Rogue Wave Multiplets in the Complex KdV Equation
RRogue wave multiplets in the complex KdV equation
M. Crabb and N. Akhmediev
Optical Sciences Group, Department of Theoretical Physics, Research School of Physics,The Australian National University, Canberra, ACT, 2600, Australia
We present a multi-parameter family of rational solutions to the complex Korteweg–de Vries(KdV) equations. This family of solutions includes particular cases with high-amplitude peaks atthe centre, as well as a multitude of cases in which high-order rogue waves are partially split intolower-order fundamental components. We present an empirically-found symmetry which introducesa parameter controlling the splitting of the rogue wave components into multi-peak solutions, andallows for nonsingular solutions at higher order.
PACS numbers: 05.45.Yv, 42.65.Tg, 42.81.qb
INTRODUCTION
Rogue waves are known to exist on deep ocean sur-faces [1, 2], within water in the form of internal roguewaves [3, 4], in optical fibres in supercontinuum genera-tion [5, 6], in the vacuum in the form of quantum fluctu-ations [7] and even in the theory of gravitational waves[8]. Their universality has been confirmed by water tankexperiments [9], in quadratic nonlinear crystals [10], and,most strikingly, in our encounters with extreme naturalphenomena [11]. The most common approach to the de-scription of rogue waves is using the exact solutions of in-tegrable evolution systems such as three-wave interaction[12], nonlinear Schr¨odinger (NLS) [13, 14], Kadomtsev-Petviashvili [15, 16], and Davey-Stewartson [17] equa-tions, among others [18]. Rational Peregrine-like solu-tions of these equations provide a good approximation todescribing rogue wave formation in a variety of physicalsituations [19, 20].The real Korteweg-de Vries (KdV) equation [21] is thebasis of the most common tool for the (1+1)-dimensionalmodelling of shallow water waves, which has been in usesince the work of Boussinesq [22]. Numerical modellingdone by Zabusky and Kruskal revealed the presence ofsoliton solutions of this equation [23], and the inversescattering technique developed for the KdV equation en-abled the derivation of analytic solutions for given initialconditions with zeros at infinity [24]. Being the historicfirst among the integrable nonlinear evolution equations,the KdV equation attracted significant attention fromboth physicists and mathematicians [25–28].Despite such extensive interest, until very recently, theKdV equation was thought to lack rogue wave solutions.This is true, but only if the wave described by the KdVequation is purely real. If we consider complex-valuedsolutions to the KdV equation, it is possible to deriverogue wave solutions [29]. Being the first work on thissubject, the paper [29], however, presented only selected(fixed-parameter) rogue wave solutions and did not re-veal the large variety of possible features of this impor-tant class of solutions. In this work, we provide a more detailed mathematical treatment and derive families ofrogue wave solutions with free parameters that determinea range of features. The presence of several parametersin our equations makes our approach much more power-ful than in previous work [29]. Here, by use of a simplesymmetry of the n -fold Darboux transformation, we showthat rational solutions to the KdV equation can be sub-stantially generalised to describe a much larger variety ofrogue waves. In principle, depending on the choice of pa-rameters involved, these rational solutions may be eithersingular or nonsingular. We also show that higher orderrogue waves in the complex KdV equation can appearin multi-peak formations, in a similar way to the roguewaves of the NLS equation [30–32]. THE n -TH ORDER RATIONAL SOLUTION FORTHE COMPLEX KDV EQUATION We will consider the complex KdV equation in the form ∂u∂t − u ∂u∂z + ∂ u∂z = 0 . (1)where z = x + iy is a complex variable, and u = u ( z, t )a complex function. Regardless of whether u is real orcomplex, (1) is also the condition of compatibility of thesystem ∂ψ∂t = − ∂ ψ∂z + 6 u ∂ψ∂z + 3 ∂u∂z ψ, (2) − ∂ ψ∂z + uψ = λψ. (3)This equivalence has several consequences. One of themost important for our purposes is that the system (2,3) is Darboux covariant, giving us a dressing method toconstruct nontrivial solutions to (1) from simple ones.Given an initial (seed) solution u = u to the KdV equa-tion, and n linearly independent solutions ψ , . . . , ψ n tothe associated linear system (2, 3), with corresponding a r X i v : . [ n li n . S I] S e p spectral parameters λ = λ , . . . , λ = λ n , the n -fold Dar-boux transformation of u is given by [33] u n = u − ∂ ∂z log W n +1 , (4)where W n is the Wro´nskian determinant of the functions ψ , . . . , ψ n with respect to z : W n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ψ . . . ψ n ∂ z ψ ∂ z ψ . . . ∂ z ψ n ... ... . . . ... ∂ n − z ψ ∂ n − z ψ . . . ∂ n − z ψ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (5)and u = u n will be another solution to the KdV equation(1). To be more concise we omit writing explicitly thedependence of W n on the functions ψ , . . . , ψ n .In order that the transformation (4) be non-trivial, theparameters λ k must be distinct. In order to obtain aDarboux transformation in the degenerate case λ k → λ for all k = 1 , , . . . , n , we define ψ , . . . , ψ n such that ψ ( z, t ; λ k ) = ψ k ( z, t ) . Then, expanding the matrix ele-ment ∂ iz ψ j as a Taylor series with respect to λ k , we havelim λ k → λ (cid:54) k (cid:54) n W n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ∂ λ ψ . . . ∂ n − λ ψ∂ z ψ ∂ λ ∂ z ψ . . . ∂ n − λ ∂ z ψ ... ... . . . ... ∂ n − z ψ ∂ λ ∂ n − z ψ . . . ∂ n − λ ∂ n − z ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (6)i.e. in the degenerate limit W n becomes the Wro´nskianof the functions ψ, ∂ λ ψ, . . . , ∂ n − λ ψ. So if we take, for example, the simple constant seedsolution u ( z, t ) = c, we can take as linearly independentsolutions to the system (2, 3) the functions ψ k ( z, t ) = cosh ω k { z + 2(3 c − ω k ) t } (7)where ω k = √ c − λ k for λ k (cid:54) = c . In (7), each eigenvalue λ k is distinct.We will also note here that due to the translationalinvariance u ( z, t ) (cid:55)→ u ( z − z , t − t ) of the KdV equation(1), we can introduce a second constant, via t (cid:55)→ t − t . This will not affect the choice of ψ in any substantialway, but this will be relevant later, so we allow for ar-bitrary shifts in z and t . For simplicity’s sake, we set abackground c = 0 , and the function ψ in (7) becomes ψ ( z, t ; λ k ) = cos (cid:112) λ k { z + 4 λ k ( t − it ) } , (8)from which we get W ( z, t ) = − z − λ ( t − it ) − sin 2 √ λ { z + 4 λ ( t − it ) } √ λ , with W ( z, t ) → − z as λ → . The imaginary part of z ensures that the Wro´nskian W ( z, t ) has no zeros in z and t except for the origin, and in the limit as λ → c, in this case as λ → , the Wro´nskianbecomes a polynomial in x and t . The correspondingdegenerate solution u to the complex KdV equation willthus always be a non-singular rational function as λ → y is chosen appropriately. In the simplest case, u ( x, t )becomes u ( z, t ) = 2 z = 2( x + iy ) . The 3 × λ → W ( x, t ) = − ( x + iy ) − t − it ) . For better clarity, we will write K n for the limit of theWro´nskian W n +1 as λ → , i.e. K n ( x + iy , t − it ) = lim λ → W n +1 , (9)so that in general, the n -th order rational solution of theKdV equation is given by u n ( x, t ) = − ∂ ∂x log K n ( x, t ) . (10)Singularities do not appear in all parts of the complexplane. If we choose the parameters of the solution suchthat the system (cid:60){ K n ( x + iy, t − it ) } = 0 , (cid:61){ K n ( x + iy, t − it ) } = 0has no solution in real values of x and t. To find a non-singular second-order solution, we observe that if t isstrictly real, then (cid:61){ K ( x + iy, t − it ) } = 2( x y − y + 4 t ) , and this is a quadratic in x with no real zero in x in theregion y + 12 t y < . In this region of the complex plane, the solutions will nothave any singularities for any values of x. The constant t allows us to avoid any real zeros in thedenominator of the second-order solution, since if t = 0 , the equation K ( x + iy, t − it ) = 0 always has roots inreal values of x and t for any choice of y. The second-order rational solution of the complex KdVequation is u ( z, t ) = 6 z z − t − it ) { z + 12( t − it ) } . (11)The plot of this function for fixed parameters y and t isshown in Fig. 1. With these restrictions on y , this hasthe form of a rogue wave with the maximal amplitude atthe origin, being given in terms of y and t by | u (0 , | = 6 | y + 24 yt | ( y − t ) . FIG. 1: (a) The plot of the second-order rogue wave(11) with y = and t = . The maximal amplitude is | u (0 , | = 20 . . (b) The plot of the third-orderrogue wave, defined by (13). Parameters are y = ,t = − , and g = − i . The maximal amplitude is | u (0 , | = 50 . The background for both cases is c = 0 . It is a straightforward exercise to write rational solu-tions of any order n . To give a few more examples, theWro´nskians K n as λ → K ( z, t ) = 415 ( z + 60 z t − t ) ,K ( z, t ) = 32515 ( z + 180 z t + 302400 zt ) ,K ( z, t ) = 25633075 ( z + 420 z t + 25200 z t ++ 2116800 z t − z t −− t ) . up to translations z (cid:55)→ z + z , t (cid:55)→ t − it . Since K n ( z, t ) is an ( n + 1) × ( n + 1) determinant,the explicit formulae quickly become more cumbersomewith increasing n , although the description in terms ofthe determinant holds for all n . The second-order roguewave in Fig. 1(a) is the simplest one among the hierar-chy of KdV rogue waves. It has a single maximum, andsmoothly growing and decaying fronts.The symmetry u ( z, t ) (cid:55)→ c + u ( z + 6 ct, t ) (12)of the KdV equation allows us to chose the background c arbitrarily, but at the expense of a travelling velocity of6 c. The two lowest order solutions with this adjustmentbecome u ( z, t ) = c − z + 6 ct ) ,u ( z, t ) = c − ∂ ∂z log { ( z + 6 ct ) + 12( t − it ) } . These are generalisations of previously obtained roguewave solutions. Namely, when c = − , y = − and t = , the above solutions coincide with those derivedin [29] by the complex Miura transformation. MULTI-PEAK SOLUTIONS
The higher-order solutions of the hierarchy are morecomplicated. Moreover, the usual expressions for themare always singular and may not describe physical situ-ations. In order to obtain solutions which can be non-singular, we have to go beyond the standard dressingtechnique.Solutions of order n (cid:62) K n with linear combi-nations of K n and K n − : If V n ( z, t ; g ) = K n ( z, t ) − gK n − ( z, t ) , n = 3 , , . . . , (13)where g is an arbitrary constant, then u = u n ( z, t ; g ) = − ∂ ∂z log V n ( z, t ; g ) ( n (cid:62)
3) (14)is also a solution of the KdV equation (1). This sym-metry can be verified by direct substitution as we havedone for all cases found here, i.e. up to n = 5 , and weconjecture it holds in general for all n .If we were to extend this symmetry to the n = 2 case,then it would make sense to identify K as simply a con-stant, since this would generate the solution u = 0 . Then(13) for n = 2 would just be introducing a complex ad-ditive constant. If (cid:60) ( g ) = 0 , it is identical to the substi-tution t (cid:55)→ t − it . When n = 3 , we recover a third-order rational solution, u ( z, t ; g ) = P ( z, t − it ; g ) { V ( z, t − it ; g ) } , (15) P ( z, t ; g ) = 450 g + 2160 gz + 8294400 zt ++ 1036800 z t + 192 z , (16)up to the symmetry (12). The parameters t , g and c here allow us to control the shape of the rogue wave.One example of this solution for background c = 0, with y = , t = − and g = − i is shown in Fig. 1(b).This same choice of parameters with background c = − g is purely imaginary, the roguewave is dominated by its central peak. The imaginarypart of g affects the relative heights of the peaks andthe tails. When (cid:61) ( g ) becomes large, the peaks reducein size relative to the tails, and for sufficiently large val-ues, the peaks may be even smaller than the tails. Onthe other hand, the real part of g causes splitting of thelower order components, so that they do not directly col-lide at the origin. Instead, with (cid:60) ( g ) (cid:54) = 0 , we see growthof multiple peaks. In Fig. 2(a), the central peak splitsinto two smaller ones, their locations and amplitudes de-pending on g and t . Another example is shown in Fig.2(b), in which we see that the effect of the parameter t , and position y on the imaginary axis, can be to transferamplitude from one peak to another.FIG. 2: Two plots of the third-order rogue wave. (a)Parameters are the same as in Fig. 1(b) except now g = − − i. A second peak has grown and the largerpeak has decreased in total amplitude and moved outfrom the origin. (b) Parameters are the same as in (a)except now t = − . One of the peaks has faded andthe other peak has gained more amplitude.
The general fourth-order rational solution in explicitform is given by u ( z, t ; g ) = P ( z, t − it ; g ) { K ( z, t ) − gK ( z, t ) } , (17) P ( z, t ; g ) = 50 { z + 4032 z t + 967680 t + 105 gz } −− z { g + 48 z ( z + 84 t ) }{ z ++ 2880 z t + 4838400 zt + 175 g ( z + 12 t ) } . (18)The explicit form of the fifth-order rational solution is u ( z, t ; g ) = P ( z, t − it ; g ) { K ( z, t ) − gK ( z, t ) } , (19) P ( z, t ; g ) = 60 z { g (43200 t + 5400 t z + z )++ 2048( − t ++ 19357238476800000 z t + 184354652160000 z t ++ 17667320832000 z t + 512096256000 z t ++ 1447891200 z t + 120960 z t + 504 z t + z ) }−− g (9144576000 t − z t −− z t + 22680 z t + 252 z t + z ) . (20)Higher order rational solutions can also be written insimilar form, but quickly exceed reasonable limits of pre-sentability.Two examples of the fourth-order solution for givensets of parameters and zero background c are shown inFig. 3. The profile of this solution can again take amultiplicity of forms. When the parameter g is purelyimaginary, as in Fig. 3(a), most of the rogue wave am-plitude is concentrated in the central peak, although twosmall, symmetrically located side peaks are also present. The maximal amplitude of the central peak here is 82.An example of the fourth order rogue wave for g withnonzero real part is shown in Fig. 3(b). Again, the realpart of g causes multiple peaks to grow. There we havethree distinct large peaks but of smaller amplitudes, eachroughly 20 to 30. Their relative locations and values ofvelocity are again determined by g and t .FIG. 3: Two plots of the fourth-order rogue wave. (a)Parameters are y = − , t = , g = − i and c = 0 .The maximal velocity is | u (0 , | = 82 . (b) Parametersare the same except now g = 1888 − i. With g havingnonzero real part, the main peak has decreased inamplitude and the smaller peaks have grown and movedaway from the origin. This extreme localisation shown in all graphs is thecharacteristic feature of rogue waves [34].
CONCLUSION
The crucial step taken in our work is the generalisation(13). Despite being as simple as a linear superpositionlaw for the Wro´nskians, it has important nontrivial conse-quences for the whole family of rational solutions, allow-ing them to be nonsingular i.e. physically relevant roguewaves. It also adds the complex parameter g that is es-sential in the higher order rational solutions for removingsingularities and for splitting the higher-order rogue waveinto its fundamental components. When the real part of g is zero, then the rogue wave has the highest peak atthe origin and smaller local maxima around it, as shownin Fig. 3(a). On the other hand, when g has nonzeroreal part, the central large peak decreases and smallerside peaks grow while separating from each other. Thistype of splitting of higher order rogue waves into multi-plet structures has also been observed in the case of NLSrogue waves [30–32] and their extensions [35]. However,the splitting of the higher-order rogue waves of the com-plex KdV equation is more complicated. The completeclassification of all forms of rogue waves here remainsopen for investigation.Lastly, we point out that complex solutions of theKdV equation are also applicable to unidirectional crys-tal growth [36] and complex KdV-like equations serve tomodel dust-acoustic waves in magnetoplasmas [37]. TheKdV hierarchy itself also finds applications in moderntheories of quantum gravity [38].As such, these new solutions presented in this work,previously not thought to exist, may find much wideruse in various areas of physics. [1] C. Kharif, E. Pelinovsky, A. Slunyaev, Rogue Waves inthe Ocean. (Springer, Berlin-Heidelberg, 2009).[2] A. Osborne, Nonlinear Ocean Waves and the InverseScattering Transform (Elsevier, Amsterdam, 2010).[3] R. Grimshaw, E. Pelinovsky, T. Taipova, and A.Sergeeva, Eur. Phys. J. Spec. Top. , 195 (2010).[4] M. H. Alford, Nature , 65 (2015).[5] D. R. Solli, C. Ropers, P. Koonath & B. Jalali, Opticalrogue waves, Nature , 1054 (2007).[6] J. M. Dudley, et al , Optics Express, , 21497 (2009).[7] M. Manceau, K. Yu. Spasibko, G. Leuchs, R. Filip, M.V. Chekhova, Phys. Rev. Lett., , 123606 (2019).[8] C. Bayindir and M. Arik, Rogue quantum gravitationalwaves, arxiv.org/abs/1908.02601v1 (2019).[9] A. Chabchoub, N. P. Hoffmann, and N. Akhmediev,Phys. Rev. Lett., , 204502 (2011).[10] R. Schiek, and F. Baronio, Phys. Rev. Res., (2019),Manuscript LE 17126 accepted for publication.[11] I. Nikolkina and I. Didenkulova, Nat. Hazards EarthSyst. Sci. , 2913 – 2924, (2011).[12] F. Baronio, A. Degasperis, M. Conforti, S. Wabnitz,Phys. Rev. Lett. , 044102 (2012).[13] P. Gaillard, J. Phys. A , 435204 (2011).[14] K. Trulsen, K. B. Dysthe, Wave motion, , 281 (1996).[15] P. Dubard, V. B. Matveev, Nat. Hazards Earth. Syst.Sci. , 667 (2011).[16] Y. Kodama, J. Phys. A: Math. Theor. , 434004 (2010).[17] Y. Ohta and J. Yang, Phys. Rev. E 86, 036604,(2012).[18] Y. Zhang, D. Qiu, D. Mihalache, and J. He, Chaos ,103108 (2018).[19] V. I. Shrira, V. Geogjaev, J. Eng. Math. , 11, (2010).[20] M. Onorato, S. Residori, U. Bortolozzo, A. Montina, F.T. Arecchie, Sci. Rep. , 47-89 (2013).[21] D. J. Korteweg & G. De Vries, Phil. Mag. , 422 (1895).[22] J. Boussinesq, Acad. Sci. Inst. Nat. France, XXIII , 1 –680 (1877).[23] N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. ,240 (1965)[24] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura,Phys. Rev. Lett. , 1095 – 1097 (1967).[25] R. M. Miura, SIAM Review, , No. 3, 412 – 459, (1976).[26] R. K. Bullough, and P. J. Caudrey, Acta Appl. Math., , 193 – 228, (1995).[27] J. W. Miles, J. Fluid Mech., , 131–147, (1981).[28] M. Lakshmanan, S. Rajasekar, Advanced Texts inPhysics, (Springer, Berlin, Heidelberg, 2003)[29] A. Ankiewicz, M. Bokaeeyan, and N. Akhmediev, Phys.Rev. E , 050201(R) (2019).[30] D. J. Kedziora, A. Ankiewicz, and N. Akhmediev, Phys.Rev. E , 013207 (2013).[31] A. Ankiewicz and N. Akhmediev, Rom. Rep. Phys. ,104 (2017)[32] A. Ankiewicz, D. J. Kedziora, N. Akhmediev, Phys. Lett.A , 2782 (2011). [33] V. B. Matveev, M. A. Salle, Darboux Transformationsand Solitons, (Springer, Berlin-Heidelberg, 1991)[34] N. Akhmediev, A. Ankiewicz and M. Taki, Phys. Lett. A , 675 (2009).[35] M. Crabb & N. Akhmediev, Nonlin. Dyn. , 245 (2019).[36] M. Kerszberg, Phys. Lett. A, 4, 5 (1984)[37] A. Misra, Appl. Math. Comp.256