Spatiotemporal superposed rogue-wave-like breathers in a ( 3 + 1 )-dimensional variable-coefficient nonlinear Schrödinger equation ∗

Abstract. A one-to-one relation between a variable-coefficient (3 + 1)-dimensional nonlinear Schrödinger equation with linear and parabolic potentials and the standard nonlinear Schrödinger equation is presented, and then superposed rogue-wave-like breather solution is obtained. These explicit expressions, describing the evolution of the amplitude, width, center and phase, imply that the diffraction, nonlinearity and gain/loss parameters interplay together to influence evolutional characteristics above. Moreover, the controllable mechanism for fast excitation, maintenance, restraint and recurrence of breather is studied. We also provide an experimental scheme to observe these phenomena in future experiments.

Among these structures, rogue waves (RWs, also known as freak waves, monster waves and extreme waves) from ocean sometimes can be more times higher than the average wave crests.The rational solution (also called Peregrine soliton [26]) of NLSE is the most common prototype to describe the dynamics of RWs.RW events appear from nowhere and disappear without a trace [1].RWs in higher-order NLSE were also discussed [3].Akhmediev et al. reported how to excite a rogue wave [2].Experimentally, Solli et al. [32] reported a randomly created optical rogue wave in a photonic crystal fiber.Dudley et al. [15] investigated the harnessing and control of optical rogue waves in supercontinuum generation.Chabchoub et al. [4] observed rogue wave in a water wave tank.These processes in ocean and nonlinear optics need the controllable RWs, thus many authors theoretically investigated the controllable behaviors of RWs.The management and control of self-similar picosecond [8,34] and femtosecond [13] RWs has been discussed respectively.Moreover, controllable RW triplets have been reported [7].
However, all investigations above based on the theoretical analysis to rational solutions [7,8,13,34].In fact, besides the rational solution, breather [12] (periodic in time or space and localized in space or time) is also regarded as a potential prototype to describe the possible formation mechanisms for RWs.Breather solutions have played vital roles in the electronic, magnetic, vibrational and transport properties of the systems.Recently, researchers have also found breathers in the experiments [14,18].Analytical and numerical evidence has also demonstrated that the management of breathers can be achieved [23].Therefore, controlling and making use of breathers are also needed.
All controllable behaviors for RWs based on rational solutions are studied in (1 + 1)-dimensional (1D) cases [7,8,13,34].Rational solutions are some special cases of breather solutions [19].The real world is higher dimensional case, thus direct knowledge of the control for RWs, especially based on breathers, in higher dimension would be more helpful in terms of understanding the physical phenomena related to RWs.Although Yan et al. [35] and Ma et al. [22] studied dynamical behaviors of 3D RWs, they have not discussed the important controllable behaviors of RWs, which will be investigated here.The novelty of this paper lies in: (i) Spatiotemporal superposed breather solution built from first-order and second-order RWs is firstly obtained, and (ii) the manipulation for higher dimensional RWs based on breathers such as fast excitation, maintenance, restraint and recurrence is firstly reported.
In the following, we focus on new type of breather for Eq. ( 1).Via relation (2) and Darboux transformation (DT) method [19] for Eq. ( 5), an analytical expression for breather solution for Eq. ( 1) reads where Here X and T satisfy Eq. ( 3), φ is given by Eq. ( 4), v is an arbitrary constant, n j are complex eigenvalues in DT, T 0 , T 0 and X j determine the center of solution in (T, X) coordinates.
For simplicity, we firstly analyze breather solution (7) in the framework of the standard NLSE (5), and the dynamical properties of solution (7) for Eq. ( 1) will be discussed in the next section.In fact, breather solution (7) in the framework of the standard NLSE ( 5) is two breathers built from first-order RWs [19], and two kinds of structures is demonstrated in Fig. 1.Figures 1a and 1b show two parallel breathers, whose number of RWs determined by the ratio of k 1 and k 2 .Here we choose k 1 : k 2 = 2 : 3, thus the ratio of the numbers of RWs in breathers is also 2 : 3.
In Figs.1a and 1b, one array of two breathers has different positions in X direction, that is, the centers of breathers at T = −6 in Figs.1a and  Next we analyze the conformation from Figs. 1a to 1c and from Figs. 1b to 1d.In Figs.1a-1d, there are white lines to separate figures into two similar parts.In the left part of Fig. 1a, two RWs near white line possess the same X value, and other three RWs generate a triangular distribution.When five RWs share the same origin (cf.Fig. 1c), two RWs have not overlapped but produced RW-pair, and other three first-order RWs recombine into a second-order RW.Five RWs in the right part of Fig. 1a have similar case.In each part of Fig. 1b, when five RWs share the same origin (cf.Fig. 1d), RW triplets and RW-pair generate respectively.Note that there exists X-directional shift together for RWs during the process of nonlinear superposition.To comprehend these two kinds of superposed RW-like breathers, we show sectional view at different T corresponding to Figs. 1c and 1d in Figs.1e and 1f.

Controllable superposed RW-like breathers
From solution (7), the peak is modulated by ( √ l 2 /W ) β/χ, the width W (t) and center position ξ c (t) are given in (5).The chirp of phase and phase shift are determined by W t /(2βW ) and (3/2) t 0 β(τ )b 2 (τ ) dτ with linear phase b(t) existing constraint in (3).From these expressions, we know that the diffraction β(t), nonlinearity χ(t) and gain/loss γ(t) parameters interplay together to impact evolutional characteristics such as the amplitude, width, center and phase.
Besides these controllable factors above, a vital factor for the propagation type is the relation (5) between the accumulated time T and the real time t, where the diffraction, nonlinearity and gain/loss coefficients play an important role.In the following, we discuss this kind control for the propagation type in two systems.The first system is the diffraction decreasing medium (DDM) with the Logarithmic profile [5,17] where 1/C describes the compression ratio, and L is the setting time in BECs or length of the medium in nonlinear optics with the natural logarithm e, constants β 0 and χ 0 describe initial diffraction and nonlinearity, and the gain/loss parameter γ = γ 0 (const).
The second one is a periodic diffraction amplification system (PDAS) [16,37] where β 0 and χ 0 are the parameters related to the initial diffraction and nonlinearity in system, and σ, λ and δ are the parameters about varying degree of diffraction and nonlinearity.In particular, when δ = 0, Eq. ( 9) corresponds to the exponentially modulated control parameters [9,31], which is a typical case in DDM for σ > 0.
Note that the similarity variable X and the accumulated time T are not real spatial and time variables x, y, z, t.Specially, from Eq. ( 4 in Fig. 1 will appear.Figure 2 exhibits the integral relation between T and t in Logarithmic DDM (8) and PDAS (9).Two different systems have a common property, that is, the accumulated time T exists a maximum value T m .In the framework of Eq. ( 5), T can choose arbitrary values and breathers reach their maximum amplitudes at T = T 0 and then disappear.Therefore, we can adjust the relation between T m and T 0 to realize the control for superposed breathers.
When T m is remarkably bigger than T 0 (see the red cross in Fig. 2a), the full secondorder RWs and first-order RW-pairs are excited quickly (cf.Figs.3a and 1c).The pattern at a trap in (x, y) plane is shown in Fig. 3b, where the trap from Eq. ( 6) slopes from the upper left to bottom-right corner.If T m = T 0 , second-order RWs in breathers can maintain a long time, and their amplitude and width self-similarly vary (cf.Fig. 3c).Moreover, a RW in first-order RW-pair also sustain its shape but another disappears.At last, if T m < T 0 , wave in the framework of Eq. ( 1) have not sufficient time to excite second-order RWs, and restraint of second-order RWs will happen (cf.Fig. 3d).Only part of second-order RWs and one RW in first-order RW-pair are produced, and another in first-order RW-pair is annihilated.
For another kind of superposed breather in Fig. 1d, there are some similar results.When T m is notably bigger than T 0 (see the red cross in Fig. 2a), the full RW triplets and first-order RW-pairs are excited quickly (cf.Figs.4a and 1d).The pattern at a trap in (x, y) plane is shown in Fig. 4b, where the trap is same as that in Fig. 3b.If T m = T 0 , two RWs in triplets and one RW in first-order RW-pair maintain a long lasting time in a self-similar manner but one RW in triplets and one RW in RW-pair both annihilate.At last, if T m < T 0 , the threshold of exciting triplets and pairs are both never reached and the excitations of them are both restrained (cf.Fig. 4d).Only part of superposed RW-like breather is produced.Different from the controllable behaviors in the Logarithmic DDM (8), breather in the PDAS (9) can happen recurrence behavior.In the PDAS (9), from the second expression in Eq. (3), we have with ∆ = 2λ − 4γ 0 − σ.This relation indicates that T changes within the domain Thus, as shown in Fig. 2b, the value of T decreases periodically with oscillating behavior and the maximum T m appears in the first period.This periodic change of T can produce recurrence of breather.
Here we discuss two different cases: T m > T 0 and T m < T 0 .For these two different cases, breathers will demonstrate different dynamical behaviors, and can realize the dynamical manipulation.This controllability for breather in the PDAS ( 9) is remarkably different from that in the Logarithmic DDM.For breather in Fig. 1c, when T m > T 0 (red cross in Fig. 2b), breather will recur periodically (cf.Fig. 5a).It exhibits this recurred behavior of breather in detail, the gap between second-order RWs and first-order RWpairs decreases, and RWs become concentrated with the increase of time t.Fig. 5b shows the evolution of center and width for breather.The center oscillates some periods and gradually tends to a fixed value, and width decreases by degrees when t adds.Different from Figs. 3b and 4b, there is a parabolic trap from Eq. ( 6) in Fig. 5c, where the layout of second-order RWs and first-order RW-pairs appears in (x, y) plane under the action of this trap.If T m < T 0 , wave in the framework of Eq. ( 1) have not sufficient time to be excited.As shown in Fig. 5d, second-order RWs are completely restrained, and only initial M-shaped part are produced.Moreover, RW-pairs are also only excited to one RW.Similarly, when t adds, restrained superposed breather becomes concentrated, and its amplitude dies out quickly.
Similar cases happen for the second kind of superposed breather.When T m > T 0 , breather also recurs periodically in Fig. 6a.From the enlarged plot in Fig. 6b, RW triplets and pairs both periodically appear, and the space between them gradually reduces.When T m < T 0 , the threshold of exciting full triplets and pairs are both never reached and the excitations of them are both restrained.Only one RW in RW-pairs is excited (cf.Fig. 6c and Fig. 1d).From the enlarged plot in Fig. 6d, RWs in breather are remarkably restrained when t adds.

Stability and possible observation
The stability of analytical solutions is important in the realistic application, namely, how they evolve with time when they are disturbed from their analytically given forms.We perform direct numerical simulations with initial white noise for Eq. ( 1) using split-step pulse propagation method, with initial fields coming from Eq. ( 7) in DDM and PDAS.Numerical calculations show no collapse.Instead, stable propagation with a long time is observed.Two examples of such behaviors are displayed in Fig. 7, which essentially presents a numerical rerun of Figs.3c and 5a.
Moreover, we consider the comparison between 1D and 3D cases.Note that the transition from 3D to 1D NLSE is well established in nonlinear optics and BECs [27].Solutions are similar to (7) except for the width W 1D = β/(χ exp(2Γ )) from the calculation for 1D case of Eq. ( 1).From Fig. 7, the 1D case is stable and has smaller period along x-axis than 3D case, and the amplitude's oscillation in parts between RWs in 3D case is more severe than that in 1D case.The white noise has more obviously impact for 3D case in DDM than that in PDAS.
This difference for stability implies that 3D breather is observed more difficult than 1D case.In nonlinear optics, Dudley et al. [14] observed breather in the experiment by putting the initial 1ns pulse at 1064 nm with the power P 0 = 43 W into fiber with parameters β = −75 ps 2 km −1 , χ = 60 W −1 km −1 .However, waves here carry infinite energy because +∞ −∞ |u(x, y, z, t)| 2 dx dy dz diverges.Thus an aperture at the source plane is required to observe the proposed solutions in a laboratory.Similar to method for realizing linear optical bullets [30], we use the envelope of u 0 in the source plane in the form u 0 = Θ(L x − |x|)Θ(L y − |y|)u(px + qy + rz), where Θ(ζ) is a unit step-function, 2L x and 2L y are the dimensions of the source aperture.The Fourier transform and inverse transform with the Fresnel diffraction theory make a finite aperture do not significantly affect their intensity profiles of idealized (infinite-energy) breathers.Thus, we can use this experimental protocol to create breathers here.

Conclusions
In short, we review the main points offered in this paper: • Spatiotemporal superposed breather solution built from first-order and second-order RWs are obtained.3D vcNLSE with linear and parabolic potentials is investigated.A relation between this equation and the standard NLSE is found, and exact spatiotemporal breather solution is obtained.Superposed breathers are constructed when two parallel breathers with different numbers of RWs, decided by ratio of k 1 and k 2 , share the origin.When choosing k 1 : k 2 = 2 : 3, two kinds of superposed RW-like breathers are constructed, that is, breathers constructed by second-order RWs or RW triplets and first-order RW-pairs.• Controllable behaviors of these breathers in different systems can be studied by modulating the relation between the maximum accumulated time T m from the integral relation and the accumulated time T 0 with the center of RW-like breathers.These results are listed in Table 1.We also give an experimental protocol to observe these phenomena in future experiments.These results add to our comprehension on the manipulation for breather, and stimulate novel experiments in the context of the optical communications, plasma physics and Bose-Einstein condensations, and so on.

Figure 4 .
Figure 4. (a)-(d) all cases are corresponding to Fig. 3 except for this control to breathers in Fig. 1b.(Online version in color.)

Figure 5 .
Figure 5. (a) Recurrence and (d) restraint of RW-like breathers in Fig. 1a in the PDAS, (b) the evolution of center and width for breather, and (c) RW-like breather corresponding to (a) in a trap at t = 5.6, z = 2 in (x, y) coordinates.Parameters in (a) and (d) correspond to β 0 = 0.25 (red cross) and 0.9 (blue dash) in Fig. 2b with b 0 = 0.5.(Online version in color.)

Figure 6 .
Figure 6.(a) Recurrence and (c) restraint of self-similar RW-like breathers in Fig. 1b in the PDAS, (b) and (d) different domain views for (a) and (c), respectively.Parameters are same as Fig. 5. (Online version in color.)

Figure 7 .
Figure 7.Comparison of stability between 1D and 3D cases: (a) maintenance for Figs.3c and 3b recurrence corresponding to Fig. 5a.Only the dependence on x is shown for 3D case.An added 5% white noise are added to the initial values.(Online version in color.)

Table 1 .
Controllable behaviors in different systems.