General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear equations
RRogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations
General Rogue waves and their dynamics in severalreverse time integrable nonlocal nonlinear equations
Bo Yang
1, 2 and Yong Chen
1, 2, 3, a) Shanghai Key Laboratory of Trustworthy Computing, East China Normal University,Shanghai, 200062, People’s Republic of China MOE International Joint Lab of Trustworthy Software, East China Normal University,Shanghai, 200062, People’s Republic of China Department of Physics, Zhejiang Normal University, Jinhua, 321004,China
A study of general rogue waves in some integrable reverse time nonlocal non-linear equations is presented. Specifically, the reverse time nonlocal nonlinearSchr¨odinger (NLS) and nonlocal Davey-Stewartson (DS) equations are inves-tigated, which are nonlocal reductions from the AKNS hierarchy. By usingDarboux transformation (DT) method, several types of rogue waves are con-structed. Especially, a unified binary DT is found for this nonlocal DS system,thus the solution formulas for nonlocal DSI and DSII equation can be writtenin an uniform expression. Dynamics of these rogue waves is separately ex-plored. It is shown that the (1+1)-dimensional rogue waves in nonlocal NLSequation can be bounded for both x and t , or develop collapsing singularities.It is also shown that the (1+2)-dimensional line rogue waves in the nonlo-cal DS equations can be bounded for all space and time, or have finite-timeblowing-ups. All these types depend on the values of free parameters intro-duced in the solution. In addition, the dynamics patterns in the multi- andhigher-order rogue waves exhibits more richer structures, most of which haveno counterparts in the corresponding local nonlinear equations.Keywords: Reverse time nonlocal nonlinear equations, Darboux transforma-tion, Rogue waves I. INTRODUCTION
The integrable nonlinear evolution equations are exactly solvable models which play animportant role in the field of nonlinear science, especially in the study of nonlinear physicalsystems, including nonlinear optics, Bose-Einstein condensates, plasma physics and oceanwater waves. Most of these integrable equations are local equations, that is, the solutionsevolution only depends on the local solution value. In recent years, numbers of new inte-grable nonlocal equations were proposed and studied . The first such nonlocal equationwas the PT -symmetric nonlocal nonlinear Schr¨odinger (NLS) equation :i q t ( x, t ) = q xx ( x, t ) + 2 σq ( x, t ) q ∗ ( − x, t ) . (1)Here, σ = ± PT -symmetric systems have attracted a lot of attention in optics and otherphysical fields in recent years .Following this nonlocal PT -symmetric NLS equation, some new reverse space-time andreverse time type nonlocal nonlinear integrable equations were also introduced and quicklyreported . They are integrable infinite dimensional Hamiltonian dynamical systems,which arise from remarkably simple symmetry reductions of general ZS-AKNS scatteringproblems where the nonlocality appears in both space and time or time alone. These ex-amples including the reverse space-time and/or the reverse time only nonlocal NLS , the a) Electronic mail: (Corresponding author.) [email protected]. a r X i v : . [ n li n . S I] D ec ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 2(complex) modified Korteweg-deVries, sine-Gordon, three-wave interactions, derivative NLS(dNLS), Davey-Stewartson, discrete NLS type , Sasa-Satsuma (S-S) and nonlocal complexshort pulse (CSP) equations , and many others .These nonlocal equations, because of their novel space and/or time coupling, are distinctlydifferent from local equations. Indeed, solution properties in some of these nonlocal equa-tions have been analyzed by the inverse scattering transform method, Darboux transforma-tion or the bilinear method. On the one hand, those new systems could reproduce solutionpatterns which already have been discovered in their local counterparts. On the other hand,some interesting behaviors such as blowing-up solutions with finite-time singularities andthe existence of even more richer structures have also been revealed . Although thesenonlocal equations are mathematically interesting. In the view of further potential appli-cations, it links to an unconventional system of magnetics , and relates to the concept of PT -symmetry, which is a hot research area in contemporary physics .Rogue waves have attracted a lot of attention in recent years due to their dramatic andoften damaging effects, such as in the ocean and optical fibers . The first analyticalexpression of a rogue wave for the NLS equation was derived by Peregrine in 1983 . Later,the analytical rogue-wave solutions have been derived and interesting dynamical patternsbeen revealed for a large number of integrable systems .As an unexplored and interesting subject, rogue waves in the nonlocal integrable sys-tems have received much attention. For local integrable equations, the evolution for mostrogue-wave solutions depends only on the local solution value with its local space andtime derivatives. However, for the nonlocal equations, the sates of rogue-wave solutionsat distinct locations x and − x are directly related . Hence, these facts are basicallyimportant and further motivate us ask an interesting open question: If one only considerthe connections between the sates of rogue-wave solutions at reverse time points t and − t ,whether there are rogue waves existing in some reverse time nonlocal nonlinear equations?In this article, to give an answer to this question, we study rogue waves in several reversetime integrable nonlocal nonlinear equations. As typically concrete examples, we focus onthe reverse time nonlocal NLS equation:i q t ( x, t ) = q xx ( x, t ) + 2 σq ( x, t ) q ( x, − t ) , (2)and the reverse time nonlocal DS equations: iq t + 12 γ q xx + 12 q yy + ( qr − φ ) q = 0 , (3) φ xx − γ φ yy − qr ) xx = 0 , (4)where r ( x, y, t ) = σq ( x, y, − t ), q , r and φ are functions of x, y, t , γ = ± γ = 1 being the DS-I and γ = − and partially PT -symmetric DS equations . In addition, dynamicsof these rogue waves is further analyzed. For these reverse time nonlocal equations, it isshown that general rogue waves can be bounded for all space and time. More importantly,they can also develop collapsing singularities (for nonlocal NLS) or finite time blowing-ups(for nonlocal DS), which have no counterparts for the local equations. In addition, undercertain parameter conditions, the dynamics of multi-rogue waves and higher-order roguewaves can exhibit more patterns. Most of them haven’t been found before in the integrablenonlocal nonlinear equations.ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 3 II. ROGUE WAVES IN THE REVERSE TIME NONLOCAL NONLINEAR SCHR ¨ODINGEREQUATION
In this section, we consider rogue waves in the focusing reverse time nonlocal NLS equa-tion (2) (with σ = 1), which approach the unit constant background when x, t → ±∞ . Inthe local NLS equation, general rogue waves in the form of rational solutions have beenreported in , which are bounded solutions for both t and x . For the nonlocal NLS equa-tion (1), we will show that the reverse time nonlocal NLS equation admits a wider varietyof rogue waves. In addition to the non-collapsing patterns, there are also other types ofrogue waves which can develop collapsing singularities.It has been pointed that Eq.(2) is an integrable Hamilton evolution equation that admitsan infinite number of conservation laws . The first four conserved quantities are verifiedand given by I = (cid:90) ∞−∞ q ( x, t ) q ( x, − t ) dx,I = (cid:90) ∞−∞ q ( x, − t ) q x ( x, t ) dx,I = (cid:90) ∞−∞ (cid:2) q ( x, − t ) q xx ( x, t ) + q ( x, t ) q ( x, − t ) (cid:3) dx,I = (cid:90) ∞−∞ q ( x, − t ) (cid:8) q xxx ( x, t ) + [ q ( x, t ) q ( x, − t )] x +2 q ( x, − t ) q ( x, t ) q x ( x, t ) } dx. For this equation, the evolution at time t depends on not only the local solution at t , butalso the nonlocal solution at the reverse time point − t . That is, solution states at reversetime points t and − t are directly related. Especially, N-solitons are derived recently via theRiemann-Hilbert solutions of AKNS hierarchy under the reverse-time nonlocal reduction.In this part, we focus on the general rogue waves.We begin with the following ZS-AKNS scattering problem :Φ x = U ( q, r, λ )Φ , (5)Φ t = V ( q, r, λ )Φ , (6)where Φ is a column-vector, U ( q, r, λ ) = − i λσ + Q, (7) V ( q, r, λ ) = 2i λ σ − λQ − i σ (cid:0) Q x − Q (cid:1) , (8) σ = diag(1 , − , Q ( x, t ) = (cid:18) q ( x, t ) r ( x, t ) 0 (cid:19) , x, t ∈ R . The compatibility condition of these equations give rise to the zero-curvature equation U t − V x + [ U, V ] = 0 , (9)which yields the following coupled system for potential functions ( q, r ) in the matrix Q ( x, t ):i q t = q xx − q r, (10)i r t = − r xx + 2 r q. (11)The focusing reverse-time nonlocal NLS equation (2) is obtained from the above coupledsystem (10)-(11) under the symmetry reduction: r ( x, t ) = − q ( x, − t ) . (12)As it is stressed by Ablowitz and Musslimani :“ This reduction is new, remarkably simple,which has not been noticed in the literature and leads to a nonlocal in time NLS hierarchy.”Under reduction (12), the potential matrix Q satisfies the following symmetry condition: Q T ( x, t ) = − Q ( x, − t ) . (13)ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 4To construct the Darboux transformation, we need to introduce the adjoint-spectral prob-lem for (5)-(6): − Ψ x = Ψ U ( q, r, ζ ) , (14) − Ψ t = Ψ V ( q, r, ζ ) , (15)here Ψ is a row-vector. A. N -fold Darboux transformation and reverse-time reduction For the ZS-AKNS scattering problem, there is a general Darboux transformation proposedin , which can be represented as: T = I + ζ − λ λ − ζ P , P = Φ Ψ Ψ Φ , (16)where Φ = ( φ , φ ) T is a solution with spectral parameter λ = λ , and Ψ = ( ψ , ψ ) T is a solution of conjugation system with spectral parameter ζ = ζ . This DT can convert(5)-(6) into a new system (cid:0) Φ [1] (cid:1) x = U ( q [1] , r [1] , λ )Φ [1] , (cid:0) Φ [1] (cid:1) t = V ( q [1] , r [1] , λ )Φ [1] , with the transformation between potential matrix: Q [1] = Q + i( ζ − λ ) [ σ , P ] . (17)Next, we consider the reduction of DT for system (5)-(6). Due to the potential symmetry(13), it is shown by direct calculation that matrix U satisfy the symmetry: U T ( x, t, λ ) = − U ( x, − t, − λ ) . (18)For the given form (7) in matrix U , the matrix V which satisfies the zero-curvature equation(9) with the specific form of λ -dependence as in Eq. (8) is unique. Therefore, by utiliz-ing symmetry (18) and the zero-curvature equation (9), we can derive the correspondingsymmetry of the matrix V : V T ( x, t, λ ) = V ( x, − t, − λ ) . (19)Using these U and V symmetries, we can derive the symmetries of wave functions Φand adjoint wave functions Ψ, and hence the symmetry of the Darboux transformation forthe reverse-time nonlocal NLS equation (1). Applying these symmetries to the spectralproblems (5)-(6), we get − Φ Tx ( x, − t ) = Φ T ( x, − t ) U ( x, t, − λ ) , (20) − Φ Tt ( x, − t ) = Φ T ( x, − t ) V ( x, t, − λ ) . (21)Thus, if Φ( x, t ) is a wave function of the linear system (5)-(6) at λ , then Φ T ( x, − t ) is anadjoint wave function of the adjoint system at ζ = − λ . In this case, if Ψ ( x, t, λ ) andΦ T ( x, − t, − λ ) are linearly dependent on each other, then matrix (16) would preserve thepotential reduction (12) and thus be a Darboux transformation for the reverse-time nonlocalNLS equation (1). Specifically, we have the following results. Proposition 3.1 . For any spectral λ , ζ ∈ C , if ζ = − λ , Ψ ( x, t, ζ ) = α Φ T ( x, − t, λ ) , (22)where α is a complex constant. Then the Darboux matrix (16) accords with a Darbouxtransformation for the focusing reverse-time nonlocal NLS equation (1).ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 5This proposition can be readily proved by checking that the new potential matrix Q [1] from Eq. (17) satisfies the symmetry (12) under conditions (22). Via a direct calculation,this property can be easily verified.The N-fold Darboux transformation is a N times iteration of the elementary DT withcorresponding reductions. For the local NLS equation, a N-fold Darboux matrix has beengiven in . Here, with the reduction given above, we have the following N-fold Darbouxmatrix. Proposition 3.2 . The N-fold Darboux transformation matrix for the focusing PT -symmetric NLS equation can be represented as: T N = I − Y M − D − X, (23) where , Y = [ | y (cid:105) , | y (cid:105) , . . . , | y N (cid:105) ] , | y k (cid:105) = Φ k ( λ k ) ,X = [ (cid:104) x | , (cid:104) x | , . . . , (cid:104) x N | ] , (cid:104) x k | = Ψ k ( ζ k ) ,M = (cid:16) m ( N ) i,j (cid:17) ≤ i,j ≤ N , m ( N ) i,j = (cid:104) x i | y j (cid:105) λ j − ζ i ,D = diag ( λ − ζ , λ − ζ , . . . , λ − ζ N ) , where | y i (cid:105) solves the spectral equation (5)-(6) at λ = λ i , and (cid:104) x j | solves the adjoint spec-tral equation (14)-(15) at ζ = ζ j . Here, for any k ∈ N + , we should have ζ k = − λ k with (cid:104) x k ( x, t ) | = | y k ( x, − t ) (cid:105) T . Moreover, the B¨acklund transformation between potential functions is: u [ N ] = u + 2i (cid:12)(cid:12)(cid:12)(cid:12) M X Y (cid:12)(cid:12)(cid:12)(cid:12) | M | , (24) where Y represents the first row of matrix Y , and X represents the second column ofmatrix X . Expression (24) can be found , which appears as Riemann-Hilbert solution. The proofof this theorem has already been given. B. Derivation of general rogue-wave solutions
In this section, we derive a general formula for rogue waves. First of all, we need thegeneral eigenfunctions solved from the linear system (5)-(6) and its adjoint system (14)-(15).Choosing a plane wave solution u [0] = e − t to be the seed solution, we can derive a generalwave function for the linear system (5)-(6) asΦ( x, t ) = D φ ( x, t ) , (25)where D = diag (cid:0) e − i t , e i t (cid:1) , φ ( x, t ) = (cid:18) c e A + c e − A c e A + c e − A (cid:19) , (26) A = (cid:112) − λ − x − λt + θ ) ,c = c (cid:16) i λ − (cid:112) − λ − (cid:17) , c = c (cid:16) i λ + (cid:112) − λ − (cid:17) , and c , c , θ are arbitrary complex constants. Imposing the conditions of λ being purelyimaginary, i.e., λ = ih , | λ | > θ is complex. Furthermore, to obtain the adjoint wavefunction, one can use symmetry condition (22). Through a simple Gauge transformationon the complex constant c , c , we can normalize α = 1. Hence, the adjoint wave functionto Eqs.(14)-(15) at ζ = − λ satisfying (22) can be given as:Ψ( x, t, ζ ) = ψ ( x, t, ζ ) D ∗ , (27)ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 6where ψ ( x, t, ζ ) = φ T ( x, − t, λ ) . (28)Moreover, with a constant normalization on free complex constants c and c , the wavefunction φ ( x, t ) becomes φ ( x, t, λ ) = 1 √ h − (cid:18) sinh (cid:2) A + ln (cid:0) h + √ h − (cid:1)(cid:3) sinh (cid:2) − A + ln (cid:0) h + √ h − (cid:1)(cid:3) (cid:19) . (29)Here, the scaling constant √ h − h → λ → i). In the view of wave function and Proposition 3.2,this scaling of the wave function does not affect the solution except for a simple shifting √ h − ln (cid:16) | c c | (cid:17) occurs on the x-direction.Next, to derive rogue-wave solutions from above wave functions and adjoint wave func-tions, we need to choose spectral parameters λ k and ζ k so that the exponents A can vanishunder certain limits. Thus, we can take λ k to be i, and correspondingly, ζ k to be − i.Specifically, we choose spectral parameters λ = λ k , ζ = ζ k and complex parameters θ = θ k ,where λ k = i(1 + (cid:15) k ) , ζ k = − i(1 + (cid:15) k ) , (30) θ k = N − (cid:88) j =0 s j (cid:15) jk , ≤ k ≤ N, (31) s , s , . . . , s n − are k -independent complex constants.Thus, with λ = i(1 + (cid:15) ) , ζ = − i(1 + (cid:101) (cid:15) ), we have the following expansions: φ ( x, t, λ ) = ∞ (cid:88) k =0 φ ( k ) (cid:15) k , ψ ( x, t, ζ ) = ∞ (cid:88) k =0 ψ ( k ) (cid:101) (cid:15) k , (32) ψ ( x, t, ζ ) φ ( x, t, λ ) λ − ζ = ∞ (cid:88) k =0 ∞ (cid:88) l =0 m k, l (cid:101) (cid:15) k (cid:15) l , (33)where φ ( k ) = lim (cid:15) → ∂ k φ ( x, t, λ )(2 k )! ∂(cid:15) k , ψ ( k ) = lim (cid:15) → ∂ k φ T ( x, − t, λ )(2 k )! ∂(cid:15) k , (34) m k,l =lim (cid:15), (cid:101) (cid:15) → k − l − ∂ k +2 l − ∂ (cid:101) (cid:15) k − ∂(cid:15) l − (cid:20) φ T ( − t, ζ ) φ ( t, λ ) λ − ζ (cid:21) . (35)Here, we use a different notation (cid:101) (cid:15) instead of (cid:15) in the adjoint wave function φ for aboveexpansions. In the sense of limit (cid:15) → (cid:101) (cid:15) →
0, they still meets the reduction condition (22).Therefore, applying these expansions to each matrix element in the B¨acklund transformation(24), performing simple determinant manipulations and taking the limits of (cid:15) k , (cid:101) (cid:15) k → ≤ k ≤ N ), we derive general rogue waves in the following Theorem. Theorem 3.1 . The N-th order rogue waves in the focusing reverse-time nonlocal NLSequation can be formulated as: u N ( x, t ) = e − it (cid:18) τ τ (cid:19) , (36) where τ = det ≤ i,j ≤ n ( m i,j ) , τ = det (cid:18) ( m i,j ) ≤ i,j ≤ N Y (2) X (1) (cid:19) , ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 7 X = (cid:104) φ (0) , φ (1) , ..., φ ( N − (cid:105) , Y = (cid:104) ψ (0) , ψ (1) , ..., ψ ( N − (cid:105) T . Here, X (1) stands the 1-st row in matrix X , while Y (2) represents the 2-nd column in Y .Functions φ ( λ ) and ψ ( ζ ) are defined in the form as (28)-(29), the matrix element m i,j isgiven through limitation (35). Since wave functions and adjoint wave functions in the Darboux transformation for thereverse-time nonlocal NLS equation (1) are related via the reverse-time reduction, this newreduction will lead to different types of rogue waves, which will be discussed in the following.
C. Dynamics in the rogue-wave solutions
In this section, we give a discussion on the dynamics of these rogue-wave solutions.
1. Fundamental rogue-wave solution
The first-order (fundamental) rogue wave is obtained by setting N = 1 in Eq. (36), wherethe matrix elements can be obtained from Theorem 3.1. In this case, τ = m , = − i2 (cid:0) s x + 4 s + 16 t + 4 x + 1 (cid:1) ,τ = (cid:12)(cid:12)(cid:12)(cid:12) m , Y X (cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:0) s x + 4 s + 16 t + 8i t + 4 x − (cid:1) , where s is complex constants. Thus, the first-order rogue-wave is given by u ( x, t ) = e − i t (cid:18) τ τ (cid:19) = − e − i t (cid:0) s x + 4 s + 16 t + 16 i t + 4 x − (cid:1) s x + 4 s + 16 t + 4 x + 1 . Denoting x = Re ( s ) and y = Im ( s ), this rogue wave can be rewritten as u ( x, t ) = − e − i t (cid:20) t − t + 4 (ˆ x + i y ) + 1 (cid:21) , (37)where ˆ x = x + x . Hence, this solution has one non-reducible real parameter y , since theparameter x can be removed by space-shifting. Moreover, it is interesting to find that thisfundamental rogue-wave solution also admits the first order rogue wave recently derived in the PT -symmetric nonlocal NLS equation. This is due to an important property whichsolution (37) satisfies: u ∗ ( − ˆ x, t ) = u (ˆ x, − t ) (38)For this rogue wave solution, when y < /
4, it is shown to be nonsingular. If y = 0,it degenerates into the classical Peregrine soliton for the local NLS equation. [note thatany solution of the local NLS equation satisfying u ∗ ( x, t ) = u ( x, − t ) would satisfy thereverse-time nonlocal equation (1)]. The peak amplitude of this Peregrine soliton is 3, i.e.,three times the level of the constant background. But if y (cid:54) = 0, its peak amplitude wouldaccurately becomes (cid:12)(cid:12)(cid:12) s +34 s − (cid:12)(cid:12)(cid:12) , which is higher than 3. One of such solutions is shown in Fig.1. However, once y ≥ /
4, this rogue wave would blow-up at x = 0 with two time points t c = ± (cid:112) (4 y − /
16. One such blowing-up solution is displayed in Fig.2. Since wavecollapse has been reported in bright solitons for the nonlocal NLS equation (1). Here wesee that collapse occurs for rogue waves in the reverse-time nonlocal equation as well.ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 8
FIG. 1. The nonsingular first-order rogue wave solution (37). (a). s = r = i6 (corresponding to y = 1 / s = 2i; (a) 3D plot; (b) density plot.
2. Higher-order rogue-wave solution
Next, we consider the second order rogue waves, which are given in (36) with N = 2. Inthis case, the general second order rogue wave can be obtained as q ( x, t ) = e − i t (cid:18) τ τ (cid:19) (39)where, τ = − { t + 3072 t x + 768 t x + 64 x + 6912 t − t x + 1584 t + 48 x +108 x + 384 s x + 48 s (cid:0) t + 20 x + 1 (cid:1) + 64 s (cid:2) x (cid:0) t + 20 x + 3 (cid:1) − s (cid:3) + 64 s − s x (cid:0) − t + 4 x − (cid:1) + 12 s (cid:2) − s x + 256 t + 96 t (cid:0) x − (cid:1) + 80 x + 24 x + 9 (cid:3) +144 s + 24 s (cid:2) s (cid:0) t − x + 1 (cid:1) + x (cid:0) t + 128 t x + 16 x + 8 x − t + 9 (cid:1)(cid:3) + 9 } ,τ = − (cid:2) t + i (cid:0) t + 16 x + 64 s x + 24 x + 96 s x + 48 s x + 64 s x + 48 s x (cid:1) +i t (cid:0) x + 768 s x + 384 s + 288 (cid:1) + t (cid:0) x + 1024 s x + 512 s + 128 (cid:1) + t (cid:0) s x + 384 s x +256 s x − s x + 192 s x + 64 s − s + 192 s s + 64 x − x − (cid:1) + i (cid:0) s + 16 s + 48 s s − (cid:1)(cid:3) , By choosing free complex parameters s and s in (39), we can get both nonsingular andsingular (blowing-up) solutions. One nonsingular triangular pattern solution is observed anddisplayed in Fig. 3(a). Even though the present solution does not admit u ∗ ( x, t ) = u ( x, − t )and thus does not satisfy the local NLS equation, this pattern resemble that in the localNLS equation . Moreover, this triangular pattern features the double temporal bumpswith a single temporal bump, which is also different from the pattern found in the PT -symmetric nonlocal NLS equation . For the second-order rogue waves, the blowing-upsolutions are shown to have more interesting and complex patterns, which have not beenobserved before. Five of them are displayed in Fig. 3. In panel (b), the solution containsogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 9 FIG. 3. Six second-order rogue waves (39) with different patterns. (a). s = i ; s = 20; s = 0; (b). s = 2 i ; s = 30 i ; (c). s = 2 i ; s = 10; (d). s = 5 i ; s = 0; s = 0; (e). s = 2 i ; s = 40; s = 0;(f). s = 0; s = 30 i ; s = 0; two singular (blowing-up) peaks on the horizontal x axis, and two nonsingular “Peregrine-like” humps on the vertical t axis. In panel (c), these is one “Peregrine-like” nonsingularhump in the middle of two singular peaks on the vertical t axis, plus another two singularpeaks locate also on the vertical t axis. The other three panels each contain six singularpeaks, which are arranged in different patterns. The maximum number of singular peaksin these solutions is six. FIG. 4. Four third-order rogue waves (top row: nonsingular solutions; lower row: collapsingsolutions). (a). s = i ; s = 20; s = 0 (b). s = i ; s = 0; s = − s = 0; s = 0; s =300 i ; s = 4 i ; s = 84; s = 0; (d). s = 2 i ; s = 0; s = 300 i ; Third-order rogue waves would exhibit an even wider variety of solution patterns. Here,four of them are chosen and displayed in Fig. 4. The upper row shows two nonsingular so-lutions, which contain six “Peregrine-like” humps arranged in triangular and pentagon pat-terns, resemble those solutions in the local NLS equation , but not in the PT -symmetricnonlocal NLS equation , while the spatial-temporal structures displayed there are differentfrom these two nonsingular patterns.The lower row in Fig. 4 contains two blowing-up solutions. In panel (c), there is oneogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 10“Peregrine-like” nonsingular hump surrounded by ten singular peaks. In panel (d), twelvesingular peaks arranging in a very exotic pattern. Again, this maximum number of singularpeaks is found to be twelve.The results given above can be apparently extended to higher order rogue waves. Byspecial choices of the free parameters s k ( k ∈ N + ), we can yield even richer spatial-temporalpatterns, in the form of nonsingular humps, singular peaks or their hybrid patterns, butwith more intensity. Moreover, for the fundamental pattern rogue waves(i.e., only s are FIG. 5. The 2D plot of | u N ( x, | from the first- to the third-order rogue waves, with parameters: s = i ; s = 0; s = 0; taken as nonzero value while the rest of the parameters are taken as zero.) in the local NLSequation, the central maximum amplitude for each solutions is given by | u N (0 , | = 2 N +1.However, this value becomes higher for nonlocal equation (2). Taking the first to the thirdorder fundamental rogue-wave pattern as an example. It is shown that the elevations forthem are numerically attained at | u (0 , | = 3 . > | u (0 , | = 6 . >
5, and | u (0 , | = 12 . > III. ROGUE WAVES IN THE REVERSE-TIME NONLOCAL DS SYSTEMS
In this part, we consider rogue waves in the reverse-time nonlocal DS equations, which isrecently introduced in refs. . Eqs.(3)-(4) can be regarded as an integrable multidimensionalversion of the reverse-time nonlocal nonlinear Schr¨odinger equation.We start with the following auxiliary linear system: L Φ = 0 , L = ∂ y − J∂ x − P, (40) M Φ = 0 , M = ∂ t − (cid:88) j =0 V − j ∂ j , (41) V = i γ − J, V = i γ − P, V = i2 γ (cid:0) P x + γ JP y + Q (cid:1) ,J = γ − (cid:18) − (cid:19) , P = (cid:18) q − r (cid:19) , Q = (cid:18) φ φ (cid:19) , with, φ = qr − γ ( φ − φ ) . (42)ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 11The compatibility condition [ L, M ] = 0 leads to the following equation: P t = V ,y − JV ,x − [ P, V ] + (cid:88) j =1 V − j ∂ j ( P ) , (43)which yields to the following coupled system for potential functions q ( x, y, t ), r ( x, y, t ) and φ ( x, y, t ): iq t + 12 γ q xx + 12 q yy + ( qr − φ ) q = 0 , (44) φ xx − γ φ yy − qr ) xx = 0 . (45)Then, the reverse-time nonlocal DS equations (3)-(4) are obtained from the above coupledsystem (44)-(45) under the symmetry reduction: r ( x, y, t ) = σq ( x, y, − t ) . (46)Under reduction (46), it is noticed that potential matrix P satisfies the following symmetrycondition: τ σ P ( x, y, t ) τ − σ = − P † ( x, y, − t ) , τ σ = (cid:18) σ (cid:19) . (47)Here ¯ represents the complex conjugation of the function. In this case, we should have φ ( x, y, t ) = φ ( x, y, − t ) , φ ( x, y, t ) = φ ( x, y, − t ). Thus, φ ( x, y, t ) is an even-function on t , and V ( x, y, t ) have the property τ σ V ( x, y, t ) τ − σ = V † ( x, y, − t ) + i γP † x ( x, y, − t ) . (48)Afterwards, symmetry conditions (47)-(48) further lead to the symmetry in L and M : τ σ L † τ − σ = − L ( t →− t ) , τ σ M † τ − σ = M ( t →− t ) . (49)Here, † stands the (formal) adjoint on an operator. Especially, for a matrix A , where A † denotes the Hermitian conjugate of A . A. A unified binary Darboux transformstion and its reductions
In this section, we construct the corresponding Darboux transformation in a unified wayfor the reverse-time nonlocal DSI and DSII equation. It is known that the DT for DSI andDSII equations (local or nonlocal) appears either in the differential form or in the integralform, which are totally different. However, for this reverse-time nonlocal DS system, wecan find a unified way to tackle their DTs and represent the solutions in one formula.The standard scheme for binary DT was firstly introduced by Matveev and Salle. Es-pecially, for operators L and M given in (40)-(41), a binary DT has been constructed, which can be written explicitly as G θ,ρ = I − θ Ω − ( θ, ρ ) ∂ − ρ † , Ω( θ, ρ ) = ∂ − ( ρ † θ ) , (50)where θ satisfy L ( θ ) = 0, and ρ admits the adjoint operator: L † ( ρ ) = 0. Then the operatorˆ L = G θ,ρ LG − θ,ρ (51)can be directly verified to be a new operator which has the same form as L , and so is for M → (cid:99) M . Furthermore, this Darboux transformation makes sense for any m × k matrices θ and ρ , and we only need Ω( θ, ρ ) to be an invertible square matrix . By the combinationof an elementary DT with its inverse, one derives the B¨acklund transformation between thepotential matrices: ˆ P = P + [ J, θ Ω − ( θ, ρ ) ρ † ] . (52)ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 12Next, to reduce (50) and (52) to the Binary DT for the reverse time nonlocal DS equations(3)-(4), the wave function and the adjoint wave function are restricted to satisfy the followingimportant relation: ρ ( x, y, t ) = τ σ θ ( x, y, − t ) . (53)With condition (53), it can be verified that the new potential matrix in (52) indeedpreserves the symmetry: τ σ ˆ P ( x, y, t ) τ − σ = − ˆ P † ( x, y, − t ) . (54)At the same time, ˆ V will also preserves symmetry (48) due to (54). Hence new opera-tors ˆ L and ˆ M can further satisfy symmetry (49). Therefore, under this reduction, matrix(50) becomes the binary DT for the reverse-time nonlocal DS equations. The B¨acklundtransformations (52) for generating new solutions are given asˆ q = q + γ − [ θ Ω − ( θ, ρ ) ρ † ] , , (55)ˆ φ = φ + 2 γ − [ tr ( θ Ω − ( θ, ρ ) ρ † )] x . (56)Taking n different wave-functions θ i and adjoint wave-functions ρ i which satisfy reduction(53). Denoting Θ = ( θ , · · · , θ n ) , P = ( ρ , · · · , ρ n ) . Introducing 1 × n row vectors ψ i and ϕ i , i = 1 ,
2. The above 2 × n matrices can be furtherrepresented by Θ = (cid:18) ψ ψ (cid:19) , P = (cid:18) ϕ ϕ (cid:19) . Defining the n × n matrix Ω(Θ , P ) as:Ω(Θ , P ) = (cid:16) ∂ − ( ρ † i θ j ) (cid:17) ≤ i,j ≤ n . Hence, the n-fold Buckl¨und transformations between potential functions is given as: q [ n ] ( x, y, t ) = u [0] ( x, y, t ) − γ (cid:12)(cid:12)(cid:12)(cid:12) Ω(Θ , P ) ϕ † ψ (cid:12)(cid:12)(cid:12)(cid:12) det(Ω(Θ , P )) , (57) φ [ n ] ( x, y, t ) = w [0] ( x, y, t ) + 2 γ ∂ x { log [det (Ω(Θ , P ))] } . (58)This n-fold Buckl¨und transformation has been reported by using quasi-determinant tech-nique with simple linear algebra. The proof of expressions (57)-(58) has already beengiven . B. General binary Darboux transformation
To obtain the high-order solutions, we need to perform the general DT scheme. The ideawas firstly introduced by Matveev and further developed by Ling to construct high-order rogue waves in local NLS equation. For this nonlocal DS system, the general binaryDT can be also constructed via a direct limitation technique, those are summarised as thefollowing results. Theorem 3
The generalized binary Darboux matrix for the reverse time nonlocal DSequations (3)-(4) can be represented as: G [ n ] = I − ΘΩ − (Θ , P )Ω( · , P ) , ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 13 where, θ i = θ i ( k i + (cid:15) i ) , ρ j = ρ j (˜ k j + ˜ (cid:15) j ) , (cid:98) Θ = (Θ , Θ , . . . , Θ s ) , Θ i = (cid:16) θ [0] i , . . . , θ [ r i − i (cid:17) , (cid:98) P = ( P , P , . . . , P s ) , P j = (cid:16) ρ [0] j , . . . , ρ [ r j − j (cid:17) , Ω( (cid:98) Θ , (cid:98) P ) = (cid:16) Ω [ ij ] (cid:17) ≤ i,j ≤ s , Ω [ ij ] = (cid:16) Ω [ ij ] m,n (cid:17) r i × r j ,θ i = r i − (cid:88) k =0 θ [ k ] i (cid:15) ki + O ( (cid:15) r i i ) , ρ j = r j − (cid:88) k =0 ρ [ k ] j ˜ (cid:15) kj + O (˜ (cid:15) r j j ) , Ω( θ j ( k j + (cid:15) j ) , ρ i (˜ k i + ˜ (cid:15) i )) = r i (cid:88) m =1 r i (cid:88) m =1 Ω [ ij ] m,n + O (˜ (cid:15) r i i , (cid:15) r j j ) . Moreover, the general Buckl¨und transformations between potential functions are given as: q [ n ] = q [0] − γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω( (cid:98) Θ , (cid:98) P ) (cid:98) P † (cid:98) Θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det(Ω( (cid:98) Θ , (cid:98) P )) , (59) φ [ n ] = φ [0] − ∂ x { log (cid:104) det (cid:16)(cid:98) Ω(Θ , P ) (cid:17)(cid:105) } , (60) Here, (cid:98) Θ represents the 1-st row of matrix (cid:98) Θ, and (cid:98) P stands the 2-nd row in matrix (cid:98) P .The proof for above results can be obtained by directly applying the expansions to eachmatrix element in the n-fold Buckl¨und transformation, performing simple determinant op-erations and taking the limits in (57)-(58). C. Rogue-wave solutions and their dynamics
It is shown that a family of rational solutions were derived by the bilinear method,which generate the rogue waves for the local DS equations. In this work, rogue wavesolutions for the reverse time nonlocal DS equations can be constructed via a unified binaryDarboux transformation method.Firstly, choosing a constant q [0] = c, φ [0] = σc ( c ∈ R ) as the seeding solution. Thenthe eigenfunction solved from Eqs. (40)-(41) has the form ξ i ( x, y, t ) = c i exp [ ω i ( x, y, t )] ,η i ( x, y, t ) = λ i c i c exp [ ω i ( x, y, t )] ,ω i ( x, y, t ) = α i x + β i y + γ i t, γ i = i γ − α i β i ,α i = − γ (cid:18) λ i + σc λ i (cid:19) , β i = 12 (cid:18) λ i − σc λ i (cid:19) , where λ i = r i exp (i ϕ i ), r i and ϕ i are free real parameters, c i is set to be complex.Generally, to derive rational type solutions, we choose a more general eigenfunction viasuperposition principle, which can be written in the form as: θ ( x, y, t ) := { f k + ∂ ϕ k } ( ξ k , η k ) T , f k ∈ C . (61)ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 14
1. Fundamental rogue waves
To derive the first order rational solution, we set n = 1, c = 1 with c = 1 in formula(57)-(58). Then the first-order rational solution in this reverse-time nonlocal system is: q ( x, y, t ) = 1 − F ( t ) F ( x, y, t ) , (62) φ ( x, y, t ) = σ + 2 γ [ln( F ( x, y, t ))] xx , (63)where, F ( t ) = 4 iγ ( λ − + λ ) t + 2 , λ = r e i ϕ , F ( x, y, t ) = (cid:2) σλ − ( γx + y ) − λ ( γx − y ) − f + 1 (cid:3) + 4 (cid:0) λ − + λ (cid:1) t + 1 . The structure of this rational solution is quite different from that in the partially PT -symmetric nonlocal Davey-Stewartson equations. By analysing the denominator in solution(62), it is shown that this rational solution has different dynamical patterns according tothe parameter values of r and ϕ . Those results are discussed separately in the followingcontext.Reverse-time nonlocal DSI ( γ = 1) equation.(i). When ϕ = kπ , λ = ( − k r is a real number. In this case, q ( x, y, t ) is a line roguewave which approaches a constant background, i.e., q → φ → σ as t → ±∞ . Defining m = r − σr − , m = r + σr − , then F ( t ) and F ( x, y, t ) becomes: F ( t ) = 2 i ( m + m ) t + 2 , F ( x, y, t ) = [ H ( x, y )] + (cid:0) m + m (cid:1) t + 1 ,H ( x, y ) = m γx − m y + ( − k (2i f − . For this solution, if Re( f ) (cid:54) = 0 and (Re( f )) ≥ , then function F ( x, y, t ) becomes zero atsymmetrical critical time t c = ± √ f )) − m + m , the singularity occurs along on the line: m γx − m y − ( − k (2Im( f ) + 1) = 0in the ( x, y ) plane. FIG. 6. Fundamental rogue waves in reverse-time nonlocal DS-I equation with parameters: k = 2, σ = 1, r = 2, f = . For other choice in the parameter f , this solution describes the nonsingular line roguewaves, The imaginary part of f can can be further shifted by x or y . The line in thissolution oriented in the ( m , γm ) direction of the ( x, y ) plane. The orientation angleogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 15 β of this solution is β = γ arctan( m /m ) , ( γ = ± (cid:112) m + m = √ σ cos 2 β , so it is angel dependent. Moreover, if (Re( f )) < , the solutionkeeps as constant along the line direction with m γx − m y − ( − k (2Im( f ) + 1) = 0 fixed.As t → ±∞ , the solution q uniformly approaches the constant background 1 everywhere inthe spatial plane. But in the intermediate times, | q | reaches maximum amplitude f ) +34(Re f ) − (more than three times the background amplitude) at the center of the line wave at time t = 0. The speed at which this line approaches its peak amplitude is σ cos 2 β , which is alsoangle dependent. Moreover, as what has been discussed in , for a given σ , the line roguewaves in the reverse-time nonlocal DSI equation have a limited range of orientations. When σ = 1, since r is a real number, one can see that | m /m | <
1, thus the orientation angleof this line rogue wave is between − ◦ and 45 ◦ . If σ = −
1, the satiation is opposite andthe orientation angle is between 45 ◦ and 135 ◦ . We plot this fundamental line rogue wavein Fig.6 with parameters taken as k = 2, σ = 1, r = 2 and f = .Especially, when Re( f ) = 0, this solution has the property: q ( x, y, t ) = q ( x, y, − t ) , (64)which indicates that q ( x, y, t ) also admits the local DSI equation.(ii). When ϕ = ( k − ) π , then λ = i( − k − r is a purely imaginary number. Thisgenerates the rational travelling waves with: F ( t ) = − i ( m + m ) t + 2 , F ( x, y, t ) = (cid:0) m + m (cid:1) t − [ G ( x, y )] + 1 ,G ( x, y ) = m γx − m y + ( − k − (2 f + i) . The ridge of the solution lays approximately on the following two parallel [ x ( t ) , y ( t )] tra-jectories: m γx − m y + ( − k − f ) ± (cid:0) m + m (cid:1) t = 0.Reverse-time nonlocal DSII equation( γ = −
1; Here, we take γ = i in the followingdiscussion).(i). For the first case, if σ = 1, r = 1, i.e., | λ | = 1. We obtain the fundamental linerogue wave. In this case, q ( x, y, t ) approaches a constant background as t → ±∞ . Defining m = r − σr − , m = r + σr − , then F ( t ) and F ( x, y, t ) becomes: F ( t ) = − t cos 2 ϕ + 2 , F ( x, y, t ) = [ F ( x, y )] + 16 t cos ϕ + 1 ,F ( x, y ) = 2 x sin ϕ + 2 y cos ϕ − f + 1 . This line rogue wave solution, if Re( f ) (cid:54) = 0 and (Re( f )) ≥ , then function F ( x, y, t )becomes zero at symmetrical critical time t c = ± √ f )) −
14 cos 2 ϕ . This singularity occurs onthe ( x, y ) plane: 2 x sin ϕ + 2 y cos ϕ + 2Im( f ) + 1 = 0 . For other choice in the parameter f , this solution is the nonsingular line rogue wave. Theline oriented in the (cos ϕ , − sin ϕ ) direction of the spatial plane, the orientation angle is − ϕ . The width in line rogue wave is angel-independent. For any given time, this solutionis a constant along the line with fixed x sin ϕ + y cos ϕ , and approached the constantbackground 1 with x sin ϕ + y cos ϕ → ±∞ . When t → ±∞ , the solution q uniformlyapproaches the constant background 1 everywhere in the spatial plane. In the intermediatetimes, | q | reaches maximum amplitude f ) +34(Re f ) − (more than three times the backgroundamplitude) at the center of the line wave at time t = 0, which is parameter-dependent. Thisline rogue wave is displayed in Fig.7 with parameters chosen as ϕ = π , r = 1 , f = .ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 16 FIG. 7. Fundamental line rogue waves in reverse-time nonlocal DS-II equation with parameters: ϕ = π , r = 1 , f = . It is noted that when cos 2 ϕ = 0, this line wave is oriented in 45 ◦ or − ◦ . In thiscase, this rational solution is not a rogue wave. Instead, it becomes a stationary line solitonsitting on the constant background. Moreover, when Re( f ) = 0, this solution also satisfies(64) so that q ( x, y, t ) admits the local DSII equation .(ii). For the second case, if σ = − r = 1, it generates the rational travelling waveswith: F ( t ) = − t cos 2 ϕ + 2 , F ( x, y, t ) = − [ H ( x, y )] + 16 t cos ϕ + 1 ,H ( x, y ) = 2 x cos ϕ − y sin ϕ + 2 f + i , and the ridge of the solution lays approximately on the following two parallel [ x ( t ) , y ( t )]trajectories: 2 x cos ϕ − y sin ϕ + 2Re( f ) ± t cos 2 ϕ = 0,Noticing that the partially PT -symmetric nonlocal DS system admits the cross-shapedtravelling wave solution .
2. Muliti rogue waves
One of the important subclass of non-fundamental rogue waves is the multi-rogue waves,which describe the interaction between n individual fundamental rogue waves. These roguewaves are obtained when we take n > n real parameters r , ..., r n or ϕ , ..., ϕ n . When t → ±∞ , the solution ( q, φ ) uniformly approaches the constantbackground 1 in the entire spatial plane. In the intermediate times, n line rogue waves arisefrom the constant background, intersect and interact with each other, and then disappearinto the background again. For the nonlocal DSI equation, its multi rogue-wave solutionconsists of n separate line rogue waves in the far field of the spatial plane. However, inthe near field, the wavefronts of the rogue wave solution are no longer lines, and therewould be some interesting curvy wave patterns. For the nonlocal DSII equation, the multirogue-waves appear in the form of exploding rogue waves, which develop singularities undersuitable choice of parameters.To demonstrate these multi rogue-wave solutions in the reverse-time nonlocal DS system,we first consider the case of n = 2. In this case, two rogue-wave solutions for this reverse-time nonlocal DS system are uniformly contained in only one expression, where λ , λ , ϕ , ϕ are free real parameters, f , f are free complex parameters.ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 17 FIG. 8. A two-rogue waves in reverse-time nonlocal DS-I equation with parameters (65).
When γ = 1, i.e., for the nonlocal DSI equation, a two-rogue-wave solution with param-eters λ = 1 , λ = 2 , f = 0 . , f = − .
01 (65)is shown in Fig. 8.Firstly, these two line rogue waves, arising from the constant background, possess higheramplitude in the intersection region at t = −
1. Afterwards, these higher amplitudes in theintersection region fade, then in the far field, two line rogue solutions rise to higher amplitudeat t = 0. Afterwards, the solution goes back to the constant background again at large times(see the t = 10 panel). During this process, the interaction between two fundamental linerogue waves does not generate very high peaks. Actually, the maximum value of solution | q | does not exceed 4 for all times (i.e., four times the constant background). It is notedthat this kind of wave pattern is first found in the local DSI equation . Interestingly, if f , f in (65) are purely imaginary numbers, we have q ∗ ( x, y, t ) = q ( x, y, − t ). In that case,this two-rogue-wave solution also admits the local DSI equation.However, if we choose the value of real parameter r not to be one. For example, if weset λ = 1 / , λ = 2 , f = 0 . , f = − . , (66)as what is shown in Fig.9, the maximum value of solution | q | becomes higher and exceedsthe value 4, which is different from the previous pattern.Thus, for larger n , multi-rogue waves in the reverse-time nonlocal DSI equation havequalitatively similar behaviors, more fundamental line rogue waves will arise and interactwith each other, and more complicated wave fronts will emerge in the interaction region.For example, with n = 3 and parameter choices the corresponding solution is shown inFig.10.As can be seen, the transient solution patterns become more complicated. Also, as wecould see, wether the maximum value of solution | q | stays below or beyond 4 for all timesdepends on the choice of the real parameters λ i . For example, the parameters for Fig.10are chosen as: λ = 1 , λ = 1 . , λ = 2 , f = 0 . , f = 0 , f = − . . (67)In this case, the transient solution patterns become more complicated. But the maximumvalue of this solution | q | still stays below 4 in all times. However, if one takes λ = 1 / FIG. 9. A two-rogue waves which generate higher amplitude in reverse-time nonlocal DS-I equationwith parameters (66).FIG. 10. A three-rogue waves in reverse-time nonlocal DS-I equation with parameters (67).
Next, when γ = 1, we find some exploding rogue-wave solutions for the reverse-timenonlocal DSII equation. These exploding rogue waves, which arise from the constant back-ground 1, can blow up to infinity in a finite time interval at isolated spatial locations undercertain parameter conditions. In this case, the solutions go to a constant background, q → , φ → σ , as t → −∞ . To demonstrate, we consider a two-rogue-wave solution whoseexpression are given by taking n = 2 in (57)-(58). Choosing parameters as λ = 1 , λ = i , f = 0 . , f = 0 . . (68)This exploding rogue-wave solution is displayed in Fig.11.One can see a cross-shape wave firstly appears in the intermediate times (see the t = − . t = − x -direction (corresponding to parameter λ ),the other line oriented along the y -direction (corresponding to parameter λ ). Afterwards,the peak amplitude in this cross-shape rogue wave becomes much higher instantaneously.In Fig.11, we plotted solution up to time t = − .
8, shortly before the blowing-up. One cansee the maximum amplitude for this rogue wave solution becomes extremely high.ogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 19
FIG. 11. The exploding rogue waves in reverse-time nonlocal DS-II equation with parameters (68).
For the local DSII equation, multi and higher-order collapsing wave solutions withconstant boundary conditions have been derived in . Moreover, for the partially PT -symmetric nonlocal system, similar exploding rogue waves also exist, where the blowupscan occur on an entire hyperbola of the spatial plane.
3. High-order rogue waves
Another important subclass of non-fundamental rogue waves is the higher-order roguewaves, which are obtained from the higher-order rational solution with a real value of λ .For example, if n = 2 in (59)-(60), and we get the second-order rogue-wave solution. Todemonstrate these dynamic behaviours, we consider above 2-nd order rogue waves withparameters γ = 1 , σ = 1 , λ = 1 , (69)while f is a free complex parameter. Thus we get q ( x, y, t ) = g h , (70)where ζ ( y ) := y + y + , h = (cid:0) t − x − ζ (cid:1) + ζ (cid:0) t + 4 (cid:1) + 4i f (2 y + 1 − i f ) (cid:2) − (2 y + 1 − i f ) − t + 2 ζ − x − (cid:3) ,g = h − t ) (cid:104) ( − f + 2 y + 1) − (1 + 4i t ) − x (cid:105) . For solution (70), if f is taken as a purely imaginary number, then denominator function h is real, thus relation (64) is satisfied. However, if f is chosen as real number, then h is complex, in that case, this solution do not satisfy the local DSI equation. By choosingsuitable value in f , this solution can be nonsingular.An interesting dynamic behaviour for this solution is that, these higher-order rogue wavesdo not uniformly approach the constant background as t → ±∞ . Instead, only parts of theirwave structures approaches background as t → ±∞ . For instance, if we we set f = 0 . | t | (cid:29)
1, this solution becomes a localized lump sitting on the constant background1 (see the t = ± t →
0. At the same time, aparabola-shaped rogue wave generates from the background. Moreover, when t = 0, thisparabola is approximatively located atogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 20 x = − y − y − .Visually, the solution displayed in Fig.12 can be described as an incoming lump beingreflected back by the appearance of a parabola-shaped rogue wave. This interesting patternis firstly obtained via the bilinear method in ref . It is indeed surprising that a similarpattern can be produced in this reverse-time nonlocal DSI equation, although the expressionfor this solution can be different. FIG. 12. The second order rogue waves in reverse-time nonlocal DS-I equation with parameters(69).
However, when γ = −
1, we only derive some higher-order rational solutions with almostfull-time singularities. Hence, it is still unknown whether there exists nonsingular or finite-time blowing-up high-order rogue waves in the reverse time nonlocal DS-II equation.
IV. SUMMARY AND DISCUSSION
In this article, general rogue waves have been derived for the reverse time nonlocal NLSequation (2) and the reverse time nonlocal DSI and DSII equations (3)-(4) by Darbouxtransformation method under certain symmetry reductions. It is interesting to show that anew unified binary DT has been constructed for this reverse time nonlocal DS system. Thus,the rogue-wave solution in reverse time nonlocal DSI and DSII equation can be written ina unified formula.New dynamics of rogue waves is further explored. It is shown the (1+1)-dimensionalfundamental rogue waves can be bounded for both x and t , or develop collapsing singu-larities. It is also shown that the (1+2)-dimensional fundamental line rogue waves can bebounded for all space and time, or have finite-time blowing-ups. All these types dependon the values of free parameters contained in the solution. In addition, the dynamics pat-terns in the multi- and higher-order rogue waves exhibits richer structures, most of whichare found in the nonlocal integrable equations for the first time. For example, the (1+1)-dimensional higher-order rogue waves exhibit more hybrid of collapsing and non-collapsingpeaks, arranging in triangular, pentagon, circular and other erotic patterns. The (1+2)-dimensional multi-rogue waves describe the interactions between several fundamental linerogue waves, and some curvy wave patterns with higher amplitudes appear due to the in-teraction. However, for the (1+2)-dimensional higher-order rogue waves, only part of thewave structure rises from the constant background and then retreats back to it, which pos-sesses the parabolas-like shapes. While the other part of the wave structure comes fromfar distance as a localized lump, which interacts with the rogue waves in the near field andthen reflects back to the large distance again. These rogue-wave to the nonlocal equationsgeneralize the rogue waves of the local equation into the reverse-time satiation, which couldogue Waves in the Reverse Time Integrable Nonlocal Nonlinear Equations 21play a role in the physical understanding of rogue water waves in the ocean. In addition, theDT reduction method in our paper can be generalized to look for rogue waves in other typesof integrable nonlocal equations, for example, the reverse-space, or the reverse-space-timenonlocal equations . ACKNOWLEDGMENT
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