Appearing (disappearing) lumps and rogue lumps of the two-dimensional vector Yajima-Oikawa system
aa r X i v : . [ n li n . PS ] D ec Appearing (disappearing) lumps and rogue lumps of thetwo-dimensional vector Yajima–Oikawa system
N.V. Ustinov a,b, ∗ a Department of Informational Technologies, Kaliningrad Institute of Management,236001 Kaliningrad, Russia b Department of Photonics and Microwave Physics, Lomonosov Moscow State University,119991 Moscow, Russia
Abstract
The solutions of the two-dimensional multicomponent Yajima–Oikawa sys-tem that have the functional arbitrariness are constructed by using the Dar-boux transformation technique. For the zero and constant backgrounds,different types of solutions of this system, including the lumps, line roguewaves, semi-rational solutions and their higher-order counterparts, are con-sidered. Also, the generalization of the lump solutions (namely, appearing ordisappearing lumps) is obtained in the two-component case under the specialchoice of the arbitrary functions. Then, the suitable ansatz is used to findthe further generalization of these lumps (appearing-disappearing lumps orrogue lumps).
Keywords: lump, rogue wave, Yajima–Oikawa system
1. Introduction
The considerable attention was paid in the recent years to an investigationof the two-dimensional multicomponent Yajima-Oikawa (YO) system [1–11].This system comprises multiple (say N ) short-wave components and a singlelong-wave one. It generalizes the scalar ( N = 1) two-dimensional YO sys-tem [12] and is often called the 2D coupled long-wave–short-wave resonanceinteraction system. ∗ Corresponding author
Email address: [email protected] (N.V. Ustinov)
Preprint submitted to Elsevier December 24, 2019 he two-dimensional multicomponent YO systems can arise in differentphysical contexts. In particular, the two-component ( N = 2) system andthe multicomponent one were derived by applying the reductive perturba-tion method in Refs. [1] and [5], respectively, as the governing equation forthe interaction of dispersive waves in a weak Kerr-type nonlinear medium inthe small amplitude limit. In these systems, the short waves propagate inanomalous dispersion regime while the long wave propagates in the normaldispersion regime. Also, a generation of the terahertz radiation by opticalpulses in a medium of asymmetric quantum particles is described under thequasi-resonance conditions by the two-dimensional two-component YO sys-tem [11]. It is worth to note that the two-dimensional scalar YO system wasderived for a two-layer fluid model by using the multiple scale perturbationmethod [13]. Interestingly, this system can be also deduced from the govern-ing equations for two-dimensional two-wave interaction [14, 15] by means ofthe reductive perturbation method (see Refs. [1, 16–18]).It is well-known [19] that the solutions of the multicomponent systems ex-hibit some novel properties that have not been observed in the scalar (single-component) counterpart. It was found, in particular, that the bright soli-tons of the two-dimensional two-component YO system undergo the energy-sharing (inelastic) collisions [3]. The unusual properties of the interaction ofthe multimode dromions of this system was revealed in Ref. [2]. Note thatthe solutions of the one-dimensional multicomponent YO system display alsosome specific features [20–22].Recently, the study of the rational solutions of the two-dimensional equa-tions and their generalizations (such as the lump-type solutions and the semi-rational ones) has attracted a lot of attention [8, 10, 23–53]. Some of thesesolutions are suitable in describing the behavior of rogue (or freak) wavesthat were intensively investigated in the last decades [14, 54–59]. In partic-ular, the rogue wave solutions were found for the Kadomtsev–Petviashviliequation [25, 28, 44], Davey–Stewartson eqiation [26, 27], Nizhnik–Novikov–Veselov equation [29, 40] and some other (2+1)-dimensional equations. Asa rule, the solutions in the form of rogue waves are obtained from the lumpsolutions by a reduction of variables.The rational and semi-rational solutions of the two-dimensional multicom-ponent YO system were investigated in Refs. [8] and [10], respectively. Therational solutions were obtained by means of the bilinear method and includethe fundamental (simplest) and general (multi- and higher-order) lumps andline rogue waves. It was shown that the fundamental lumps and rogue waves2ave three different patterns: bright, intermediate and dark states [8]. Thefundamental semi-rational solutions obtained also by the bilinear method candescribe the fission of a dark soliton into a lump and a dark soliton or thefusion of one lump and one dark soliton into a dark soliton [10]. The non-fundamental semi-rational solutions were shown to fall into three subclasses:higher-order, multi- and mixed-type semi-rational solutions.The rich structure of the solutions of the two-dimensional multicomponentYO system is due to its higher-dimensional and multi-component nature. Be-ing motivated by this, we exploit the Darboux transformation (DT) technique[60, 61] to obtain the solutions of this system. Note that the DT techniquewas applied to the multicomponent YO systems in the one-dimensional casein Refs. [62–65].The paper is organized as follows. The overdetermined system of linearequations of the two-dimensional multicomponent YO system of the generalform is given in Section 2. The DT technique is applied also in this section tothe systems considered. This technique allows us to construct the solutions ofthe two-dimensional multicomponent YO system that contain the arbitraryfunctions. In Section 3, the solutions of this system on the zero and constantbackgrounds are discussed. The case of the two-component two-dimensionalYO system is studied in details in Section 4. The solutions in the form ofthe appearing (disappearing) lumps are obtained in this section by taking thearbitrary functions in a special manner. Then, the ansatz is used to constructthe generalization of these solutions in the form of appearing-disappearinglump (or rogue lump). The estimation of the lifetime of this lump is given.The main results are summarized in Section 5.
2. Overdetermined linear system and Darboux transformation
Consider the two-dimensional multicomponent YO system i ∂ϕ n ∂y + ∂ϕ n ∂x ! = − ∂ ϕ n ∂τ − uϕ n ,n = 1 , . . . , N,∂u∂y = ∂∂τ N X n =1 σ n | ϕ n | . (1)Here ϕ n is the n th short-wave component, u is the long-wave one, σ n = ± n = 1 , . . . , N ). 3f the wave components are independent of variable x , then Eqs. (1) arereduced to the one-dimensional multicomponent YO system. This systemhas various physical applications [62, 63, 66–69]. In particular, it describesin the case N = 2 and σ = σ = 1 the propagation of the vector electromag-netic and acoustic pulses [62, 63, 68, 69]. In these physical applications, thevariable τ plays the role of the dimensionless ”local” time, while the variable y is the dimensionless spatial coordinate. Here, such an interpretation ofthe independent variables is understood also. Besides, this corresponds tothe roles of the independent variables in the two-dimensional two-componentYO system studied in Ref. [11] under a consideration of the terahertz radi-ation generation by the optical pulses in a medium of asymmetric quantumparticles.Eqs. (1) are integrable by the inverse scattering transformation method[66] and admits a representation as the compatibility condition of the overde-termined system of linear equations ∂ ψ ∂τ = − i ∂ψ ∂y + ∂ψ ∂x ! − uψ ,∂ψ n +1 ∂τ = σ n +1 ϕ ∗ n ψ ,n = 1 , . . . , N, (2)and ∂ψ ∂y = − N X n =1 ϕ n ψ n +1 ,∂ψ n +1 ∂y + ∂ψ n +1 ∂x = iσ n ϕ ∗ n ∂ψ ∂τ − ∂ϕ ∗ n ∂τ ψ ! ,n = 1 , . . . , N, (3)where ψ k = ψ k ( τ, y, x ) ( k = 1 , . . . , N + 1) is the solution of Eqs. (2) and (3).Let χ k = χ k ( τ, y, x ) ( k = 1 , . . . , N +1) be a solution of the overdeterminedsystem (2) and (3). Then the differential 1-form d δ ( χ, ψ ) = δ τ ( χ, ψ ) dτ + δ y ( χ, ψ ) dy + δ x ( χ, ψ ) dx, where δ τ ( χ, ψ ) = χ ∗ ψ ,δ y ( χ, ψ ) = − N X n =1 σ n χ ∗ n +1 ψ n +1 , x ( χ, ψ ) = i χ ∗ ∂ψ ∂τ − ∂χ ∗ ∂τ ψ ! + 2 N X n =1 σ n χ ∗ n +1 ψ n +1 , is closed; i.e., for a contour Γ connecting the points ( τ , y , x ) and ( τ, y, x ),an integral δ ( χ, ψ ) = Z Γ d δ ( χ, ψ ) + C (4)( C is a constant) depends only on the initial and final points and does notdepend on a specific choice of the contour.Let us apply the DT technique to obtain the solutions of the two-dimen-sional multicomponent YO system (1). The overdetermined system (2) and(3) of linear equations is covariant with respect to the DT ψ k → ψ k [1] ( k =1 , . . . , N + 1), ϕ n → ϕ n [1] ( n = 1 , . . . , N ), u → u [1], where the transformedquantities are defined in the following manner [11]: ψ k [1] = ψ k − δ ( χ, ψ ) δ ( χ, χ ) χ k , k = 1 , . . . , N + 1 , (5) ϕ n [1] = ϕ n − χ ∗ n +1 χ δ ( χ, χ ) , n = 1 , . . . , N, (6) u [1] = u + 2 ∂ ∂τ log δ ( χ, χ ) . (7)Relations (6) and (7) define a new (”dressed”) solution of the system (1),while the expressions (5) give corresponding solutions of the overdeterminedsystem of linear equations.Having expressions (5) for the solutions of the transformed overdeter-mined system, we can perform the iterations of the DT considered. Let χ ( l ) k = χ ( l ) k ( τ, y, x ) ( k = 1 , . . . , N + 1, l = 2 , . . . , L ) be the solutions of theoverdetermined system (2) and (3). Taking into account the identity δ ( χ ( l ) [1] , ψ [1]) δ ( χ, χ ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( χ ( l ) , ψ ) δ ( χ, ψ ) δ ( χ ( l ) , χ ) δ ( χ, χ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , we obtain after the L -fold iteration of the DT (5)–(7) the following expres-sions for the transformed solutions of the two-dimensional multicomponentYO system (1): ϕ n [ L ] = ϕ n − D [ L ] L X l,m =1 D [ L ] ( l,m ) χ ( l ) n +1 ∗ χ ( m )1 , n = 1 , . . . , N, (8)5 [ L ] = u + 2 ∂ ∂τ log D [ L ] , (9)where D [ M ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( χ (1) , χ (1) ) . . . δ ( χ (1) , χ ( M ) )... . . . ... δ ( χ ( M ) , χ (1) ) . . . δ ( χ ( M ) , χ ( M ) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,D [ L ] ( l,m ) is the algebraic complement of the element in the l th row and m thcolumn of the matrix of the determinant D [ L ]. Here we put χ k = χ (1) k ( k = 1 , . . . , N + 1).
3. The zero and constant backgrounds
The DT formulas (6), (7) and (8), (9) allows us to construct solutions ofthe two-dimensional YO system (1), which have the functional arbitrariness.To illustrate this we consider the zero and constant backgrounds.Let the initial solution of the YO system (1) be the zero background: ϕ = . . . = ϕ N = u = 0 . A solution of the overdetermined system (2) and (3) can be chosen in thiscase in the form χ = ZZ D ∞ X j =0 Λ j ( µ ) ∂ j ∂µ j e ∆ ( µ ) dµ R dµ I , (10) χ n +1 = f n ( y − x ) , n = 1 , . . . , N, (11)where µ is a complex parameter, µ R = ℜ ( µ ), µ I = ℑ ( µ ), D is a domain onthe parameter µ plane, ∆ ( µ ) = µτ + iµ x + f ( µ ) , Λ j ( µ ) ( j = 0 , , , . . . ), f ( µ ) and f n ( y − x ) ( n = 1 , . . . , N ) are arbitraryfunctions of their arguments.In the simplest case, whenΛ ( µ ) = δ ( µ R − λ ) δ ( µ I − ν ) , Λ j ( µ ) = 0 , j ∈ N , where δ ( µ R − λ ) and δ ( µ I − ν ) are Dirac’s delta functions, λ and ν are realconstants, such that λ + iν ∈ D , expression (10) is reduced to χ = e ∆ ( λ + iν ) . (12)6hen, we have from Eq. (4) δ ( χ, χ ) = 12 λ e ∆ ( λ + iν )+∆ ( λ + iν ) ∗ − y − x Z y − x N X n =1 σ n | f n ( ζ ) | dζ + a , (13)where a is a real constant.Substitution of the expressions (11)–(13) into the DT formulas (6) and(7) gives the simplest solution of the two-dimensional YO system (1) thathas the functional arbitrariness. If f n ( y − x ) ∼ exp[ ε ( y − x )] ( n = 1 , . . . , N ; ε is a real constant), and constant a is defined properly, then this solution isa line soliton. Condition ε = 2 λν , corresponds to the one-dimensional case.If, for example, f n ( y − x ) ∼ / cosh[ ε ( y − x )] ( j = n, . . . , N ), and constant a is chosen in a proper manner, then the solution of system (1) has localizedshort-wave components propagating along the lines y − x = const.Let us consider the initial solution of the YO system (1) in the form ofthe constant background: ϕ n = a n , n = 1 , . . . , N,u = 0 . (14)Without loss of generality, we assume here that the constants a n ( n =1 , . . . , N ) are real and a N = 0. Then, a solution of the overdeterminedsystem (2) and (3) can be represented in the following manner: χ = ZZ D ∞ X j =0 Λ j ( µ ) ∂ j ∂µ j e ∆( µ ) dµ R dµ I ,χ n +1 = σ n a n ZZ D ∞ X j =0 Λ j ( µ ) ∂ j ∂µ j e ∆( µ ) µ ! dµ R dµ I + f n (˜ y − ˜ x ) ,n = 1 , . . . , N, (15)where ∆( µ ) = µ ˜ τ + iµ ˜ x − ˜ y − ˜ x µ N X n =1 σ n a n , ˜ τ = τ − ˜ τ , ˜ x = x − ˜ x , ˜ y = y − ˜ y , τ , ˜ x and ˜ y are real constants, Λ j ( µ ) ( j = 0 , , , . . . ) and f n (˜ y − ˜ x ) ( n =1 , . . . , N −
1) are arbitrary functions of their arguments, f N (˜ y − ˜ x ) = − a N N − X n =1 a n f n (˜ y − ˜ x ) . Different types of the solutions of the two-dimensional YO system (1) areobtained by the substitution of the expressions (15) into the DT formulas(6), (7). In particular, the line soliton corresponds to the following choice:Λ = δ ( µ R − λ ) δ ( µ I − ν ) , Λ j ( µ ) = 0 , j = 1 , , . . . ,f n (˜ y − ˜ x ) = 0 , n = 1 , . . . , N. (16)This line soliton is dark.The lump and line rogue wave solutions [8], semi-rational ones [10] areobtained if Λ ( µ ) = 0 , Λ ( µ ) = δ ( µ R − λ ) δ ( µ I − ν ) , Λ j ( µ ) = 0 , j = 2 , , . . . , (17)and conditions (16) are imposedIn the cases of the lump and line rogue wave solutions, the constant C inEq. (4) is chosen in a such manner depending on the initial point of contour Γthat δ ( χ, χ ) is the product of the exponential function with the polynomial ofthe second degree with respect to the independent variables. These solutionswere studied in details in Ref. [8]. In the case of the semi-rational solutions, δ ( χ, χ ) is a sum of the constant and the product of the exponential functionwith the polynomial of the second degree. These solutions describe a fissionof the line dark soliton into the lump and the line dark soliton or a fusion ofone lump and one line dark soliton into the line dark soliton [10].The higher-order lump and line rogue wave solutions, higher-order semi-rational solutions are obtained if only one arbitrary function Λ m ( µ ) ( m ∈ N , m ≥
2) in Eqs. (15) differs from the zero:Λ m ( µ ) = δ ( µ R − λ ) δ ( µ I − ν ) . The solutions describing an interaction of the line solitons, usual and/orhigher-order lumps, line rogue waves and semi-rational solutions are con-structed by using the formulas (8), (9) of the iterated DT.8ote that the conditions (16) on the functions f n (˜ y − ˜ x ) ( n = 1 , . . . , N )have to take place in the cases of the line solitons, lumps, line rogue wavesand the semi-rational solutions. In the next section, these conditions will notbe imposed. This will give an opportunity to construct the generalizationsof the lumps.
4. The two-dimensional two-component YO system
Consider the YO system (1) in the simplest vector case: N = 2, σ = σ = 1. Let the initial solution of this system be the constant background(14). Then, the solution localized on ( x, y )-plane can be obtained by usingthe DT formulas (6), (7) if conditions (17) are valid, ν = 0 and arbitraryfunction f (˜ y − ˜ x ) is defined in the following manner: f (˜ y − ˜ x ) = a P (˜ y − ˜ x ) exp a + a λ (˜ x − ˜ y ) + iθ (˜ y − ˜ x ) ! , where P (˜ y − ˜ x ) is arbitrary function of its argument, θ (˜ y − ˜ x ) is arbitraryreal function. In the simplest case P (˜ y − ˜ x ) = r, where r is a constant, this gives us the solution of the form ϕ n = a n − ρ n ρ + + | r | e − λ ˜ τ , n = 1 , , (18) u = 2 ∂ ∂τ log (cid:16) ρ + + | r | e − λ ˜ τ (cid:17) , (19)where ρ = a ( ρ − − i ˜ x ) + a r ∗ ρ e ξ ,ρ = a ( ρ − − i ˜ x ) − a r ∗ ρ e ξ ,ρ ± = ˜ τ + a + a λ (˜ y − ˜ x ) ! + 4 λ ˜ x ± λ ,ρ = ˜ τ + a + a λ (˜ y − ˜ x ) + 2 iλ ˜ x + 12 λ ,ξ = − λ ˜ τ + iλ ˜ x − iθ (˜ y − ˜ x ) . θ (˜ y − ˜ x ) and real valuedexponents having crucial influence on its dynamics. When the exponentsare absent ( r = 0), this solution is nothing but the usual lump propagatingalong y axis without a change of its form. If r = 0, then solution (18), (19)describes an appearance ( λ >
0) or disappearance ( λ <
0) of such lump.The profiles of the long-wave component u of the disappearing lump fordifferent values of ˜ τ are presented in Fig. 1. Corresponding profiles of theabsolute value of the short-wave component ϕ are given in Fig. 2. Thedynamics of u and one of the absolute value of the short-wave componentsare different (compare Figs. 1d, 1e with Figs. 2d, 2e). At the same time, thedynamics of | ϕ | + | ϕ | is very similar to that of the long-wave component u . The structures shown in Figs. 2d, 2e will be discussed below. Figure 1: Profiles of variable u of the disappearing lump with parameters a = a = 1, r = 1, θ ( y − x ) = 0, λ = − τ = − τ = − . τ = − . τ = 1 . τ = 3 . τ = 6 (f). The process of the appearance of the lump can be understood from Figs. 1and 2. Note, that the phases of the short-wave components after the disap-10 igure 2: Profiles of the absolute value of variable ϕ of the disappearing lump withparameters as in Fig. 1 and ˜ τ = − τ = − . τ = − . τ = 1 . τ = 3 . τ = 6 (f). pearance (appearance) of the lump differs of π from the initial ones.It is seen from Eqs. (18) that the short-wave components ϕ and ϕ arenot proportional for any real values of parameters a and a if r = 0. Forthis reason, solution (18), (19) is not reduced to the solution of the scalartwo-dimensional YO system except for the trivial case r = 0.The formulas (8), (9) allows us to construct the solutions describing aninteraction of appearing and/or disappearing lumps. However, an existenceof the lumps that appear only or disappear only rises a problem of searchingthe solutions of the two-dimensional YO system (1) that join both the typesof dynamics (namely, appearing-disappearing lumps). To solve this problem,we will use the direct approach.Given the formulas (18), (19) and expressions for ρ n ( n = 0 , ,
2) and ρ ± ,11e apply the following ansatz to find the appearing-disappearing lump: ϕ n = a n − Q n + r R ( − ) n e − λ ˜ τ + iλ ˜ x + r R (+) n e λ ˜ τ + iλ ˜ x Q + | r | e − λ ˜ τ + | r | e λ ˜ τ , n = 1 , , (20) u = 2 ∂ ∂τ log (cid:16) Q + | r | e − λ ˜ τ + | r | e λ ˜ τ (cid:17) , (21)where r and r are the arbitrary constants, Q n ( n = 0 , ,
2) and R ( ± ) n ( n =1 ,
2) are the polynomials of the second and first degree of variables ˜ τ , ˜ y and˜ x , respectively: Q n = b n ˜ τ + c n ˜ y + d n ˜ x + e n ˜ τ ˜ y + f n ˜ τ ˜ x + g n ˜ y ˜ x + h n ˜ τ + i n ˜ y + j n ˜ x + k n , n = 0 , , , (22) R ( ± ) n = A ( ± ) n ˜ τ + B ( ± ) n ˜ y + C ( ± ) n ˜ x + D ( ± ) n , n = 1 , , (23)whose yet unknown coefficients have to be determined. This ansatz corre-sponds to the case θ (˜ y − ˜ x ) = 0.Substituting the expressions (20) and (21) into the YO system (1), col-lecting the coefficients of monomials ˜ τ , ˜ y , ˜ x and different exponents, equal-izing each coefficient in the left- and right-hand sides, we obtain the overde-termined system of the algebraic equations to define the unknowns of thepolynomials Q n ( n = 0 , ,
2) and R ( ± ) n ( n = 1 , b = 1 , c = ( a + a ) | s | ρλ , d = σ λ ρ ,e = − f = σ ( a + a ) λ ρ [1 + 2 λ ( r r ∗ + r ∗ r )] ,g = σ ( a + a )2 λ ρ [8 iλ ( r r ∗ − r ∗ r ) − ( a + a )(1 − λ | r r | )] ,h = i = j = 0 ,k = 1 + 16 λ | r r | λ ,b n = a n b , c n = a n c , d n = a n d , e n = a n e ,f n = a n f , g n = a n g , h n = 0 , n = ia n d + g − λ λ ,j n = − ia n d + g λ ,k n = a n λ | r r | − λ ,n = 1 , ,A ( ± )1 = − a λe ∓ i (2 d + g )4 λ √ d ,B ( ± )1 = a g ± iλe √ d , C ( ± )1 = a q d ,D (+)1 = a λ (1 − r r ∗ λ ) e − i (1 + 4 r r ∗ λ )(2 d + g )8 λ √ d ,D ( − )1 = − a λ (1 − r ∗ r λ ) e + i (1 + 4 r ∗ r λ )(2 d + g )8 λ √ d ,A ( ± )2 = a λe ∓ i (2 d + g )4 λ √ d ,B ( ± )2 = − a g ± iλe √ d , C ( ± )2 = − a q d ,D (+)2 = − a λ (1 − r r ∗ λ ) e − i (1 + 4 r r ∗ λ )(2 d + g )8 λ √ d ,D ( − )2 = a λ (1 − r ∗ r λ ) e + i (1 + 4 r ∗ r λ )(2 d + g )8 λ √ d , where σ = 16 λ + ( a + a ) (1 − λ | r r | ) ,ρ = 16 λ + ( a + a ) (1 + 16 λ | r r | ) + 4( a + a ) × λ [( a + a )( r r ∗ + r ∗ r ) + 4 iλ ( r r ∗ − r ∗ r )] ,s = 4 λ (1 + 4 λ r r ∗ ) + i ( a + a )(1 − λ | r r | ) . The substitution of the expressions written above into Eqs. (20)–(23)gives the solution of the two-dimensional YO system (1) in the form of theappearing-disappearing lump. If r = 0 (or r = 0), then this solution is13othing but the appearing (disappearing) lump considered in the beginningof this section.Note that the ansatz similar to that for the long-wave component (21),(22) was used in Refs. [41], [46] and [50] in the cases of the (2+1)-dimensionalSawada–Kotera equation, Kadomtsev–Petviashvili equation and (2+1)-di-mensional reduced Yu–Toda–Sasa–Fukuyama equation, respectively. Thecorresponding solutions of these equations describe an interaction of the lumpwith a pair of line solitons.The profiles of the long-wave component u and the absolute value of theshort-wave component ϕ of the appearing-disappearing lump for differentvalues of ˜ τ are presented in Fig. 3 and 4, respectively. The duration ofthe interval on the variable τ axis between the end of the appearance ofthe lump and the beginning of its disappearance (i.e., the lump lifetime) isroughly estimated as τ ∗ = 1 λ log k | r r | . (24)The amplitude of variable u of the developed lump is equal approximately to u max = 16 λ (see Figs. 3d and 3e).To clarify the structures displayed in Figs. 4b, 4g (and Figs. 2d, 2e)consider the profiles given in Fig. 5 of the long-wave component u and theabsolute value of the short-wave component ϕ of the appearing-disappearinglump for the fixed value of ˜ y . It is seen that the lump appearance (disap-pearance) is accompanied by the disappearance (appearance) of two wavepackets. The distance between these wave packets along ˜ x axis is propor-tional to exp( | λ ˜ τ | ). The minimal interval between the wave packets along ˜ τ axis is about τ ∗ . Note that the wave packets considered here are bright-dark(see Fig. 5b), while the solitons studied in Ref. [10] are dark. It is remark-able that the profiles of the sum of the squares of the absolute values ofthe short-wave components reveal no the disappearing (or appearing) wavepackets (see Fig. 5c).The amplitude of variable u of the wave packets is equal to u max /
8, i.e., itis less on the order almost than u max (see Fig. 5a). The amplitude of | ϕ | ofthe developed lump is greater than the one of the wave packets in two timesapproximately (Fig. 5b). So, the amplitudes of the appearing-disappearinglump exceed ones of the preceding and subsequent wave packets considerably.14his feature is typical for rogue waves [54–59]. For this reason, the appearing-disappearing lump can be called as rogue lump.The rogue lump has free parameters λ , r , r , τ , y and x . The firstthree parameters determine the velocity of the lump and its lifetime. Theremaining ones determine the location of the lump.It should be noted that the solution (18), (19) is not reduced to thesolution of the scalar (one-component) YO system. This points out that theansatz (20)–(23) is inapplicable in the scalar case.The ansatz (20)–(23) used to obtain the rogue lump corresponds to thesimplest case: P (˜ y − ˜ x ) is a constant, θ (˜ y − ˜ x ) = 0. One may assume thatthe solutions of the vector YO system in the form of the localized structuresexist when these constraints are not imposed.
5. Conclusion
In the present report, we applied the DT technique to the two-dimensionalmulticomponent YO system. This technique gives an infinite hierarchy ofthe solutions of the system considered that are expressed in the terms ofthe solutions of the associated overdetermined linear system. It is importantthat the solutions obtained this way have the functional arbitrariness.The solutions of the two-dimensional multicomponent YO system on thezero and constant backgrounds were considered. The special attention waspaid to the two-component case. Here, the generalizations of the lumps(namely, appearing or disappearing lumps) were obtained under the suitablechoice of the arbitrary functions. Moreover, the further generalization ofthese solutions in the form of the appearing-disappearing lump (or roguelump) was found by using the proper ansatz. The lifetime of this lumpdepends on its parameters and is estimated by the relation (24).It follows from the consideration in Sec. 4 of the generalizations of thelumps that there exists a class of the initial conditions of the vector YO sys-tem, whose evolution leads to an appearance of the localized patterns havingfinite lifetime. The amplitudes of the long- and short-wave components ofthese patterns can exceed significantly the ones of the initial background.Such kind of the dynamics resembles that of rogue waves.
6. Acknowledgement
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Generation of an acoustic supercontinuum un-der conditions of the hypersound intrapulse scattering mode. JETP2011;112:401–13. 22 igure 3: Profiles of variable u of the appearing-disappearing lump with parameters a = a = 1, r = r = 0 . λ = 1 and ˜ τ = −
12 (a), ˜ τ = − τ = − τ = − τ = 1 (e), ˜ τ = 4 (f), ˜ τ = 6 (g), ˜ τ = 12 (h). igure 4: Profiles of the absolute value of variable ϕ of the appearing-disappearing lumpwith parameters as in Fig. 3 and ˜ τ = −
12 (a), ˜ τ = − τ = − τ = − τ = 1(e), ˜ τ = 4 (f), ˜ τ = 6 (g), ˜ τ = 12 (h). igure 5: Profiles of variable u (a), the absolute value of variable ϕ (b) and the sum ofthe squares of the absolute values of variables ϕ and ϕ (c) of the appearing-disappearinglump with parameters as in Fig. 3 and ˜ y = 0.= 0.