Universal patterns of rogue waves
UUniversal patterns of rogue waves
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 (NLS) equation and the derivative NLS equa-tion are analytically studied. It is shown that when the free parameters in the analytical expressionsof these rogue waves are large, these waves would exhibit the same patterns, comprising fundamentalrogue waves forming clear geometric structures 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 free parameter. This connection of rogue wave patterns to theroot structures of the Yablonskii-Vorob’ev polynomial hierarchy goes beyond the NLS and derivativeNLS equations, and it gives rise to universal rogue wave patterns in integrable systems.
PACS numbers: 05.45.Yv, 42.65.-k
Rogue waves are large and spontaneous nonlinear waveexcitations that “appear from nowhere and disappearwith no trace” [1]. They are a threat to ships in the oceanand cause various extreme events in optical systems, andhave thus received intensive studies in recent years (see[2–5] for reviews). So far, analytical expressions of roguewaves have been derived in a wide array of integrablephysical models, such as the nonlinear Schr¨odinger (NLS)equation for wave-packet propagation in the ocean andoptical systems [6–12], the derivative NLS equation forcircularly polarized nonlinear Alfv´en waves in plasmas[13–17], and the Manakov equations for light transmis-sion in randomly birefringent fibers [18, 19]. Some of thepredicted rogue wave solutions have also been observedin both water-wave and optics experiments [20–23].The study of rogue wave patterns is important as itallows for the prediction of later rogue wave events fromearlier wave forms. Although rogue wave solutions havebeen obtained in many physical integrable equations, andlow-order rogue wave graphs in those systems have beenplotted, systematic studies of rogue wave patterns, espe-cially the richer patterns arising from high-order roguewave solutions, is still very limited. For the NLS equa-tion, preliminary investigations on rogue patterns werereported in [10, 24, 25] through Darboux transformationand numerical simulations. It was observed in [10] that ifa N -th order rogue wave exhibits a ring structure, thenthe center of the ring is a ( N − u t + 12 u xx + | u | u = 0 . (1)Analytical expressions for general rogue waves in thisequation have been derived in [9, 11, 12] by various meth-ods. However, those expressions are not the best for so-lution analysis. Here, we present a simpler expression forthese solutions, which can be readily derived by incor-porating a new parameterization [17] into bilinear roguewaves in [12]. These simpler expressions, for rogue waveswith unit-amplitude boundary conditions of u ( x, t ) → e i t as x, t → ±∞ , are u N ( x, t ) = σ σ e i t , (2)where the positive integer N represents the order of therogue wave, σ n is a N × N Gram determinant σ n = det ≤ i,j ≤ N (cid:16) m ( n )2 i − , j − (cid:17) , (3)the matrix elements in σ n are defined by m ( 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 ) , a r X i v : . [ n li n . PS ] S e p 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 , with the asterisk * representing complex conjugation, s = ( s , s , · · · ) are coefficients from the expansion ∞ (cid:88) r =1 s r λ r = ln (cid:20) λ tanh (cid:18) λ (cid:19)(cid:21) , the Schur polynomials S k ( x ), with x = ( x , x , . . . ), aredefined 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) , (4)and a k +1 ( k = 0 , , · · · , N −
1) are free complex con-stants. Of these N free complex constants, we will nor-malize a = 0 by a shift of the x and t axis. Thus, theabove general rogue waves have N − a , a , · · · , a N − .Clear and recognizable rogue wave patterns will emergewhen some of these N − O (1). It turns outthat the resulting rogue patterns are completely deter-mined by the Yablonskii-Vorob’ev polynomial hierarchy,and this polynomial hierarchy will be introduced first.Yablonskii-Vorob’ev polynomials arose in rational so-lutions of the second Painlev´e equation [26, 27], and theirdeterminant expressions were derived in [28]. 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) . Then, Yablonskii-Vorob’ev polynomials Q N ( z ) are givenby a N × N determinant [28, 29] Q N ( z ) = γ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p N ( z ) p N +1 ( z ) · · · p N − ( z ) p N − ( z ) p N − ( z ) · · · p N − ( z )... ... ... ... p − N +2 ( z ) p − N +3 ( z ) · · · p ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where γ N = (cid:81) Nj =1 (2 j − p k ( z ) = 0 if k <
0. Todefine the Yablonskii-Vorob’ev polynomial hierarchy, welet 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) , (5) where m is a positive integer. Then, the Yablonskii-Vorob’ev hierarchy Q [ m ] N ( z ) are given by the N × N de-terminant Q [ m ] N ( z ) = γ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p [ m ] N ( z ) p [ m ] N +1 ( z ) · · · p [ m ]2 N − ( z ) p [ m ] N − ( z ) p [ m ] N − ( z ) · · · p [ m ]2 N − ( z )... ... ... ... p [ m ] − N +2 ( z ) p [ m ] − N +3 ( z ) · · · p [ m ]1 ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where p [ m ] k ( z ) = 0 if k <
0. When m = 1, Q [1] N ( z ) are theoriginal Yablonskii-Vorob’ev polynomials Q N ( z ). When m > Q [ m ] N ( z ) give higher members of this polynomialhierarchy. Root structures of the Yablonskii-Vorob’evpolynomial hierarchy have been studied in [29].Through this Yablonskii-Vorob’ev polynomial hierar-chy, the pattern of the N -th order NLS rogue wave (2) forlarge a m +1 and the other parameters O (1) are asymp-totically described as follows.Far away from the origin, with √ x + t = O (cid:0) | a m +1 | / (2 m +1) (cid:1) , there are N p fundamental (Pere-grine) rogue waves, where N p = [ N ( N + 1) − N ( N + 1)] / , (6)and N ( N + 1) / Q [ m ] N ( z ). This N value is given explicitlyby the equation N ≡ N mod (2 m + 1) , or (7) N ≡ − N − m + 1) , (8)under the restriction of 0 ≤ N ≤ m . These Peregrinewaves are u ( x − ˆ x ( k )0 , t − ˆ t ( k )0 ), where u ( x, t ) = (cid:18) − t )1 + 4 x + 4 t (cid:19) e i t , (9)and their positions (ˆ x ( k )0 , ˆ t ( k )0 ) are given asymptoticallyby ˆ x ( k )0 + i ˆ t ( k )0 = ˆ z k , (10)ˆ z k = z k (cid:18) − m + 12 m a m +1 (cid:19) m +1 , (11)with z k being any non-zero root of Q [ m ] N ( z ). Thus, therogue wave pattern formed by these Peregrine compo-nents has the same geometric shape as the root structureof the polynomial Q [ m ] N ( z ), except for a dilation and ro-tation due to the multiplication factor on the right sideof Eq. (11). In particular, the rotation-induced orienta-tion of the rogue pattern is controlled by the phase of thecomplex parameter a m +1 .In the neighborhood of the origin, where √ x + t = O (1), lies a N -th order rogue wave u N ( x, t ). This is alower-order rogue wave given by Eq. (2), with its internalparameters a , a , · · · , a N − the same as those in theoriginal rogue wave. If N = 0, then there will not besuch a lower-order rogue wave at the origin.Before proving these analytical results, let us com-pare these analytical predictions with true rogue wavepatterns. For this purpose, we first show in Fig. 1true rogue wave solutions (2) from the 3rd to 5th or-der, with large a , a , a and a in the first to fourthcolumns respectively. The specific large-parameter val-ues in Fig. 1 are a = ( − , − , − a = ( − , − , − a = ( − , − a = − a , pentagon patternsfor large a , heptagon patterns for large a , and nonagonpatterns for large a . In addition, some of these roguewaves contain a lower-order rogue wave at their centers.These rogue patterns resemble those plotted in [25] fromAkhmediev breathers in the rogue-wave limit. FIG. 1: NLS rogue wave patterns | u N ( x, t ) | of 3rd to 5thorders from true solutions (2) when one of the solution pa-rameters is large and the other parameters set as zero. Now, we compare these true rogue patterns with ouranalytical predictions. Our predicted solution | u ( p ) N ( x, t ) | 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) k =1 (cid:16)(cid:12)(cid:12)(cid:12) u ( x − ˆ x ( k )0 , t − ˆ t ( k )0 ) (cid:12)(cid:12)(cid:12) − (cid:17) , (12)where u ( x, t ) is the Peregrine wave given in (9), and u N ( x, t ) is the lower-order rogue wave (2) with its in-ternal parameters all zero, as inherited from the originalrogue waves in Fig. 1. For the 5th order rogue waves, ourpredicted solutions for the same parameters of Fig. 1 aredisplayed in Fig. 2. These predicted patterns are strik-ingly similar to the true ones. We have also comparedthe predicted rogue waves of the 3rd and 4th orders to the true ones in Fig. 1, and the predictions are visuallyalmost identical to the true solutions as well. FIG. 2: Analytical predictions (12) for the 5-th order roguewaves in Fig. 1. The x and t intervals here are identical tothose in Fig. 1, bottom row (for 5-th order). It is illuminating to compare rogue patterns in Figs. 1and 2 to root structures of the Yablonskii-Vorob’ev hier-archy Q [ m ] N ( z ) reported in [29]. Their geometric shapesare clearly the same, except for a rotation and dilationcaused by the multiplication factor in our formula (11).This connection between rogue patterns and Yablonskii-Vorob’ev polynomials was never realized before to ourbest knowledge.Now, we sketch the proof of our analytical results onrogue patterns. Our proof is based on an asymptoticanalysis of the rogue wave solution (2), or equivalently,the determinant σ n in Eq. (3), in the large a m +1 limit,with the other parameters being O (1).At large ( x, t ) where √ x + t = O (cid:0) | a m +1 | / (2 m +1) (cid:1) ,the leading order term of S k ( x + ( n ) + ν s ) is S k ( v ), where v = ( x + i t, , · · · , , a m +1 , , · · · ). From the definitionof Schur polynomials (4), 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) . Thus, it is related to the polynomial p [ m ] k ( z ) in (5) as S k ( v ) = A k/ (2 m +1) p [ m ] k ( z ) , where A = − m + 12 m a m +1 , z = A − / (2 m +1) ( x + i t ) . (13)Using these formulae, we find thatdet (cid:2) S i − j ( x + ( n ) + j s ) (cid:3) ∼ γ − N A N ( N +1)2(2 m +1) Q [ m ] N ( z ) . Similarly,det (cid:2) S i − j ( x − ( n ) + j s ) (cid:3) ∼ γ − N ( A ∗ ) N ( N +1)2(2 m +1) Q [ m ] N ( z ∗ ) . Here, S k = 0 when k < σ n in Eq. (3) as [12] σ n = (cid:88) 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) , where the summation is over all possible integers of 0 ≤ ν < ν < · · · < ν N ≤ N −
1. Since the highest orderterm 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) , where α is a ( m, N )-related non-zero constant. Thisequation shows that σ /σ ∼
1, i.e., the solution u ( x, t )is on the unit background, except at or near ( x, t ) loca-tions (cid:0) ˆ x , ˆ t (cid:1) where z 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. (10) in viewof Eq. (13). Performing further asymptotic analysis,we can show that in the neighborhood of each of these (cid:0) ˆ x , ˆ t (cid:1) locations, the solution u ( x, t ) is asymptotically aPeregrine soliton (9) centered at (cid:0) ˆ x , ˆ t (cid:1) .Next, we analyze the solution u ( x, t ) in the neigh-borhood of the origin, where √ x + t = O (1), when a m +1 is large and the other parameters being O (1). Inthis case, we first rewrite the σ n determinant (3) into a3 N × N determinant [12] σ n = (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) , (14)where Φ i,j = 2 − ( j − S i − j [ x + ( n ) + ( j − s ], andΨ i,j = 2 − ( i − S j − i [ x − ( n ) + ( i − s ]. Defining y ± to be the vector x ± without the a m +1 term, i.e., let x ± = y ± + (0 , · · · , , a m +1 , , · · · ), we find that theSchur 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 ) , where [ a ] represents the largest integer less than or equalto a . Using this relation, we express matrix elements of Φand Ψ in Eq. (14) through Schur polynomials S k ( y ± + ν s ) and powers of a m +1 . Then, we perform a seriesof row operations to the Φ matrix so that certain high-power terms of a m +1 are eliminated. Afterwards, wekeep only the highest power terms of a m +1 in each row ofthe remaining matrix. Similar column operations are alsoperformed on the matrix Ψ. With these manipulations,we find that σ n is asymptotically reduced to σ n ∼ β (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) , (15)where β is a ( m, N )-dependent constant, (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) , and ν is a certain integer. 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 ) . Using this relation, σ n ’s determinant (15) can be reducedto one with ν = 0 in the above (cid:98) Φ and (cid:98)
Ψ matrices. Such σ n then gives a N -th order rogue wave, whose internalparameters a j are identical to those in the original N -thorder rogue wave in (2).As a small application of our analytical results, we ex-plain the numerical observations in [10]. Under our bilin-ear rogue solution (2), a N -th order rogue wave exhibitsa ring structure when a N − is large (see Fig. 1). Inthis case, m = N −
1, and N = N − N -th order rogue wave is a ( N − N p = 2 N − u t + 12 u xx + i( | u | u ) x = 0 , (16)which arises in very different physical situations from theNLS equation [13, 14]. Bilinear forms of rogue wavesin this equation have been presented in [17] under theboundary conditions of u ( x, t ) → e − i(1+ α ) x − i( α − t/ as x, t → ∞ , where α is a background wavenumber parame-ter. This bilinear rogue wave of N -th order also has freecomplex parameters a , a , . . . , a N − , and we will nor-malize a = 0 through shifts of x and t as for the NLSequation. Then, when one of these complex parameters, a m +1 , is large, we find that far away from the origin,the rogue pattern also comprises N p fundamental DNLSrogue waves located at ( x, t ) = (cid:16) ˆ x ( k )0 , ˆ t ( k )0 (cid:17) , whereˆ x ( k )0 = 1 √ α Re (ˆ z k ) − α − α Im (ˆ z k ) , ˆ t ( k )0 = 1 α Im (ˆ z k ) , ˆ z k is as given in Eq. (11), and “Re, Im” represent thereal and imaginary parts of a complex number. When α = 1, these locations are identical to those in (10) for theNLS equation, which means that the DNLS rogue patternformed by these fundamental DNLS rogue waves wouldbe identical to those shown in Fig. 1 for the NLS equa-tion. When α (cid:54) = 1, the above (cid:16) ˆ x ( k )0 , ˆ t ( k )0 (cid:17) locations arerelated to the root structure of the polynomial Q [ m ] N ( z )through a stretching (shear), in addition to dilation androtation. In the neighborhood of the origin, the N -thorder DNLS rogue wave also reduces to a lower N -thorder DNLS rogue wave, with N as given in Eqs. (7)-(8). To verify these predictions, we set α = 1 and plot inFig. 3 true DNLS rogue waves from the 3rd to 5th or-der, with one of the parameters large and the others setas zero. It is seen that these rogue patterns are indeedidentical to those in Fig. 1 for the NLS equation. Theonly difference is the local shapes of DNLS fundamen-tal rogue waves away from the origin and local shapesof lower-order DNLS rogue wave at the origin. We havealso compared these true DNLS rogue patterns with ouranalytical predictions (shown in Fig. 4). It is seen thatthe predictions closely match true DNLS rogue patterns. FIG. 3: True DNLS rogue patterns | u N ( x, t ) | of 3rd to 5thorders when one of the solution parameters is large and theother parameters set as zero. We have also explored rogue patterns in other equa- tions such as the Manakov equations and the Boussi-nesq equation, and found that those patterns under largeparameters are described by the root structures of theYablonskii-Vorob’ev polynomial hierarchy as well. Thus,rogue waves in different equations develop universal pat-terns.
FIG. 4: Analytical predictions for 5-th order DNLS roguewaves of Fig. 3.
In summary, we have shown that universal rogue wavepatterns appear in different physical systems, and theserogue patterns are analytically predicted by the rootstructures of the Yablonskii-Vorob’ev polynomial hier-archy. These results significantly improve our analyticalunderstanding of rogue patterns, and they can be usefulfor the quantitative prediction of physical rogue events.This work was supported in part by AFOSR (FA9550-18-1-0098) and NSF (DMS-1910282). [1] N. Akhmediev, A. Ankiewicz and M. Taki, Phys. Lett. A373, 675 (2009).[2] K. Dysthe, H.E. Krogstad and P. M¨uller, Annu. Rev.Fluid Mech. 40, 287 (2008).[3] C. Kharif, E. Pelinovsky and A. Slunyaev,
Rogue Wavesin the Ocean (Springer, Berlin, 2009).[4] D.R. Solli, C. Ropers, P. Koonath and B. Jalali, Opticalrogue waves, Nature 450, 1054 (2007).[5] S. Wabnitz (Ed.),