials were conjectured tobe monic polynomials with integer coefficients as well [42]. The first few Q [2] N ( z ) polynomials are Q [2]2 ( z ) = z ,Q [2]3 ( z ) = z ( z − ,Q [2]4 ( z ) = z − z − . These Q [ m ] N ( z ) polynomials, through relations similar to (11)-(12), provide the unique rational solution for the P II hierarchy [42, 43]. It is noted that the determinant (17) for Q [ m ] N ( z ) is a Wronskian, because it is easy to see fromEq. (16) that p [ m ] k ( z ) = [ p [ m ] k +1 ] (cid:48) ( z ) . (18)Root structures of the Yablonskii-Vorob’ev polynomial hierarchy have been studied before [42–46]. Regarding thezero root, its multiplicity in Q N ( z ), Q [2] N ( z ) and Q [3] N ( z ) was presented in [42, 45]. Generalizing those results, we havethe following theorem. Theorem 2.
The general Yablonskii-Vorob’ev polynomial Q [ m ] N ( z ) has degree N ( N + 1) /
2, and is of theform Q [ m ] N ( z ) = z N ( N +1) / q [ m ] N ( ζ ) , ζ = z m +1 , (19)where q [ m ] N ( ζ ) is a polynomial with a nonzero constant term, and the integer N is given by the equation N ≡ N mod (2 m + 1) , or (20) N ≡ − N − m + 1) , (21)under the restriction of 0 ≤ N ≤ m . Due to this restriction on N , only one of Eqs. (20) and (21) canhold, and thus the resulting N value is unique.The proof of this theorem will be provided in Appendix B. This theorem gives the multiplicity of the root zero in any Q [ m ] N ( z ) polynomial. It also shows that the root structure of Q [ m ] N ( z ) is invariant under 2 π/ (2 m + 1)-angle rotationin the complex z plane. In the particular case of the original Yablonskii-Vorob’ev polynomials Q N ( z ) where m = 1,the above theorem shows that 0 ≤ N ≤
1. This means that zero is either not a root or a simple root for Q N ( z ), inagreement with previous results in [44, 45].On the determination of the unique N value in the above theorem, let us give an example. When N = 5 and m = 4, the N value under the restriction of 0 ≤ N ≤ N = 3.Regarding nonzero roots, it was shown in [44] that for the original Yablonskii-Vorob’ev polynomials Q N ( z ), allnonzero roots are simple. For the higher Yablonskii-Vorob’ev polynomial hierarchy Q [ m ] N ( z ), it was conjectured in [42]that all nonzero roots are also simple. In view of Theorem 2, this conjecture implies that the polynomial Q [ m ] N ( z ) has N p = 12 [ N ( N + 1) − N ( N + 1)] (22)nonzero simple roots. We have checked this conjecture for a myriad of ( N, m ) values and found it to hold in all ourexamples. Thus, we will assume it true and utilize it in our later analysis [see the sentence below Eq. (54) in Sec. 5].Roots of many Yablonskii-Vorob’ev polynomials Q [ m ] N ( z ) were plotted in [42], and highly regular and symmetricpatterns were observed. Due to the importance of these root structures to our work, we reproduce some of thoseroot plots in Fig. 1 for N = 6 and 1 ≤ m ≤
5. Boundaries of the roots in Q [ m ] N ( z ) in the large- N limit have beendetermined in [43, 46]. FIG. 1: Plots of the roots of the Yablonskii-Vorob’ev polynomial hierarchy Q [ m ] N ( z ) for N = 6 and 1 ≤ m ≤
3. ANALYTICAL PREDICTIONS OF ROGUE WAVE PATTERNS FOR A SINGLE LARGE INTERNALPARAMETER
Rogue wave solutions in Theorem 1 contain N − a , a , · · · , a N − . In thissection, we consider asymptotics of these rogue solutions when one of these internal parameters is large, while theother parameters remain O (1). Generalizations to cases where multiple internal parameters are large but satisfycertain constraints will be made in Sec. 6.Suppose a m +1 is large, where 1 ≤ m ≤ N −
1, and the other parameters are O (1). Then our results on thelarge- a m +1 asymptotics of rogue waves in Theorem 1 are summarized by the following two theorems. Theroem 3.
Far away from the origin, with √ x + t = O (cid:0) | a m +1 | / (2 m +1) (cid:1) , the N -th order rogue wave u N ( x, t ) in Eq. (3) separates into N p fundamental (Peregrine) rogue waves, where N p is given in Eq. (22).These Peregrine waves are ˆ u ( x − ˆ x , t − ˆ t ) e i t , whereˆ u ( x, t ) = 1 − t )1 + 4 x + 4 t , (23)and their positions (ˆ x , ˆ t ) are given byˆ x + i ˆ t = z (cid:18) − m + 12 m a m +1 (cid:19) m +1 , (24)with z being any one of the N p simple nonzero roots of Q [ m ] N ( z ). The error of this Peregrine wave ap-proximation is O ( | a m +1 | − / (2 m +1) ). Expressed mathematically, when (cid:2) ( x − ˆ x ) + ( t − ˆ t ) (cid:3) / = O (1),we have the following solution asymptotics u N ( x, t ; a , a , · · · , a N − ) = ˆ u ( x − ˆ x , t − ˆ t ) e i t + O (cid:16) | a m +1 | − / (2 m +1) (cid:17) , | a m +1 | (cid:29) . (25)When ( x, t ) is not in the neighborhood of any of these N p Peregrine waves, or √ x + t is larger than O (cid:0) | a m +1 | / (2 m +1) (cid:1) , u N ( x, t ) asymptotically approaches the constant background e i t as | a m +1 | → ∞ . Theroem 4.
In the neighborhood of the origin, where √ x + t = O (1), u N ( x, t ) is approximately a lower N -th order rogue wave u N ( x, t ), where N is given in Theorem 2 with 0 ≤ N ≤ N −
2, and u N ( x, t )is given by Eq. (3) with its internal parameters a , a , · · · , a N − being the first N − a , a , · · · , a N − ) of the original rogue wave u N ( x, t ). The error of this lower-order roguewave approximation u N ( x, t ) is O ( | a m +1 | − ). Expressed mathematically, when √ x + t = O (1), u N ( x, t ; a , a , · · · , a N − ) = u N ( x, t ; a , a , · · · , a N − ) + O (cid:0) | a m +1 | − (cid:1) , | a m +1 | (cid:29) . (26)If N = 0, then there will not be such a lower-order rogue wave in the neighborhood of the origin, and u N ( x, t ) asymptotically approaches the constant background e i t there as | a m +1 | → ∞ .These two theorems will be proved in Sec. 5. Remark 1.
Theorem 3 predicts that when a m +1 is large, the N -th order rogue wave (3) far away from the origincomprises N p Peregrine waves. The rogue pattern formed by these Peregrine waves has the same geometric shape asthe root structure of the polynomial Q [ m ] N ( z ), and thus this rogue pattern has 2 π/ (2 m + 1)-angle rotational symmetry.The only difference between the predicted rogue pattern and the root structure of Q [ m ] N ( z ) is a dilation and rotationbetween them due to the multiplication factor on the right side of Eq. (24). The angle of rotation is equal to the angleof the complex number − a m +1 divided by 2 m + 1, and the dilation factor is equal to [(2 m + 1)2 − m | a m +1 | ] / (2 m +1) . Remark 2.
On the right side of Eq. (24), we can pick any one of the (2 m + 1)-th root of − (2 m + 1)2 − m a m +1 ,because roots z of the polynomial Q [ m ] N ( z ) have 2 π/ (2 m + 1)-angle rotational symmetry, see the comment in theparagraph below Theorem 2.As a small application of the above two theorems, we explain the numerical observations in Ref. [9]. Under ourbilinear rogue solution (3), a N -th order rogue wave exhibits a ring structure when a N − is large (see Fig. 2 in thenext section). In this case, m = N −
1, and N = N − N -th order rogue wave is a ( N − N p = 2 N − π/ (2 m + 1)-, i.e., 2 π/ (2 N −
4. COMPARISON BETWEEN TRUE ROGUE PATTERNS AND OUR ANALYTICAL PREDICTIONS
In this section, we compare true rogue patterns with our analytical predictions. For this purpose, we first show inFig. 2 true rogue wave solutions (3) from the 2rd to 7th order, with large a , a , a , a , a and a in the first tosixth columns respectively. The specific value of the large parameter in each panel of this figure is listed in Table 1,and the other parameters in each solution are chosen as zero.It is seen that these rogue waves comprise a number of Peregrine waves forming triangular patterns for large a ,pentagon patterns for large a , heptagon patterns for large a , nonagon patterns for large a , hendecagon (eleven-sided polygon) patterns for large a , and tridecagon (thirteen-sided polygon) patterns for large a . In the literature,patterns on the diagonal of Fig. 2 are sometimes called single-shell ring structures [9]. In addition to these Peregrinewaves away from the origin, some of the rogue waves also contain a lower-order rogue wave at their centers. Forinstance, for the 7-th order rogue waves in the bottom row of Fig. 2, the first and fourth panels (with large a and a respectively) exhibit a Peregrine wave in their centers; the second panel (with large a ) exhibits a second-orderrogue wave in the center; the fifth panel (with large a ) exhibits a third-order rogue wave in the center; and the lastpanel (with large a ) exhibits a fifth-order rogue wave in the center. For our choices of parameters in rogue wavesof Fig. 2, these lower-order rogue waves in the center are all super-rogue waves, i.e., rogue waves with the highestpeak amplitude of their orders. We note by passing that the first five rows of rogue patterns in Fig. 2 resemble thoseplotted in Ref. [36] from Akhmediev breathers in the rogue-wave limit, although orientations between the two sets ofpatterns are very different. TABLE 1: Value of the large parameter for rogue waves in Fig. 2
N a a a a a a − − − − − − − − − − − − − − − − − − − − − Now, we compare these true rogue patterns in Fig. 2 with our analytical predictions. Our prediction | u ( p ) N ( x, t ) | from Theorems 3 and 4 can be assembled into a simple formula, (cid:12)(cid:12)(cid:12) u ( p ) N ( x, t ) (cid:12)(cid:12)(cid:12) = | u N ( x, t ) | + N p (cid:88) j =1 (cid:16)(cid:12)(cid:12)(cid:12) ˆ u ( x − ˆ x ( j )0 , t − ˆ t ( j )0 ) (cid:12)(cid:12)(cid:12) − (cid:17) , (27) FIG. 2: True NLS rogue wave patterns | u N ( x, t ; a , a , · · · , a N − ) | from solutions (3) when N ranges from 2 to 7 and one ofthe solution parameters is large (the other parameters are set as zero). The large parameter is labeled on top of each column,and its value for each panel is listed in Table 1. The center of each panel is always the origin x = t = 0, but the ( x, t ) intervalsdiffer slightly from panel to panel. For instance, in the bottom row, the left-most panel has − . ≤ x, t ≤ .
5, and theright-most panel has − ≤ x, t ≤ where ˆ u ( x, t ) is the Peregrine wave given in (23), their positions (ˆ x ( j )0 , ˆ t ( j )0 ) given by (24) with z being every oneof the N p simple nonzero roots of Q [ m ] N ( z ), and u N ( x, t ) is the lower-order rogue wave in Eq. (26) whose internalparameters ( a , a , · · · , a N − ) are the first N − a , a , · · · , a N − ) of the originalrogue wave u N ( x, t ). For true rogue waves in Fig. 2, all internal parameters except for a m +1 were chosen as zero,and N ≤ m (see Theorem 2). Then, all internal parameters in the predicted lower-order rogue wave u N ( x, t ) at theorigin are also zero.Our predicted ( N p , N ) values for rogue waves of Fig. 2 are displayed in Table 2, where m = 1 , , · · · , a , a , · · · , a respectively. These ( N p , N ) values provide our predictions for the number of Peregrine wavesaway from the origin ( x, t ) = (0 , TABLE 2: Predicted ( N p , N ) values for true rogue waves of Fig. 2 N m = 1 m = 2 m = 3 m = 4 m = 5 m = 62 (3, 0)3 (6, 0) (5, 1)4 (9, 1) (10, 0) (7, 2)5 (15, 0) (15, 0) (14, 1) (9, 3)6 (21, 0) (20, 1) (21, 0) (18, 2) (11, 4)7 (27, 1) (25, 2) (28, 0) (27, 1) (22, 3) (13, 5) We further compare our predicted whole solutions (27) with the true solutions of Fig. 2 for the same sets of( a , a , · · · ) parameter values. These predicted whole solutions (27) are displayed in Fig. 3, with identical ( x, t )intervals as in Fig. 2’s true solutions. It is seen that the predicted patterns are strikingly similar to the true ones. Inparticular, since our predicted Peregrine locations (24) in the ( x, t ) plane are given by all the non-zero roots of theYablonskii-Vorob’ev polynomials Q [ m ] N ( z ), multiplied by a fixed complex constant, predicted patterns formed by thesePeregrine waves then are simply the root structures of these Yablonskii-Vorob’ev polynomials under certain rotationand dilation, as is evident by comparing predicted rogue waves in Fig. 3 to the Yablonskii-Vorob’ev root structuresin Fig. 1 for N = 6. These predicted Peregrine patterns clearly match the true ones in Fig. 2 very well. This visualagreement shows the deep connection between NLS rogue patterns and root structures of the Yablonskii-Vorob’evhierarchy, as our theorem 3 predicts.Regarding our predictions u N ( x, t ) for centers of the rogue waves u N ( x, t ) in Fig. 2, we can show that our bilinearrogue wave solution (3) in Theorem 1 with all internal parameters set as zero gives the super-rogue wave. This meansthat our predictions u N ( x, t ) for the centers of true rogue waves are all lower-order super-rogue waves, which agreewith centers of true solutions shown in Fig. 2.Theorem 3 reveals that the orientation of the rogue pattern formed by Peregrine waves is controlled by the phase ofthe large parameter a m +1 . Specifically, the rogue-pattern orientation is the one of the root pattern of Q [ m ] N ( z ) rotatedby an angle of arg( − a m +1 ) / (2 m + 1), where “arg” represents the argument (angle) of a complex number. To checkthis prediction, we choose the 4-th order pentagon-shaped rogue waves, where a is large and the other parametersare set as zero. For three choices of the a value with the same modulus but different arguments, namely, 500 e − i π/ ,500 e i π/ and 500 e π/ , true rogue patterns from solutions (3) are displayed in the upper row of Fig. 4. As expected,orientations of these pentagon patterns indeed change as the argument of a varies. Using our formula (24), predictedlocations of Peregrine waves in the rogue pattern are shown in the lower row of Fig. 4. Comparison of the upper andlower rows of Fig. 4 shows that the predicted orientations are in perfect agreement with the true ones.Next, we make quantitative comparisons between true rogue waves and our predictions for large a m +1 , and verifythe error decay rate of O ( | a m +1 | − / (2 m +1) ) for the prediction of Peregrine-wave locations far away from the origin inTheorem 3, and the error decay rate of O ( | a m +1 | − ) for the prediction of the lower-order rogue wave at the centerin Theorem 4.For the quantitative comparison on Peregrine-wave locations away from the origin, we choose two patterns of 3rd-order rogue waves. One is a triangle pattern from large a , and we set arg( a ) = − π/
4; and the other is a pentagonpattern from large a , and we set a to be real positive. In each pattern, we choose all other parameters of the roguewave solutions to be zero. These triangul and pentagon patterns are shown schematically in Fig. 5(a, c) respectively.In each of these two patterns, we pick one of its Peregrine waves, which is marked by an arrow, and quantitativelycompare its true ( x , t ) location with our analytical prediction (24) as | a | or | a | increases. Here, the true locationof the Peregrine wave is defined as the ( x , t ) location where this Peregrine wave attains its maximum amplitude,and the error of our asymptotic prediction (ˆ x , ˆ t ) in Eq. (24) is defined aserror of Peregrine location = (cid:113) (ˆ x − x ) + (cid:0) ˆ t − t (cid:1) . These errors of Peregrine locations versus | a | or | a | are plotted as solid lines in panels (b) and (d) of Fig. 5 forthe triangular and pentagon patterns respectively. For comparison, the decay rates of | a | − / and | a | − / are alsodisplayed in these panels as dashed lines. We see that these errors of Peregrine locations indeed decay at the rate of | a m +1 | − / (2 m +1) , thus confirming the analytical error estimates (25) in Theorem 3. FIG. 3: Analytical predictions (27) for true rogue waves in Fig. 2. The x and t intervals here are identical to those in Fig. 2. To quantitatively compare our prediction in Theorem 4 on the lower-order rogue wave at the center with the truesolution, we choose a fifth-order rogue wave u ( x, t ) with large a and the other internal parameters set as zero. This | u ( x, t ) | solution with a = − N = 5 and m = 4. Since 5 ≡ − N = 3 from Eq. (21). Thus, according to Theorem 4, this u ( x, t ) solution contains a 3rd-order rogue wave u ( x, t ) in its center region, where all internal parameters ( a , a ) in this u ( x, t ) solution are zero. Such a u ( x, t )solution is a third-order super rogue wave. This predicted | u ( x, t ) | solution is displayed in Fig. 6(c), with the same( x, t ) internals as in the true center-region solution displayed in panel (b). Visually, this predicted center solution in(c) is identical to the true center solution in (b). Quantitatively, we have also obtained the errors in our predictedsolution u ( x, t ) at x = t = 0 . a increases in magnitude with arg( a ) = − π/
2. Our error isdefined as error of center region prediction = | u ( x, t ) − u ( x, t ) | x = t =0 . . FIG. 4: Orientations of 4-th order pentagon-shaped rogue waves with a = 500 e − i π/ (left column), 500 e i π/ (middle column)and 500 e π/ (right column) respectively; the other parameters in the rogue solutions are zero. Upper row: true rogue patternsfrom solutions (3); lower row: predicted locations of Peregrine waves from Eq. (24).FIG. 5: Decay of errors in our prediction (24) for the Peregrine location as | a | or | a | increases. (a) A triangle pattern of3rd-order rogue waves when | a | is large and arg( a ) = − π/
4. (b) Error versus | a | for the Peregrine location marked by anarrow in (a). (c) A pentagon pattern of 3rd-order rogue waves when | a | is large with arg( a ) = − π/
4. (b) Error versus | a | for the Peregrine location marked by an arrow in (c). The dependence of this error on | a | is plotted in Fig. 6(d). Comparison of this error decay with the | a | − decay[shown as a dashed line in panel (d)] indicates that this error is indeed of O ( | a | − ), confirming the error prediction(26) in Theorem 4.
5. PROOFS FOR THE ANALYTICAL RESULTS IN SEC. 3
In this section, we prove the analytical predictions on NLS rogue patterns in Theorems 3 and 4 of Sec. 3. Ourproof is based on an asymptotic analysis of the rogue wave solution (3), or equivalently, the determinant σ n in Eq.(4), in the large a m +1 limit. Proof of Theorem 3.
When a m +1 is large and the other parameters O (1) in the rogue wave solution (3), at1 FIG. 6: Decay of errors in our prediction u ( x, t ) for the center region of the rogue wave u ( x, t ) with large a . (a) A true 5-thorder rogue wave | u ( x, t ) | with a = − x, t ) intervals here are − ≤ x, t ≤ | u ( x, t ) | for the center region with the same ( x, t ) intervals as in (b). (d) Error decay of our predicted solution at the ( x, t )location of (0 . , .
5) as a increases in size with arg( a ) = − π/ ( x, t ) where √ x + t = O (cid:0) | a m +1 | / (2 m +1) (cid:1) , by denoting λ = a − / (2 m +1)2 m +1 (28)and recalling the expression of Schur polynomials in Eq. (9), we have S k ( x + ( n ) + ν s ) = S k (cid:0) x +1 , νs , x +3 , νs , · · · (cid:1) = λ − k S k (cid:0) x +1 λ, νs λ , x +3 λ , νs λ , · · · (cid:1) ∼ λ − k S k [( x + i t ) λ, , · · · , , , , · · · ] = S k ( x + i t, , · · · , , a m +1 , , · · · ) . (29)Thus, S k ( x + ( n ) + ν s ) ∼ S k ( v ) , | a m +1 | (cid:29) , (30)where v = ( x + i t, , · · · , , a m +1 , , · · · ) . (31)From the definition of Schur polynomials (8), S k ( v ) is given by ∞ (cid:88) k =0 S k ( v ) (cid:15) k = exp (cid:2) ( x + i t ) (cid:15) + a m +1 (cid:15) m +1 (cid:3) . (32)Thus, it is related to the polynomial p [ m ] k ( z ) in (16) as S k ( v ) = A k/ (2 m +1) p [ m ] k ( z ) , (33)where A = − m + 12 m a m +1 , z = A − / (2 m +1) ( x + i t ) . (34)Using these formulae, we find thatdet ≤ i,j ≤ N (cid:2) S i − j ( x + ( n ) + j s ) (cid:3) ∼ c − N A N ( N +1)2(2 m +1) Q [ m ] N ( z ) , | a m +1 | (cid:29) . (35)Similarly, det ≤ i,j ≤ N (cid:2) S i − j ( x − ( n ) + j s ) (cid:3) ∼ c − N ( A ∗ ) N ( N +1)2(2 m +1) Q [ m ] N ( z ∗ ) , | a m +1 | (cid:29) . (36)2Hereafter, S k = 0 when k < σ n in Eq. (4) as [11] σ n = (cid:88) ≤ ν <ν < ··· <ν N ≤ N − det ≤ i,j ≤ N (cid:20) ν j S i − − ν j ( x + ( n ) + ν j s ) (cid:21) × det ≤ i,j ≤ N (cid:20) ν j S i − − ν j ( x − ( n ) + ν j s ) (cid:21) . (37)Since the highest order term of a m +1 in this σ n comes from the index choices of ν j = j −
1, then σ n ∼ | α | | a m +1 | N ( N +1)2 m +1 (cid:12)(cid:12)(cid:12) Q [ m ] N ( z ) (cid:12)(cid:12)(cid:12) , | a m +1 | (cid:29) , (38)where α = 2 − N ( N − / c − N (cid:18) − m + 12 m (cid:19) N ( N +1)2(2 m +1) . (39)Since α is independent of n , the above equation shows that for large a m +1 , σ /σ ∼
1, i.e., the solution u ( x, t ) is onthe unit background, except at or near ( x, t ) locations (cid:0) ˆ x , ˆ t (cid:1) where z = A − / (2 m +1) (ˆ x + iˆ t ) (40)is a root of the polynomial Q [ m ] N ( z ), and such (cid:0) ˆ x , ˆ t (cid:1) locations are given by Eq. (24) in view of Eq. (34).Next, we show that when ( x, t ) is in the neighborhood of each of the (cid:0) ˆ x , ˆ t (cid:1) locations given by Eq. (24), i.e., when (cid:2) ( x − ˆ x ) + ( t − ˆ t ) (cid:3) / = O (1), the rogue wave u N ( x, t ) in Eq. (3) approaches a Peregrine wave ˆ u ( x − ˆ x , t − ˆ t ) e i t for large a m +1 . The asymptotic analysis above indicates that when ( x, t ) is in the neighborhood of (cid:0) ˆ x , ˆ t (cid:1) , thehighest power term | a m +1 | N ( N +1)2 m +1 in σ ( x, t ) vanishes. Thus, in order to determine the asymptotics of u N ( x, t ) in that( x, t ) region, we need to derive the leading order term of a m +1 in Eq. (37) whose order is lower than | a m +1 | N ( N +1)2 m +1 .For this purpose, we notice from Eq. (29) that when ( x, t ) is in the neighborhood of (cid:0) ˆ x , ˆ t (cid:1) , we have a more refinedasymptotics for S k ( x + ( n ) + ν s ) as S k ( x + ( n ) + ν s ) = λ − k S k (cid:0) x +1 λ, , · · · , , , , · · · (cid:1) (cid:2) O ( λ ) (cid:3) = S k (cid:0) x +1 , , · · · , , a m +1 , , · · · (cid:1) (cid:2) O ( λ ) (cid:3) , (41)i.e., S k ( x + ( n ) + ν s ) = S k (ˆ v ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) , (42)where ˆ v = ( x + i t + n, , · · · , , a m +1 , , · · · ) . (43)The polynomials S k (ˆ v ) are related to p [ m ] k ( z ) in (16) as S k (ˆ v ) = A k/ (2 m +1) p [ m ] k (ˆ z ) , (44)where A is as given in Eq. (34), and ˆ z = A − / (2 m +1) ( x + i t + n ).Now, we derive the leading order term of a m +1 in Eq. (37). This leading order term comes from two index choices,one being ν = (0 , , · · · , N − ν = (0 , , · · · , N − , N ).With the first index choice, in view of Eqs. (42) and (44), the determinant involving x + ( n ) in Eq. (37) is α a N ( N +1)2(2 m +1) m +1 Q [ m ] N (ˆ z ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) , (45)where α is given in Eq. (39). Expanding Q [ m ] N (ˆ z ) around ˆ z = z , where z is given in Eq. (40), and recalling Q [ m ] N ( z ) = 0, we have Q [ m ] N (ˆ z ) = A − / (2 m +1) (cid:2) ( x − ˆ x ) + i( t − ˆ t ) + n (cid:3) (cid:104) Q [ m ] N (cid:105) (cid:48) ( z ) (cid:104) O (cid:16) A − / (2 m +1) (cid:17)(cid:105) . (46)3Inserting this equation into (45) and recalling the definition of A in (34), the determinant involving x + ( n ) in Eq. (37)becomes (cid:2) ( x − ˆ x ) + i( t − ˆ t ) + n (cid:3) ˆ α a N ( N +1) − m +1) m +1 (cid:104) Q [ m ] N (cid:105) (cid:48) ( z ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) , (47)where ˆ α = α [ − (2 m + 1)2 − m ] − / (2 m +1) . Similarly, the determinant involving x − ( n ) in Eq. (37) becomes (cid:2) ( x − ˆ x ) − i( t − ˆ t ) − n (cid:3) ˆ α ∗ ( a ∗ m +1 ) N ( N +1) − m +1) (cid:104) Q [ m ] N (cid:105) (cid:48) ( z ∗ ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) . (48)Next, we consider the contribution in Eq. (37) from the second index choice of ν = (0 , , · · · , N − , N ). For thisindex choice, the determinant involving x + ( n ) in Eq. (37) isdet ≤ i ≤ N (cid:20) S i − ( x + ) , S i − ( x + + s ) , · · · , N − S i − ( N − [ x + + ( N − s ] , N S i − ( N +1) ( x + + N s ) (cid:21) . (49)Utilizing Eqs. (42)-(44), this determinant is2 − N ( N − / − A N ( N +1) − m +1) det ≤ i ≤ N (cid:104) p [ m ]2 i − (ˆ z ) , p [ m ]2 i − (ˆ z ) , · · · , p [ m ]2 i − ( N − (ˆ z ) , p [ m ]2 i − ( N +1) (ˆ z ) (cid:105) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) . (50)Recalling Eq. (18), we see that p [ m ]2 i − ( N +1) (ˆ z ) = [ p [ m ]2 i − N ] (cid:48) (ˆ z ). Thus, the determinant in the above expression is equalto c − N (cid:104) Q [ m ] N (cid:105) (cid:48) (ˆ z ), so that the determinant (49) becomes12 ˆ α a N ( N +1) − m +1) m +1 (cid:104) Q [ m ] N (cid:105) (cid:48) (ˆ z ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) . (51)When ( x, t ) is in the neighborhood of (ˆ x , ˆ t ), we expand (cid:104) Q [ m ] N (cid:105) (cid:48) (ˆ z ) around ˆ z = z to reduce this expression furtherto 12 ˆ α a N ( N +1) − m +1) m +1 (cid:104) Q [ m ] N (cid:105) (cid:48) ( z ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) . (52)Similarly, the determinant involving x − ( n ) in Eq. (37) becomes12 ˆ α ∗ ( a ∗ m +1 ) N ( N +1) − m +1) (cid:104) Q [ m ] N (cid:105) (cid:48) ( z ∗ ) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) . (53)Summarizing the above two contributions, we find that σ n ( x, t ) = | ˆ α | (cid:12)(cid:12)(cid:12)(cid:12)(cid:104) Q [ m ] N (cid:105) (cid:48) ( z ) (cid:12)(cid:12)(cid:12)(cid:12) | a m +1 | N ( N +1) − m +1) (cid:20) ( x − ˆ x ) + (cid:0) t − ˆ t (cid:1) − (2i) n (cid:0) t − ˆ t (cid:1) − n + 14 (cid:21) (cid:104) O (cid:16) a − / (2 m +1)2 m +1 (cid:17)(cid:105) . (54)Finally, we recall that nonzero roots are simple in Yablonskii-Vorob’ev polynomials Q N ( z ) [44]. In addition, nonzeroroots have also been conjectured to be simple in all the Yablonskii-Vorob’ev hierarchy Q [ m ] N ( z ) [42]. Assuming thisconjecture is true, then (cid:104) Q [ m ] N (cid:105) (cid:48) ( z ) (cid:54) = 0. This indicates that the above leading-order asymptotics for σ n ( x, t ) nevervanishes. Therefore, when a m +1 is large and ( x, t ) in the neighborhood of (cid:0) ˆ x , ˆ t (cid:1) , we get from (54) that u N ( x, t ) = σ σ e i t = e i t (cid:18) − t − ˆ t )]1 + 4( x − ˆ x ) + 4( t − ˆ t ) (cid:19) + O (cid:16) a − / (2 m +1)2 m +1 (cid:17) , (55)which is a Peregrine wave ˆ u ( x − ˆ x , t − ˆ t ) e i t , and the error of this Peregrine prediction is O (cid:16) a − / (2 m +1)2 m +1 (cid:17) . Theorem 3is then proved. Proof of Theorem 4.
To analyze the large- a m +1 behavior of the rogue wave u N ( x, t ) in the neighborhood ofthe origin, where √ x + t = O (1), we first rewrite the σ n determinant (4) into a 3 N × N determinant [11] σ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O N × N Φ N × N − Ψ N × N I N × N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (56)4where Φ i,j = 2 − ( j − S i − j [ x + ( n ) + ( j − s ], and Ψ i,j = 2 − ( i − S j − i [ x − ( n ) + ( i − s ]. Defining y ± to be thevector x ± without the a m +1 term, i.e., let x + = y + + (0 , · · · , , a m +1 , , · · · ) , x − = y − + (0 , · · · , , a ∗ m +1 , , · · · ) , (57)we find that the Schur polynomials of x ± are related to those of y ± as S j ( x + + ν s ) = [ j m +1 ] (cid:88) i =0 a i m +1 i ! S j − (2 m +1) i ( y + + ν s ) , S j ( x − + ν s ) = [ j m +1 ] (cid:88) i =0 ( a ∗ m +1 ) i i ! S j − (2 m +1) i ( y − + ν s ) , (58)where [ a ] represents the largest integer less than or equal to a . Using this relation, we express matrix elements of Φand Ψ in Eq. (56) through Schur polynomials S k ( y ± + ν s ) and powers of a m +1 and a ∗ m +1 .We need to determine the highest power term of a m +1 in the determinant (56). For that purpose, it may betempting to retain only the highest power term of a m +1 and a ∗ m +1 in each element of this determinant. That doesnot work though because it would result in multiple rows (and columns) which are proportional to each other, makingthe reduced determinant zero. The correct way is to first judiciously remove certain leading power terms of a m +1 and a ∗ m +1 from elements of the determinant through row and column manipulations, so that the remaining determinant,after retaining only the highest power term of a m +1 and a ∗ m +1 in each element, would be nonzero. These row andcolumn manipulations are described below.Suppose N ≡ N mod (2 m +1), i.e., N = k (2 m +1)+ N for some positive integer k , with 0 ≤ N ≤ m . We performthe following series of row operations to the matrix Φ so that certain high-power terms of a m +1 in its lower rows areeliminated. In the first round, we use the 1st to m -th rows of Φ to eliminate the highest-power term a ν m +1 from the[ ν (2 m + 1) + 1]-th up to the [ ν (2 m + 1) + m ]-th rows for each 1 ≤ ν ≤ k , so that the remaining terms in those rowshave the highest power a ν − m +1 . We also use the ( m + 1)-th to (2 m + 1)-th rows of Φ to eliminate the highest-powerterm a ν +12 m +1 from the [ ν (2 m + 1) + m + 1]-th to the [ ν (2 m + 1) + 2 m + 1]-th rows for each 1 ≤ ν ≤ k −
1, with theremaining terms in those rows having the highest power a ν m +1 . In each step, the highest power terms a ν m +1 or a ν +12 m +1 of each row are eliminated simultaneously, because the coefficient vector of those highest power terms in each rowbelow the (2 m + 1)-th is proportional to the coefficient vector of the highest power terms in the corresponding upperrow between the 1st and (2 m + 1)-th due to the relation (58).In the second round, we use the (2 m + 1 + 1)-th to (2 m + 1 + m )-th rows of the remaining matrix Φ to eliminate thehighest-power term a ν +12 m +1 from the [( ν +1)(2 m +1)+1]-th up to the [( ν +1)(2 m +1)+ m ]-th rows for each 1 ≤ ν ≤ k − a ν m +1 . We also use the (2 m + 1 + m + 1)-th to(2 m + 1 + 2 m + 1)-th rows of Φ to eliminate the highest-power term a ν +22 m +1 from the [( ν + 1)(2 m + 1) + m + 1]-th upto the [( ν + 1)(2 m + 1) + 2 m + 1]-th rows for each 1 ≤ ν ≤ k −
2, with the remaining terms in those rows having thehighest power a ν +12 m +1 . This process is repeated k rounds.At the end of this process, the i -th row of the remaining matrix Φ has the highest power a [( i + m ) / (2 m +1)]2 m +1 . Then, wekeep only the highest power terms of a m +1 in each row. Similar column operations are also performed on the matrixΨ. With these manipulations, we find that σ n is asymptotically reduced to σ n = β | a m +1 | k (2 m +1)+ k (2 N +1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O N × N (cid:101) Φ N × N − (cid:101) Ψ N × N I N × N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:2) O (cid:0) a − m +1 (cid:1)(cid:3) , (59)where β is a ( m, N )-dependent nonzero constant, matrices (cid:101) Φ N × N and (cid:101) Ψ N × N have the structures (cid:101) Φ N × N = (cid:32) L ( N − N ) × ( N − N ) O ( N − N ) × N O ( N − N ) × ( N − N ) M N × ( N − N ) (cid:98) Φ N × N O N × ( N − N ) (cid:33) , (60) (cid:101) Ψ N × N = U ( N − N ) × ( N − N ) (cid:99) M ( N − N ) × N O N × ( N − N ) (cid:98) Ψ N × N O ( N − N ) × ( N − N ) O ( N − N ) × N , (61) L i,j = S i − j (cid:2) y + + ( j − s (cid:3) , U i,j = S j − i (cid:2) y − + ( i − s (cid:3) , (62) (cid:98) Φ i,j = 2 − ( j − S i − j (cid:2) y + ( n ) + ( j − ν ) s (cid:3) , (cid:98) Ψ i,j = 2 − ( i − S j − i (cid:2) y − ( n ) + ( i − ν ) s (cid:3) , (63)5 ν = k (2 m + 1), and M , (cid:99) M are matrices of elements S j ( y + + ν s ) and S j ( y − + ν s ) respectively. Since L and U arerespectively lower triangular and upper triangular matrices with unit elements on the diagonal in view that S = 1and S j = 0 for j < σ n in Eq. (59) then is σ n = β | a m +1 | k (2 m +1)+ k (2 N +1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) O N × N (cid:98) Φ N × N − (cid:98) Ψ N × N I N × N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:2) O (cid:0) a − m +1 (cid:1)(cid:3) . (64)Finally, we notice that S j [ y ± + ( ν + ν ) s ] is related to S j ( y ± + ν s ) through S j (cid:2) y ± + ( ν + ν ) s (cid:3) = [ j/ (cid:88) i =0 S i ( ν s ) S j − i ( y ± + ν s ) . (65)Using this relation, the determinant in (64) can be reduced to one where ν is set to zero in the above (cid:98) Φ and (cid:98)
Ψmatrices given in Eq. (63). Such a determinant for σ n gives a N -th order rogue wave, whose internal parame-ters ( a , a , · · · , a N − ) are the first N − a , a , · · · , a N − ). Thus, in theneighborhood of the origin , u N ( x, t ; a , a , · · · , a N − ) = σ σ e i t = u N ( x, t ; a , a , · · · , a N − ) (cid:2) O (cid:0) a − m +1 (cid:1)(cid:3) , | a m +1 | (cid:29) , (66)which means that the original N -th order rogue wave u N ( x, t ) is approximated by a lower N -th order rogue wave u N ( x, t ), with the approximation error O (cid:0) a − m +1 (cid:1) .If N ≡ − N − m + 1) with 0 ≤ N ≤ m , Eq. (66) can also be derived by similar analysis, and thus thesame conclusion holds.Lastly, we recall that 1 ≤ m ≤ N −
1. In addition, 0 ≤ N ≤ m in view of Theorem 2. Furthermore, when m = N −
1, we find from Eq. (21) of Theorem 2 that N = N −
2. As a consequence, 0 ≤ N ≤ N −
2. Theorem 4 isthen proved.
6. ROGUE WAVE PATTERNS WHEN MULTIPLE INTERNAL PARAMETERS ARE LARGE
The rogue patterns in Theorems 3 and 4 were derived under the assumption that only one of the internal parametersin the rogue wave solutions (3) was large, and the other parameters were O (1). It turns out that those results can begeneralized to more general parameter conditions. We discuss these generalizations in this section.Regarding the generalization of Theorem 3, we can show that if a m +1 is large, and the other parameters a , . . . , a m − , a m +3 , . . . , a N − are also large but satisfy the conditions a j +1 = o (cid:18) a j m +1 m +1 (cid:19) , j (cid:54) = m, (67)then, far away from the origin, with √ x + t = O (cid:0) | a m +1 | / (2 m +1) (cid:1) , the rogue wave u N ( x, t ) still separatesinto N p Peregrine waves, whose positions (ˆ x , ˆ t ) are given by Eq. (24). Expressed mathematically, when (cid:2) ( x − ˆ x ) + ( t − ˆ t ) (cid:3) / = O (1), we have u N ( x, t ; a , a , · · · , a N − ) −→ ˆ u ( x − ˆ x , t − ˆ t ) e i t as | a m +1 | → ∞ . (68)The reason is that, when √ x + t = O (cid:0) | a m +1 | / (2 m +1) (cid:1) , under the same notation λ = a − / (2 m +1)2 m +1 as in Eq. (28),the condition (67) means that a j +1 = o ( λ − j ) for j (cid:54) = m . Thus, x +2 j +1 λ j +1 = o ( λ ) for j (cid:54) = m . Then, in view ofEq. (9), we have S k ( x + ( n ) + ν s ) = S k (cid:0) x +1 , νs , x +3 , νs , · · · (cid:1) = λ − k S k (cid:0) x +1 λ, νs λ , x +3 λ , νs λ , · · · (cid:1) = λ − k S k (cid:0) x +1 λ, , · · · , , , , · · · (cid:1) [1 + o ( λ )] = S k (cid:0) x +1 , , · · · , , a m +1 , , · · · (cid:1) [1 + o ( λ )] . (69)This relation is the counterpart of Eq. (41) in the proof of Theorem 3. Due to this relation and a similar one on S k ( x − ( n ) + ν s ), the calculations in the proof of Theorem 3 can still go through. The only difference is that theerror of the present Peregrine approximation may be different. Indeed, the previous analysis, combined with theabove equation (69), indicates that the error of the current Peregrine approximation (68) is the largest order among6 O (cid:16) a j +1 /a j/ (2 m +1)2 m +1 (cid:17) , where 1 ≤ j ≤ N − j (cid:54) = m . So, if a j +1 = O (cid:16) a (2 j − / (2 m +1)2 m +1 (cid:17) or smaller for all j (cid:54) = m , then the error of the current Peregrine approximation (68) would remain the same as that given in Eq. (25)of Theorem 3, i.e., O (cid:0) | a m +1 | − / (2 m +1) (cid:1) . Otherwise, this error would be larger than O (cid:0) | a m +1 | − / (2 m +1) (cid:1) , whichmeans that the error would decay to zero slower than the rate | a m +1 | − / (2 m +1) when a m +1 gets large.Regarding the generalization of Theorem 4, we can show that if a m +1 is large, and a , · · · , a m − = O (1) , a m +3 , · · · , a N − = O ( a m +1 ) , (70)then Theorem 4 remains valid. Specifically, the asymptotics (26), including its error estimates, still holds. The prooffor this is an extension of the proof for Theorem 4, and will be presented in Appendix C.To demonstrate these generalized results on rogue patterns, we consider an example of a 7th order rogue wave u ( x, t ) with parameter choices of a = 1 , a is large , a = a , a = 2 a , a = 3 a , a = 4 a . (71)This set of parameters satisfy both conditions (67) and (70). Thus, according to the above discussions, both Theo-rems 3 and 4 remain valid, including their error estimates, since a j +1 = O (cid:16) a (2 j − / (2 m +1)2 m +1 (cid:17) or smaller for all j (cid:54) = m here. These theorems predict that far away from the origin, this u ( x, t ) would split into 25 Peregrine waves, whose( x, t ) locations are given by Eq. (24). Near the origin, this u ( x, t ) would reduce to a 2nd-order rogue wave u ( x, t )with a = 1. To verify these predictions, we choose a = − | u ( x, t ) | is plotted in Fig. 7(a), and its center region is amplified and shown in panel (b). Our asymptotic predictions (27) fromTheorems 3 and 4 for the same ( x, t ) intervals as in panels (a) and (b) are displayed in panels (c) and (d) respectively.One can clearly see that our predictions are almost indistinguishable from the true solutions. FIG. 7: A 7th order rogue wave | u ( x, t ) | for generalized parameters (71) with a = − − . ≤ x, t ≤ .
5; (b) zoomed-in plot of the center region of the true solutionmarked by a dashed-line box in panel (a); (c) predicted solution with the same ( x, t ) internals as in (a); (d) zoomed-in plot ofthe center region of the predicted solution.
7. CONCLUSIONS AND DISCUSSIONS
In this paper, we have analytically studied rogue wave patterns in the NLS equation. We have shown that whenone of the internal parameters in the bilinear rogue wave solutions is large, these waves would exhibit clear geometricstructures, which comprise Peregrine rogue waves organized in shapes such as triangle, pentagon, heptagon andnonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined bythe root structures of the Yablonskii-Vorob’ev polynomial hierarchy, and their orientations are controlled by the phaseof the large parameter. We have also generalized these results and shown that, when multiple internal parameters inthe rogue waves are large but satisfy certain constraints [such as (67) and (70)], then the same rogue patterns wouldpersist. Comparison between true rogue patterns and our analytical predictions has shown excellent agreement. Asa small application of our analytical results, the numerical observation in [9] on single-shell ring structures has beenexplained. Our results reveal the deep connection between NLS rogue wave patterns and the Yablonskii-Vorob’evpolynomial hierarchy, and make prediction of sophisticated patterns in higher-order NLS rogue waves possible.7It turns out that this connection between rogue wave patterns and the Yablonskii-Vorob’ev polynomial hierarchy isnot restricted to the NLS equation. We have found that such connections persist in many other integrable equations,such as the derivative NLS equation, the Boussinesq equation, the Manakov equations and others. This generalconnection then gives rise to universal rogue wave patterns in integrable systems. This university result was brieflyreported in [47]. Its details are beyond the scope of this paper and will be pursued in future publications.In this article, NLS rogue wave patterns are determined by the complex roots of the Yablonskii-Vorob’ev polynomialhierarchy, and these roots are the pole locations of rational solutions to the P II hierarchy (see Sec. 2 2.2 and [42,43]). Interestingly, in very different contexts, somewhat similar results have also been reported. For instance, inthe semiclassical NLS equation after wave breaking, a sequence of Peregrine waves appear, and their locations aredetermined by the poles of the tritronqu´ee solution to the first Painlev´e (P I ) equation [48]. In the semiclassical sine-Gordon equation with initial conditions near the separatrix of a simple pendulum, superluminal (infinite velocity)kinks that appear in the solution are linked to the real roots of the Yablonskii-Vorob’ev polynomials associated withrational solutions of the P II equation [49]. This connection of wave phenomena to rational solutions of the Painlev´eequations may arise again in other wave systems in the future. Acknowledgment
This material is based upon work supported by the National Science Foundation under award number DMS-1910282,and the Air Force Office of Scientific Research under award number FA9550-18-1-0098.
Appendix A
In this appendix, we briefly derive the bilinear rogue waves presented in Theorem 1. These new rogue waveexpressions can be obtained by applying a new parameterization developed in Ref. [17] to the bilinear derivation ofrogue waves in Ref. [11]. Specifically, instead of the previous choice (3.11) for the matrix element m ( n ) ij in Ref. [11],which we denote as φ ( n ) ij in this paper, we now choose φ ( n ) ij = 1 i ! ( p∂ p ) i j ! ( q∂ q ) j φ ( n ) (cid:12)(cid:12)(cid:12)(cid:12) p = q =1 , (72)where φ ( n ) = ( p + 1)( q + 1)2( p + q ) (cid:18) − pq (cid:19) n exp (cid:32) ξ + η + ∞ (cid:88) k =1 a k (ln p ) k + ∞ (cid:88) k =1 b k (ln q ) k (cid:33) , (73) ξ = px + p x , η = qx − q x . (74)and a k , b k are arbitrary complex constants. Obviously, the function τ n = det ≤ i,j ≤ N (cid:16) φ ( n )2 i − , j − (cid:17) with the abovechoice of φ ( n ) ij also satisfies the bilinear equations (3.14) in [11]. Then, when we set b k = a ∗ k , x = x and x = i t/
2, this τ n function would satisfy the bilinear equations (3.1) of the NLS equation in [11] [with t switched to − t/ t rescaling]. Applying the same reduction technique of[11] to the above new τ n solution, we can remove the differential operators in the expression (72) of its matrix element φ ( n ) ij and reduce it to σ n = det ≤ i,j ≤ N (cid:16) φ ( n )2 i − , j − (cid:17) , where φ ( n ) i,j = min( i,j ) (cid:88) ν =0 ν S i − ν (ˆ x + ( n ) + ν s ) S j − ν (ˆ x − ( n ) + ν s ) , (75)vectors ˆ x ± ( n ) = (cid:0) x ± , x ± , · · · (cid:1) are defined by x +1 = x + i t + n + a , x − = x − i t − n + a ∗ , x + k = x + 2 k − (i t ) k ! + a k , x − k = x − k − (i t ) k ! + a ∗ k , k ≥ , (76)and s = ( s , s , · · · ) are coefficients from the expansion (7). Through a shift of the x and t axes, we normalize a = 0without loss of generality. Finally, we split the vectors ˆ x ± ( n ) into x ± ( n ) + w ± , where x ± ( n ) is as given in Eq. (6),8and w ± = (0 , x ± , , x ± , · · · ). Since ˆ x ± ( n ) + ν s = x ± ( n ) + ν s + w ± , it is easy to show from the definition of Schurpolynomials (8) that S k (ˆ x ± ( n ) + ν s ) = [ k/ (cid:88) j =0 S j ( ˆ w ± ) S k − j ( x ± ( n ) + ν s ) , (77)where ˆ w ± = ( x ± , x ± , · · · ). Rewriting the σ n solution det ≤ i,j ≤ N (cid:16) φ ( n )2 i − , j − (cid:17) as a 3 N × N determinant (56) andutilizing the above relation, we can apply row and column manipulations to eliminate all terms involving ˆ w ± in this3 N × N determinant. The remaining 3 N × N determinant then becomes det ≤ i,j ≤ N (cid:16) φ ( n )2 i − , j − (cid:17) , whose matrixelement φ ( n ) ij is as given in Theorem 1. Appendix B
In this appendix, we prove Theorem 2. First, we derive the multiplicity of root zero in Q [ m ] N ( z ). For this purpose,we define the Schur polynomial S [ m ] k ( z ; a ) as ∞ (cid:88) k =0 S [ m ] k ( z ; a ) (cid:15) k = exp (cid:2) z(cid:15) + a (cid:15) m +1 (cid:3) , (78)where a is a constant. Through these Schur polynomials S [ m ] k ( z ; a ), we define polynomials P [ m ] N ( z ; a ) = c N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S [ m ]1 ( z ; a ) S [ m ]0 ( z ; a ) · · · S [ m ]2 − N ( z ; a ) S [ m ]3 ( z ; a ) S [ m ]2 ( z ; a ) · · · S [ m ]4 − N ( z ; a )... ... ... ... S [ m ]2 N − ( z ; a ) S [ m ]2 N − ( z ; a ) · · · S [ m ] N ( z ; a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (79)where S [ m ] k ( z ; a ) ≡ k <
0. It is easy to see that S [ m ] k ( z ; a ) is related to the polynomial p [ m ] k ( z ) in Eq. (16) as S [ m ] k ( z ; a ) = ˆ a k/ (2 m +1) p [ m ] k (ˆ z ) , ˆ z ≡ ˆ a − / (2 m +1) z, ˆ a ≡ − (2 m + 1) 2 − m a. (80)Thus, the polynomial P [ m ] N ( z ; a ) is related to the Yablonskii-Vorob’ev polynomial hierarchy Q [ m ] N ( z ) in Eq. (17) as P [ m ] N ( z ; a ) = ˆ a N ( N +1)2(2 m +1) Q [ m ] N (ˆ z ) . (81)This equation tells us that every term in the polynomial P [ m ] N ( z ; a ) is a constant multiple of z i a j , where i +(2 m +1) j = N ( N + 1) /
2. Thus, to determine the multiplicity of the zero root z = 0 in Q [ m ] N ( z ), we need to determine the highestpower term of a in P [ m ] N ( z ; a ). To do so, we utilize the relation S [ m ] j ( z ; a ) = [ j m +1 ] (cid:88) i =0 a i i ![ j − i (2 m + 1)]! z j − i (2 m +1) , (82)which can be derived by splitting the right side of Eq. (78) into a product of two exponentials and then expandingboth exponentials into Taylor series of (cid:15) . Using this relation, we express the matrix elements in the determinant (79)through powers of z and a . Then, we need to obtain the highest power term of a in the resulting determinant. Thisproblem resembles the derivation of the highest power term of a m +1 in the σ n determinant (56) during the proofof Theorem 4, where a polynomial relation (58) similar to the above (82) was used. In this resemblance, the matrix P [ m ] N ( z ; a ) here is the counterpart of the Φ N × N matrix in Eq. (56), a here is the counterpart of a m +1 in Eq. (58),and z j in the above equation (82) is the counterpart of S j ( y + + ν s ) in Eq. (58). Performing the same row operationsas described in Theorem 4 to remove certain leading a -power terms in the lower rows of the determinant (79), we can9show that the highest-power term of a in P [ m ] N ( z ; a ) is ρ a N N − N − N m +1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z · · · z z · · · z N − (2 N − z N − (2 N − . . . z N N ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ˆ ρ a ( N − N N + N m +1) z N ( N +1) / , (83)where N is as given in Theorem 2, and ρ , ˆ ρ are ( m, N )-dependent nonzero constants. This shows that the lowestpower of z in all terms of P [ m ] N ( z ; a ) is N ( N + 1) /
2. Then, setting a = − m / (2 m + 1) where P [ m ] N ( z ; a ) becomes Q [ m ] N ( z ), the multiplicity of the zero root in Q [ m ] N ( z ) is N ( N + 1) / Q [ m ] N ( z ), we notice from the definition (16) of the polynomial p [ m ] k ( z ) thatthis polynomial admits the symmetry p [ m ] k ( z ) = ω − k p [ m ] k ( ωz ) , (84)where ω is any one of the (2 m + 1)-th root of 1, i.e., ω m +1 = 1. This symmetry of p [ m ] k ( z ) leads to the symmetry of Q [ m ] N ( z ) as Q [ m ] N ( z ) = ω − N ( N +1) / Q [ m ] N ( ωz ) . (85)Since the multiplicity of the zero root in Q [ m ] N ( z ) is N ( N + 1) /
2, let us write Q [ m ] N ( z ) = z N ( N +1) / q [ m ] N ( z ) , (86)where q [ m ] N ( z ) is a polynomial of z with a nonzero constant term. The symmetry (85) of the polynomial Q [ m ] N ( z )induces a symmetry for q [ m ] N ( z ) as q [ m ] N ( z ) = ω ( N + N − N − N ) / q [ m ] N ( ωz ) . (87)Since N + N − N − N = ( N − N )( N + N + 1), and in view of the N value given in Theorem 2, we see that( N + N − N − N ) / m + 1, which means ω ( N + N − N − N ) / = 1. Thus, the above equation reducesto q [ m ] N ( z ) = q [ m ] N ( ωz ) . (88)This symmetry of q [ m ] N ( z ) dictates that q [ m ] N ( z ) can only be a polynomial of z m +1 . Hence the form (19) of thepolynomial Q [ m ] N ( z ) is proved.Lastly, we derive the degree of the polynomial Q [ m ] N ( z ) from its definition (17). Notice from Eq. (16) that thehighest-degree term of p [ m ] k ( z ) is z k /k !. Retaining only this highest-degree term of p [ m ] k ( z ) in the determinant (17)for Q [ m ] N ( z ) and evaluating the simplified determinant by the same technique as that used in Ref. [11], we can readilyshow that the degree of the polynomial Q [ m ] N ( z ) is N ( N + 1) /
2. Thus, Theorem 2 is proved.
Appendix C
In this appendix, we prove the generalization of Theorem 4 presented in Sec. 6 when a m +1 is large and the otherparameters satisfy the conditions (70). In this parameter regime, let us denote a m +3 = β a m +1 , a m +5 = β a m +1 , · · · , a N − = β N − m − a m +1 , (89)where β , β , · · · , β N − m − are O (1) constants. We first split the vectors x ± as x + = y + + a , x − = y − + a ∗ , (90)0where a = (0 , · · · , , a m +1 , , a m +3 , , · · · , a N − ). Then, the Schur polynomials of x ± are related to those of y ± as S j ( x + + ν s ) = j (cid:88) i =0 S i ( a ) S j − i ( y + + ν s ) , S j ( x − + ν s ) = j (cid:88) i =0 S ∗ i ( a ) S j − i ( y − + ν s ) . (91)In view of the definition of a and the notations in (89), we readily see from the definition of Schur polynomialsthat S i ( a ) is a polynomial of a m +1 with coefficients dependent on ( β , β , · · · ), and its highest degree in a m +1 is[ i/ (2 m + 1)], i.e., the largest integer less than or equal to i/ (2 m + 1). Then, after a little manipulation and rearrangingterms in the above equations, we get S j ( x + + ν s ) = [ j m +1 ] (cid:88) i =0 a i m +1 [ j − (2 m +1) i ] (cid:88) k =0 c + i,k ( m, β ) S j − (2 m +1) i − k ( y + + ν s ) (92)and S j ( x − + ν s ) = [ j m +1 ] (cid:88) i =0 ( a ∗ m +1 ) i [ j − (2 m +1) i ] (cid:88) k =0 c − i,k ( m, β ) S j − (2 m +1) i − k ( y − + ν s ) , (93)where the coefficients c ± i,k are dependent on m and the vector β = ( β , β , · · · ), and c ± i, ( m, β ) = 1 /i !.These two Schur polynomial relations (92)-(93) are the counterparts of those in Eq. (58) during the proof ofTheorem 4. Using these relations, we can perform similar row and column operations to the 3 N × N determinant inEq. (56) to eliminate certain high order powers of a m +1 and a ∗ m +1 . The main difference is that, a little more sucheliminations are required here, because to eliminate a certain power of a m +1 or a ∗ m +1 in S j ( x ± + ν s ), one needs toeliminate a linear combination of polynomials S j − (2 m +1) i − k ( y ± + ν s ) now in view of the above two Schur polynomialrelations. However, these eliminations follow a clear and regular pattern, so that they can always be achieved. Anothersmall difference is that here, the row and column operations will produce some additional lower power terms of a m +1 and a ∗ m +1 . But those lower-power terms will eventually be discarded since we will retain only the highest a m +1 and a ∗ m +1 power terms in each row and column respectively. Therefore, these similar row and column operations will stillasymptotically reduce σ n to the same determinant (59) as before, and hence the generalization of Theorem 4 statedin Sec. 6 can be proved. References [1] K. Dysthe, H.E. Krogstad and P. M¨uller, “Oceanic rogue waves,” Annu. Rev. Fluid Mech. 40, 287 (2008).[2] C. Kharif, E. Pelinovsky and A. Slunyaev,
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