- Email: [email protected]
The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation
Annals of Physics 327 (2012) 512–521
Contents lists available at SciVerse ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation Chao-Qing Dai a,b,∗ , Yue-Yue Wang a , Qing Tian b , Jie-Fang Zhang b,c a
School of Sciences, Zhejiang A&F University, Lin’an, Zhejiang 311300, PR China
b
School of Physical Science and Technology, Suzhou University, Suzhou, Jiangsu 215006, PR China
c
Zhejiang University of Media and Communications, Hangzhou, 310018, PR China
article
info
Article history: Received 5 August 2011 Accepted 12 November 2011 Available online 6 December 2011 Keywords: Nonlinear Schrödinger equation Rogue wave Management and containment
abstract We present, analytically, self-similar rogue wave solutions (rational solutions) of the inhomogeneous nonlinear Schrödinger equation (NLSE) via a similarity transformation connected with the standard NLSE. Then we discuss the propagation behaviors of controllable rogue waves under dispersion and nonlinearity management. In an exponentially dispersion-decreasing fiber, the postponement, annihilation and sustainment of self-similar rogue waves are modulated by the exponential parameter σ . Finally, we investigate the nonlinear tunneling effect for self-similar rogue waves. Results show that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged or decreasing amplitudes via the modulation of the ratio of the amplitudes of rogue waves to the barrier or well height. © 2011 Elsevier Inc. All rights reserved.
1. Introduction Rogue waves (or freak waves) – single ocean waves with amplitudes significantly larger than those of the surrounding waves – were considered mysterious until recorded for the first time by scientific measurements during an encounter at the Draupner oil platform in the North Sea [1]. Although they are elusive and intrinsically difficult to monitor due to their fleeting existence, massive efforts have been devoted to studying rogue waves [2–7]. Good agreement with an approximate
∗
Corresponding author at: School of Sciences, Zhejiang A&F University, Lin’an, Zhejiang 311300, PR China. E-mail address: [email protected] (C.-Q. Dai).
0003-4916/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2011.11.016
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
513
dynamical/statistical theory was found [2]. A statistical model has been developed that predicts for a given mean sea state the probability of occurrence of extreme events [3]. Howell Peregrine identified the key role of the modulational instability (MI) in the formation of patterns resembling freak waves; in particular, as early as 1983 he drew attention to algebraic breather (also named after him—Peregrine soliton) solutions of the nonlinear Schrödinger equation (NLSE) which could serve as freak-wave weakly nonlinear prototypes [4]. Osborne [5] summarized an analytical method (the inverse scattering method) for discussing nonlinear ocean waves including rogue waves. So far, two main generic mechanisms have been identified in the absence of wave–current interaction: the Benjamin–Feir (BF) [4,6] or modulational instability and an essentially linear space–time focusing [7]. It is relatively difficult to excite rogue waves experimentally in hydrodynamical systems, although recently Akhmediev et al. [8] have studied how to excite a rogue wave and Chabchoub et al. [9] observed a rogue wave in a water wave tank. Interestingly, the NLSE not only gives a suitable description of rogue water waves, but also it is the governing equation for light pulse propagation in nonlinear optical fibers and matter waves in Bose–Einstein condensates (BECs). Thus, nonlinear optical fibers and BECs provide a good testbed for the study of rogue waves [10,11]. Unlike rogue waves, rogue solitons have often been observed experimentally in optics [12–18]; however, the rogue solitons observed do not exhibit exactly the properties of rogue waves such as the propensity to appear from nowhere and disappear without a trace [19,20]. As regards this, Solli et al. [12] have made considerable progress by observing a randomly created optical rogue soliton in a photonic crystal fiber. Kasparian et al. [13] experimentally observed optical rogue soliton statistics during high power femtosecond pulse filamentation in air. Akhmediev’s breather theory [14] and MI were also linked experimentally in nonlinear optical fibers [15,16]. Moreover, Erkintalo et al. [17] experimentally studied the characteristics of optical rogue solitons in supercontinuum generation in the femtosecond regime. More recently, the optical rogue solitons observed initially grew from noise and it is possible to control both the spectral extent and, most importantly, the noise in the supercontinuum [15,18]. Until now, experimental observation almost always reported randomly created rogue solitons [12,13,15–18], although there have been recent studies of rogue waves emerging from optical turbulence [10,11]. Thus it is significant and interesting to understand controllable rogue waves theoretically, including the restraint, annihilation, postponement and sustainment of rogue waves, which are investigated in this paper. In nonlinear optics, recent developments [21–23] have led to the discovery of new classes of waves, such as the so-called optical self-similar pulses. Thus, an interesting issue arises: can the self-similar rogue waves described by the rational solutions be managed and controlled? Of course, it is impossible to answer this question in one step. We have investigated controllable optical rogue waves without considering self-similar behaviors in our recent work [24]. In the present work, we make a further step forward in an attempt to understand this issue. The problem is governed by the variable-coefficient NLSE (vcNLSE) as follows [25,26]: iuz +
β(z ) 2
utt + χ (z )|u|2 u + α(z )t 2 u = iγ (z )u,
(1)
where u(z , t ) is the complex envelope of the electrical field, and z and t, respectively, represent the propagation distance and retarded time. β(z ), χ (z ) and γ (z ) represent the group-velocity dispersion (GVD), nonlinearity parameters and loss (or gain), respectively. The parameter α(z ) is the normalized loss rate and α(z )t 2 accounts for the chirping, indicating that the initial chirping parameter is the square of the normalized growth rate. For nonlinear waveguides, t is the spatial variable, β(z ) is the diffraction coefficient, and α(z )t 2 originates from the refractive index n(t , z ) = n0 + n1 α(z )t 2 + n2 χ(z )I (t , z ) with the optical intensity I (t , z ), the linear index n0 and Kerr parameter n2 . If variables z and t are replaced by parameters t and x, Eq. (1) is the generalized Gross–Pitaevskii (GP) equation with harmonic oscillator potentials in BECs [27,28]. When α(z ) = 0, Eq. (1) is the corresponding NLSE in [29], for which the authors investigated not only the soliton control, but also the soliton interaction. Moreover, the multi-soliton solutions in terms of the double Wronskian determinant are presented for Eq. (1) with α(z ) = 0 [30]. When all parameters are constant, Eq. (1) can describe nonlinear spin waves and magnetic solitons in a uniaxial anisotropic ferromagnet [31].
514
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
2. The similarity transformation and rational solutions In order to get the exact analytical solutions for (1), we will construct the transformation [23,32] u(z , t ) = ρ(z )U (Z , T ) exp[iϕ(z , t )],
(2)
where ρ(z ) means the amplitude, and the effective propagation distance Z (z ) and the similarity variable T (z , t ) are both to be determined. Real parameters ρ(z ), Z (z ), T (z , t ) and ϕ(z , t ) are differential functions and should be well chosen to avoid the singularity of the field u(z , t ), the parameter of system α(z ) and the form factor of the solutions W (z ), ρ(z ), ϕ(z ) (see Eqs. (3)–(6)). Note that the similarity transformation (2) with two parameters Z (z ), T (z , t ) is more general than the one with the sole parameter T (z , t ) in [28]; that is, we can discuss dynamical behaviors of multi-solitons while only single solitons could be discussed in [28]. With the substitution of Eq. (2) into Eq. (1), one gets the following expressions for the similarity variable, effective propagation distance, central position, amplitude, width and phase of the pulse: T =
t − tc (z ) W (z )
Z = 0
tc (z ) = −W02 W
z
0
β(s) W 2 (s)
W 2 (s)
ds,
ds
W0 β , χ exp(2Γ ) z Wz 2 W2 W4 β(s) ϕ(z , t ) = t − 0t− 0 ds, 2β W W 2 0 W 2 (s) z with Γ (z ) = 0 γ (ς )dς ; then Eq. (1) is reduced to the standard NLSE
ρ(z ) =
iUZ +
1 2
1
W
β , χ
β(s)
z
,
W (z ) =
UTT + |U |2 U = 0.
(3) (4)
(5)
(6)
(7)
Note that we have required the following integrability condition:
α(z ) =
β Wzz − βz Wz (4γ 2 − 2γz )β 2 + (βzz − 2βz γ )β − βz2 = 2 2β W 2β 3 +
(4χz γ − χzz )β 2 − βz χz β χz2 + . 2β 3 χ χ 2β
(8)
Note that the vanishing chirp (i.e. Wz = 0) leads to α(z ) = 0 in (8), and Eq. (1) becomes the variable-coefficient NLSE in [21,33,34]; thus the chirp is essential to the self-similar solution here. Also note that the existence of the coefficient α(z ) makes the solution more difficult than that of the corresponding equation inthe case of the generalized NLSE in [34]. For α(z ) = 0, the constraint (8) z hints that W = W0 [1 − c0 0 β(s)dz ]. Then this solution can be reduced to the corresponding solution expressed as (5)–(9) in [34]. From Eqs. (3) to (8), we know that the form factors of solutions such as the effective propagation distance, central position, amplitude, width and phase of the pulse are decided by the group-velocity dispersion and nonlinearity parameters of the system; thus this self-similar solution can be controlled under dispersion and nonlinearity management. The one-to-one correspondence (2) allows us to obtain abundant solutions, such as bright and dark single-soliton or multi-soliton solutions, Jacobian elliptic function solutions, and so on. With the help of the mapping transformation (2), the inhomogeneous NLSE (1) can be transformed into the standard NLSE (7). Then like in the procedure in [35], by means of the reverse transformation variables and
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
515
functions, we obtain the exact solutions for Eq. (1). Here we focus on rogue wave solutions. Employing the transformation (2), and the Darboux transformation [36], one can obtain rogue wave solutions (rational solutions) for Eq. (1). The first-order (n = 1) and second-order (n = 2) rational-like solutions read un =
1 W
β Gn + i(Z − Z0 )Hn v2 n × exp i 1 − (−1) + (Z − Z0 ) + v T + ϕ , χ Fn 2
(9)
where 2G1 = H1 = 8, F1 = 1 + 4[T − v(Z − Z0 )]2 + 4(Z − Z0 )2 for a one-rogue wave solution and G2 = [(T − v (Z − Z0 ))2 + (Z − Z0 )2 + 43 ][(T − v (Z − Z0 ))2 + 5 (Z − Z0 )2 + 34 ] − 43 , H2 =
}, F2 = (Z − Z0 ) {(Z − Z0 )2 − 3[T − v (Z − Z0 )]2 + 2[(T − v (Z − Z0 ))2 + (Z − Z0 )2 ]2 − 15 8 2 2 3 2 2 1 1 9 2 T − v Z − Z0 ))2 + [( T − v Z − Z + Z − Z ] + [( T − v Z − Z − 3 Z − Z ] + ( ( ( )) ( ) ( )) ( ) 0 0 0 0 3 4 16 33 3 for a two-rogue wave, T and Z satisfy Eq. (3), ϕ is given by Eq. (6), and Z0 and v are (Z − Z0 )2 + 64 16
two arbitrary constants.
3. Postponement, sustainment and annihilation of self-similar rogue waves Next we analyze how to realize the control for self-similar rogue waves. The crucial point lies in the relation between the effective propagation distance Z and the original propagation distance z. From Eq. (10), this relation can be modulated under dispersion and nonlinearity management. To demonstrate the controllable rogue waves, we take an exponentially dispersion-decreasing fiber (DDF) [37] described by
β(z ) = β0 exp(−σ1 z ),
χ (z ) = χ0 exp(σ2 z ),
γ (z ) = γ0 ,
(10)
where β0 and χ0 are the parameters related to GVD and nonlinearity, σ1 > 0 corresponds to the DDF and γ0 denotes the constant net gain (>0) or loss (<0). On one hand, from Eq. (3) the effective propagation distance Z has a relation to the original χ2
propagation distance z with Z = − W β (σ +02σ +4γ ) {1 − exp[(σ1 + 2σ2 + 4γ0 )z ]}, which implies 0 0 1 2 0 that when σ1 + 2σ2 + 4γ0 > 0, one has Z > z and Z → ∞ with z → ∞; thus rogue waves are excited at z = −ln
χ02
Z0 W02 β0 (σ1 +2σ2 +4γ0 )+χ02
/(σ1 + 2σ2 + 4γ0 ) and then vanish quickly. When χ2
σ1 + 2σ2 + 4γ0 < 0, one has Z < z and Z → Zmax = − W0 β0 (σ2 +02σ1 +4γ0 ) with z → ∞. On the other hand, in the framework of Eq. (7), rogue waves reach their maximum amplitudes when Z = Z0 and then disappear, while in the framework of Eq. (1) the full excitation of rogue waves will be sustained for quite a long distance. Therefore, at first, if Zmax is slightly bigger than Z0 , the excitations of both the one-rogue wave and the two-rogue wave are postponed (cf. Fig. 1(b) and (d)), i.e. the complete rogue waves are not excited. When Zmax adds sequentially and is close to 3Z0 , the full rogue waves are excited fleetingly (cf. Fig. 1(a) and (c)). Here we set σ2 = −σ1 . Note that during the propagation, the values reached by the maximal amplitudes of rogue waves are (a) 6.3, (b) 6.8, (c) 7.0 and (d) 10.0 in Fig. 1. It is interesting that the typical maximal amplitude of the first-order rational solution of NLSE is three times larger than the surrounding waves and the typical maximal amplitude of the second-order rational solution is five times larger than the surrounding waves [35]. Obviously the values reached by the maximum amplitudes of the fast and postponed excited rogue waves are different from those of the standard NLSE [35]. Secondly, if Zmax = Z0 , the full excitations of both the one-rogue wave and the two-rogue wave can be maintained forever with self-similar propagating behaviors (cf. Fig. 2(a) and (c)), where their amplitudes and widths vary self-similarly after a short propagation distance from the initial condition. Finally, if Zmax < Z0 , the thresholds for exciting both the one-rogue wave and the tworogue wave are never reached and their excitations are restrained or even eliminated (cf. Fig. 2(b) and (d)), looking like a bright optical similariton and separated bright similariton pairs [23,32] with very small amplitudes propagating stably along the fiber on a non-zero background.
516
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
Fig. 1. ((a), (b)) Fast and postponed excitation of one-rogue waves. ((c), (d)) Fast and postponed excitation of two-rogue waves. The parameters are v = 0.2, W0 = 0.5, χ0 = 0.12, β0 = 0.1, γ0 = 0.001, Z0 = 8 with ((a) and (c)) σ1 = 0.035 and ((b) and (d)) σ1 = 0.065. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 6.3, (b) 6.8, (c) 7.0 and (d) 10.0.
4. Nonlinear tunneling effects for controllable rogue waves The concept of nonlinear tunneling follows directly from the nonlinear wave equation that leads to the associated nonlinear dispersion relation [38]. Historically, the study of soliton tunneling effects governed by the NLSE with variable coefficients began with the pioneering work of Serkin and Belyaeva [39]. They also investigated the nonlinear tunneling of optical solitons through strong nonlinear organic thin films and polymeric waveguides, which exhibit jumplike nonadiabatic fission reactions, eventually resulting in soliton ‘‘fission reactions’’ [40]. These results hint that the concept of soliton tunneling could inaugurate a new and exciting area in the application of optical solitons. Thus, recently, the tunneling effects of solitons governed by various NLSE were extensively investigated since being predicted as early as 1978, by Newell [38]. For instance, Yang et al. [41] confirmed the compression of the optical pulse by a nonlinear barrier. Wang et al. [42] studied the tunneling effects of spatial similaritons passing through the nonlinear barrier (or well). Dai et al. [43] discussed the tunneling effects of bright and dark similaritons governed by the generalized coupled NLSE for the birefringent fiber. Zhong and Belić [44] investigated the nonlinear tunneling effects of spatial solitons in the nonlinear and diffractive barriers (wells). More recently, enigmas of optical and matter–wave soliton nonlinear tunneling were uncovered in [45]. Moreover, it should be emphasized that the nonlinear soliton tunneling effect is a subject of constantly renewed interest. In femtosecond nonlinear fiber optics, the most intriguing enigma of optical solitons is connected with the so-called soliton spectral tunneling effect (SST). This effect is characterized in the spectral domain by the passage of a femtosecond soliton through a potential barrier-like spectral inhomogeneity of GVD, including a forbidden normal dispersion barrier (or well) [46,47]. Since this phenomenon shares many analogies with that of quantum mechanical tunneling through a potential barrier, it is generally referred to as the SST effect [46,47]. However, to our knowledge, the tunneling effects of self-similar rogue waves have been less investigated. To investigate the unique property of nonlinear tunneling effects for controllable rogue waves, we focus on rogue waves propagating through a nonlinear barrier (NB) or nonlinear well (NW) on an
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
517
Fig. 2. ((a), (b)) Sustainment and annihilation of self-similar one-rogue waves. ((c), (d)) Sustainment and annihilation of selfsimilar two-rogue waves. The parameters are the same as for Fig. 1 with ((a) and (c)) σ1 = 0.076 and ((b) and (d)) σ1 = 0.1. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 9.6, (b) 3.96, (c) 16.8 and (d) 4.4.
exponential background [42,43]:
β(z ) = β0 ,
χ (z ) = χ0 {re−gz + hsech2 [a(z − z0 )]},
γ (z ) = γ0 ,
(11)
where h denotes the height of the NB or NW, respectively. a is related to the NB or NW width, g is a decaying (g > 0) or increasing (g < 0) parameter, and z0 represents the longitudinal coordinate indicating the location of NB or NW. As β(z ) > 0, we assume that h > −1, where h > 0 indicates the NB and −1 < h < 0 represents the NW. We consider the case of function α(z ) being the constant α0 ; √ then Eq. (8) hints that the widths of self-similar rogue waves are given by W (z ) = W0 exp( 2β0 α0 z ). From Eq. (3), the effective propagation distance Z has a relation to the original propagation distance √ z with Z = √
Zmax =
2 β0
4α0 W02
2β0
4α0 W02
√ [1 − exp(− 2β0 α0 z )]; thus the maximum of effective propagation distance
for α0 > 0. Like in discussion above, when Zmax > Z0 , rogue waves are postponed
(cf. Fig. 3). If Zmax = Z0 , rogue waves can be sustained (cf. Fig. 4), while for Zmax < Z0 , rogue waves are restrained or even eliminated (cf. Fig. 6). As shown in Figs. 3(a) and (c), 4(a) and (c) and 5(a) and (c), it can be seen that when the postponed, sustained and annihilated rogue waves pass through the NB at z = z0 = 100, the pulses diminish in amplitude and form the channels; then they increase in amplitude and recover their original shapes. When they propagate through the NW at z = z0 = 100, the pulses are amplified and form peaks, and then attenuate and recover their original shapes, as shown in Figs. 3(b) and (d), 4(b) and (d) and 5(b) and (d), respectively. It was found that in certain circumstances, depending on the ratio of the amplitudes of the rogue waves to the barrier or well height, rogue waves can tunnel through the barrier or well in a
518
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
Fig. 3. Postponed one-rogue and two-rogue waves passing through ((a), (c)) the DB and ((b), (d)) the DW. The parameters are g = 0.058, r = 1, z0 = 100, α0 = 0.05 with (a) a = 0.6, h = 0.3 and (b) a = 2, h = −0.2. The other parameters are the same as for Fig. 1. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 2.9, (b) 3.6, (c) 4.1 and (d) 4.2.
lossless manner. Fig. 6 illustrates the evolution of self-similar rogue waves through the NB for the decaying or increasing parameter g. When g = 0.058 for the postponed rogue wave or g = 0.045 for the sustained rogue wave, the amplitudes of rogue waves are almost unchanged as we see from the comparison of the red line with the green line in Fig. 6. When g > 0.058 for the postponed rogue wave or g > 0.045 for the sustained rogue wave and g < 0.058 for the postponed rogue wave or g < 0.045 for the sustained rogue wave, the amplitudes gradually increase (compare the red line with the black line in Fig. 6) and decrease (compare the red line with the blue line in Fig. 6) along z, respectively. This implies that the parameter g can be used to control the amplitude of the rogue wave after it passes through the barrier. Analysis shows that rogue waves passing across the NW have the same properties. 5. Conclusions In summary, we have constructed the relation between the inhomogeneous NLSE and the standard NLSE via a similarity transformation. On the basis of this transformation, we analytically obtain the one-rogue wave and the two-rogue wave for the inhomogeneous NLSE. For certain parameter conditions, we discuss the propagation behaviors of controllable rogue waves under dispersion and nonlinearity management. In an exponentially dispersion-decreasing fiber, the postponement, annihilation and sustainment of self-similar rogue waves are modulated via the exponential parameter σ . Finally, we investigate nonlinear tunneling effects for self-similar rogue waves. Results show that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged or
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
519
Fig. 4. Sustained one-rogue waves passing through ((a), (c)) the DB and ((b), (d)) the DW. The parameters are the same as for Fig. 3 with g = 0.045 and α0 = 0.056. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 1.48, (b) 1.88, (c) 2.3 and (d) 2.9.
Fig. 5. Annihilated one-rogue waves passing through ((a), (c)) the DB and ((b), (d)) the DW. The parameters are the same as for Fig. 3 with g = 0.045 and α0 = 0.06. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 0.88, (b) 1.48, (c) 1.16 and (d) 1.78.
520
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
a
b
Fig. 6. The sectional view before passing through the NB at z = 90 (dotted line) and after passing through the NB at z = 110 (solid line). (a) and (b) correspond to Figs. 3(a) and 4(a): the red line denotes the rogue wave before passing through the NB; green, black and blue denote the rogue wave after passing through the NB with g = 0.058, 0.063, 0.053 for the postponed one-rogue wave and g = 0.045, 0.05, 0.04 for the sustained one-rogue wave. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
decreasing amplitudes, through modulation of the ratio of the amplitudes of rogue waves to the barrier or well height. The results obtained in this paper supplement well our comprehension of rogue waves, that is, although they ‘‘appear from nowhere and disappear without a trace’’ [19], rogue waves can be controlled as discussed, in a similar way, in this paper. Moreover, these results may have potential values for the generation and sustainment of exceptionally high amplitude optical pulses—‘‘optical rogue waves’’ and their oceanic rogue wave counterparts. Of course, more practical implementation of these theoretical results might be an interesting task. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11005092 and No. 11072219), the Program for Innovative Research Teams of Young Teachers (Grant No. 2009RC01) and the Scientific Research and Development Fund (Grant No. 2009FK42) of Zhejiang A&F University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
W.J. Broad, Rogue Giants at Sea, The New York Times, 2006. P.A.E.M. Janssen, J. Phys. Oceanogr. 33 (2003) 863. N. Mori, P.A.E.M. Janssen, J. Phys. Oceanogr. 36 (2006) 1471. D.H. Peregrine, J. Aust. Math. Soc. Ser. B 25 (1983) 16. A.R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Method, Academic Press, San Diego, 2010. A.I. Dyachenko, V.E. Zakharov, JETP Lett. 81 (2005) 255. C. Kharif, E. Pelinovsky, A. Slyunyaev, Rogue Waves in the Ocean, Springer, Berlin, 2009. N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Rev. A 80 (2009) 043818. A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Phys. Rev. Lett. 106 (2011) 204502. K. Hammani, B. Kibler, C. Finot, A. Picozzi, Phys. Lett. A 374 (2010) 3585. B. Kibler, K. Hammani, C. Michel, C. Finot, A. Picozzi, Phys. Lett. A 375 (2011) 3149. D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450 (2007) 1054. J. Kasparian, P. Béjot, J.-P. Wolf, J.M. Dudley, Opt. Express 17 (2009) 12070. N. Akhmediev, V.I. Korneev, Theor. Math. Phys. 69 (1986) 1089. J.M. Dudley, G. Genty, F. Dias, B. Kibler, N. Akhmediev, Opt. Express 17 (2009) 21497. B. Kibler, J. Fatome, C. Finot, et al., Nat. Phys. 6 (2010) 790. M. Erkintalo, G. Genty, J.M. Dudley, Opt. Lett. 34 (2009) 2468. D.R. Solli, B. Jalali, C. Ropers, Phys. Rev. Lett. 105 (2010) 233902. N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373 (2009) 2137. Y. Xia, W.P. Zhong, M. Belic, Acta Phys. Polon. B 42 (2011) 1881. V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. E 71 (2005) 056619.
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521 [22] S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97 (2006) 013901; J.M. Dudley, C. Finot, D.J. Richardson, G. Millot, Nat. Phys. 3 (2007) 597. [23] C.Q. Dai, Y.Y. Wang, J.F. Zhang, Opt. Lett. 35 (2010) 1437; C.Q. Dai, D.S. Wang, L.L. Wang, J.F. Zhang, W.M. Liu, Ann. Phys. (NY) 326 (2011) 2356. [24] Q. Tian, Q. Yang, C.Q. Dai, J.F. Zhang, Opt. Commun. 284 (2011) 2222. [25] L. Li, Z.H. Li, S.Q. Li, G.S. Zhou, Opt. Commun. 234 (2004) 169. [26] R. Atre, P.K. Panigrahi, G.S. Agarwal, Phys. Rev. E 73 (2006) 056611. [27] V. Serkin, A. Hasegawa, T.S. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102. [28] W.P. Zhong, M.R. Belić, Y.Q. Lu, T.W. Huang, Phys. Rev. E 81 (2010) 016605. [29] W.J. Liu, B. Tian, T. Xu, K. Sun, Y. Jiang, Ann. Phys. (NY) 325 (2010) 1633. [30] X. Lü, B. Tian, T. Xu, K.J. Cai, W.J. Liu, Ann. Phys. (NY) 323 (2008) 2554. [31] Z.D. Li, Q.Y. Li, P.B. He, Z.G. Bai, Y.B. Sun, Ann. Phys. (NY) 322 (2007) 2945. [32] C.Q. Dai, X.G. Wang, J.F. Zhang, Ann. Phys. (NY) 326 (2011) 645; C.Q. Dai, R.P. Chen, G.Q. Zhou, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 145401. [33] J.F. Zhang, C.Q. Dai, Q. Yang, J.M. Zhu, Opt. Commun. 252 (2005) 408. [34] S.A. Ponomarenko, G.P. Agrawal, Opt. Express 15 (2007) 2963. [35] N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373 (2009) 675. [36] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601. [37] V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. Lett. 90 (2003) 113902; L.H. Zhao, C.Q. Dai, Eur. Phys. J. D 58 (2010) 327. [38] A.C. Newell, J. Math. Phys. 19 (1978) 1126. [39] V.N. Serkin, T.L. Belyaeva, JETP Lett. 74 (2001) 573. [40] V.N. Serkin, V.M. Chapela, J. Percino, T.L. Belyaeva, Opt. Commun. 192 (2001) 237. [41] G.Y. Yang, R.Y. Hao, L. Li, Z.H. Li, G.S. Zhou, Opt. Commun. 260 (2006) 282. [42] J.F. Wang, L. Li, S.T. Jia, J. Opt. Soc. Am. B 25 (2008) 1254. [43] C.Q. Dai, Y.Y. Wang, J.F. Zhang, Opt. Express 18 (2010) 17548. [44] W.P. Zhong, M.R. Belić, Phys. Rev. E 81 (2010) 056604. [45] T.L. Belyaeva, V.N. Serkin, C. Hernandez-Tenorio, F. Garcia-Santibanez, J. Modern Opt. 57 (2010) 1087. [46] B. Kibler, P.-A. Lacourt, F. Courvoisier, J.M. Dudley, Electron. Lett. 43 (2007) 967. [47] F. Poletti, P. Horak, D.J. Richardson, IEEE Photon. Tech. Lett. 20 (2008) 1414.
521
Contents lists available at SciVerse ScienceDirect
Annals of Physics journal homepage: www.elsevier.com/locate/aop
The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation Chao-Qing Dai a,b,∗ , Yue-Yue Wang a , Qing Tian b , Jie-Fang Zhang b,c a
School of Sciences, Zhejiang A&F University, Lin’an, Zhejiang 311300, PR China
b
School of Physical Science and Technology, Suzhou University, Suzhou, Jiangsu 215006, PR China
c
Zhejiang University of Media and Communications, Hangzhou, 310018, PR China
article
info
Article history: Received 5 August 2011 Accepted 12 November 2011 Available online 6 December 2011 Keywords: Nonlinear Schrödinger equation Rogue wave Management and containment
abstract We present, analytically, self-similar rogue wave solutions (rational solutions) of the inhomogeneous nonlinear Schrödinger equation (NLSE) via a similarity transformation connected with the standard NLSE. Then we discuss the propagation behaviors of controllable rogue waves under dispersion and nonlinearity management. In an exponentially dispersion-decreasing fiber, the postponement, annihilation and sustainment of self-similar rogue waves are modulated by the exponential parameter σ . Finally, we investigate the nonlinear tunneling effect for self-similar rogue waves. Results show that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged or decreasing amplitudes via the modulation of the ratio of the amplitudes of rogue waves to the barrier or well height. © 2011 Elsevier Inc. All rights reserved.
1. Introduction Rogue waves (or freak waves) – single ocean waves with amplitudes significantly larger than those of the surrounding waves – were considered mysterious until recorded for the first time by scientific measurements during an encounter at the Draupner oil platform in the North Sea [1]. Although they are elusive and intrinsically difficult to monitor due to their fleeting existence, massive efforts have been devoted to studying rogue waves [2–7]. Good agreement with an approximate
∗
Corresponding author at: School of Sciences, Zhejiang A&F University, Lin’an, Zhejiang 311300, PR China. E-mail address: [email protected] (C.-Q. Dai).
0003-4916/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2011.11.016
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
513
dynamical/statistical theory was found [2]. A statistical model has been developed that predicts for a given mean sea state the probability of occurrence of extreme events [3]. Howell Peregrine identified the key role of the modulational instability (MI) in the formation of patterns resembling freak waves; in particular, as early as 1983 he drew attention to algebraic breather (also named after him—Peregrine soliton) solutions of the nonlinear Schrödinger equation (NLSE) which could serve as freak-wave weakly nonlinear prototypes [4]. Osborne [5] summarized an analytical method (the inverse scattering method) for discussing nonlinear ocean waves including rogue waves. So far, two main generic mechanisms have been identified in the absence of wave–current interaction: the Benjamin–Feir (BF) [4,6] or modulational instability and an essentially linear space–time focusing [7]. It is relatively difficult to excite rogue waves experimentally in hydrodynamical systems, although recently Akhmediev et al. [8] have studied how to excite a rogue wave and Chabchoub et al. [9] observed a rogue wave in a water wave tank. Interestingly, the NLSE not only gives a suitable description of rogue water waves, but also it is the governing equation for light pulse propagation in nonlinear optical fibers and matter waves in Bose–Einstein condensates (BECs). Thus, nonlinear optical fibers and BECs provide a good testbed for the study of rogue waves [10,11]. Unlike rogue waves, rogue solitons have often been observed experimentally in optics [12–18]; however, the rogue solitons observed do not exhibit exactly the properties of rogue waves such as the propensity to appear from nowhere and disappear without a trace [19,20]. As regards this, Solli et al. [12] have made considerable progress by observing a randomly created optical rogue soliton in a photonic crystal fiber. Kasparian et al. [13] experimentally observed optical rogue soliton statistics during high power femtosecond pulse filamentation in air. Akhmediev’s breather theory [14] and MI were also linked experimentally in nonlinear optical fibers [15,16]. Moreover, Erkintalo et al. [17] experimentally studied the characteristics of optical rogue solitons in supercontinuum generation in the femtosecond regime. More recently, the optical rogue solitons observed initially grew from noise and it is possible to control both the spectral extent and, most importantly, the noise in the supercontinuum [15,18]. Until now, experimental observation almost always reported randomly created rogue solitons [12,13,15–18], although there have been recent studies of rogue waves emerging from optical turbulence [10,11]. Thus it is significant and interesting to understand controllable rogue waves theoretically, including the restraint, annihilation, postponement and sustainment of rogue waves, which are investigated in this paper. In nonlinear optics, recent developments [21–23] have led to the discovery of new classes of waves, such as the so-called optical self-similar pulses. Thus, an interesting issue arises: can the self-similar rogue waves described by the rational solutions be managed and controlled? Of course, it is impossible to answer this question in one step. We have investigated controllable optical rogue waves without considering self-similar behaviors in our recent work [24]. In the present work, we make a further step forward in an attempt to understand this issue. The problem is governed by the variable-coefficient NLSE (vcNLSE) as follows [25,26]: iuz +
β(z ) 2
utt + χ (z )|u|2 u + α(z )t 2 u = iγ (z )u,
(1)
where u(z , t ) is the complex envelope of the electrical field, and z and t, respectively, represent the propagation distance and retarded time. β(z ), χ (z ) and γ (z ) represent the group-velocity dispersion (GVD), nonlinearity parameters and loss (or gain), respectively. The parameter α(z ) is the normalized loss rate and α(z )t 2 accounts for the chirping, indicating that the initial chirping parameter is the square of the normalized growth rate. For nonlinear waveguides, t is the spatial variable, β(z ) is the diffraction coefficient, and α(z )t 2 originates from the refractive index n(t , z ) = n0 + n1 α(z )t 2 + n2 χ(z )I (t , z ) with the optical intensity I (t , z ), the linear index n0 and Kerr parameter n2 . If variables z and t are replaced by parameters t and x, Eq. (1) is the generalized Gross–Pitaevskii (GP) equation with harmonic oscillator potentials in BECs [27,28]. When α(z ) = 0, Eq. (1) is the corresponding NLSE in [29], for which the authors investigated not only the soliton control, but also the soliton interaction. Moreover, the multi-soliton solutions in terms of the double Wronskian determinant are presented for Eq. (1) with α(z ) = 0 [30]. When all parameters are constant, Eq. (1) can describe nonlinear spin waves and magnetic solitons in a uniaxial anisotropic ferromagnet [31].
514
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
2. The similarity transformation and rational solutions In order to get the exact analytical solutions for (1), we will construct the transformation [23,32] u(z , t ) = ρ(z )U (Z , T ) exp[iϕ(z , t )],
(2)
where ρ(z ) means the amplitude, and the effective propagation distance Z (z ) and the similarity variable T (z , t ) are both to be determined. Real parameters ρ(z ), Z (z ), T (z , t ) and ϕ(z , t ) are differential functions and should be well chosen to avoid the singularity of the field u(z , t ), the parameter of system α(z ) and the form factor of the solutions W (z ), ρ(z ), ϕ(z ) (see Eqs. (3)–(6)). Note that the similarity transformation (2) with two parameters Z (z ), T (z , t ) is more general than the one with the sole parameter T (z , t ) in [28]; that is, we can discuss dynamical behaviors of multi-solitons while only single solitons could be discussed in [28]. With the substitution of Eq. (2) into Eq. (1), one gets the following expressions for the similarity variable, effective propagation distance, central position, amplitude, width and phase of the pulse: T =
t − tc (z ) W (z )
Z = 0
tc (z ) = −W02 W
z
0
β(s) W 2 (s)
W 2 (s)
ds,
ds
W0 β , χ exp(2Γ ) z Wz 2 W2 W4 β(s) ϕ(z , t ) = t − 0t− 0 ds, 2β W W 2 0 W 2 (s) z with Γ (z ) = 0 γ (ς )dς ; then Eq. (1) is reduced to the standard NLSE
ρ(z ) =
iUZ +
1 2
1
W
β , χ
β(s)
z
,
W (z ) =
UTT + |U |2 U = 0.
(3) (4)
(5)
(6)
(7)
Note that we have required the following integrability condition:
α(z ) =
β Wzz − βz Wz (4γ 2 − 2γz )β 2 + (βzz − 2βz γ )β − βz2 = 2 2β W 2β 3 +
(4χz γ − χzz )β 2 − βz χz β χz2 + . 2β 3 χ χ 2β
(8)
Note that the vanishing chirp (i.e. Wz = 0) leads to α(z ) = 0 in (8), and Eq. (1) becomes the variable-coefficient NLSE in [21,33,34]; thus the chirp is essential to the self-similar solution here. Also note that the existence of the coefficient α(z ) makes the solution more difficult than that of the corresponding equation inthe case of the generalized NLSE in [34]. For α(z ) = 0, the constraint (8) z hints that W = W0 [1 − c0 0 β(s)dz ]. Then this solution can be reduced to the corresponding solution expressed as (5)–(9) in [34]. From Eqs. (3) to (8), we know that the form factors of solutions such as the effective propagation distance, central position, amplitude, width and phase of the pulse are decided by the group-velocity dispersion and nonlinearity parameters of the system; thus this self-similar solution can be controlled under dispersion and nonlinearity management. The one-to-one correspondence (2) allows us to obtain abundant solutions, such as bright and dark single-soliton or multi-soliton solutions, Jacobian elliptic function solutions, and so on. With the help of the mapping transformation (2), the inhomogeneous NLSE (1) can be transformed into the standard NLSE (7). Then like in the procedure in [35], by means of the reverse transformation variables and
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
515
functions, we obtain the exact solutions for Eq. (1). Here we focus on rogue wave solutions. Employing the transformation (2), and the Darboux transformation [36], one can obtain rogue wave solutions (rational solutions) for Eq. (1). The first-order (n = 1) and second-order (n = 2) rational-like solutions read un =
1 W
β Gn + i(Z − Z0 )Hn v2 n × exp i 1 − (−1) + (Z − Z0 ) + v T + ϕ , χ Fn 2
(9)
where 2G1 = H1 = 8, F1 = 1 + 4[T − v(Z − Z0 )]2 + 4(Z − Z0 )2 for a one-rogue wave solution and G2 = [(T − v (Z − Z0 ))2 + (Z − Z0 )2 + 43 ][(T − v (Z − Z0 ))2 + 5 (Z − Z0 )2 + 34 ] − 43 , H2 =
}, F2 = (Z − Z0 ) {(Z − Z0 )2 − 3[T − v (Z − Z0 )]2 + 2[(T − v (Z − Z0 ))2 + (Z − Z0 )2 ]2 − 15 8 2 2 3 2 2 1 1 9 2 T − v Z − Z0 ))2 + [( T − v Z − Z + Z − Z ] + [( T − v Z − Z − 3 Z − Z ] + ( ( ( )) ( ) ( )) ( ) 0 0 0 0 3 4 16 33 3 for a two-rogue wave, T and Z satisfy Eq. (3), ϕ is given by Eq. (6), and Z0 and v are (Z − Z0 )2 + 64 16
two arbitrary constants.
3. Postponement, sustainment and annihilation of self-similar rogue waves Next we analyze how to realize the control for self-similar rogue waves. The crucial point lies in the relation between the effective propagation distance Z and the original propagation distance z. From Eq. (10), this relation can be modulated under dispersion and nonlinearity management. To demonstrate the controllable rogue waves, we take an exponentially dispersion-decreasing fiber (DDF) [37] described by
β(z ) = β0 exp(−σ1 z ),
χ (z ) = χ0 exp(σ2 z ),
γ (z ) = γ0 ,
(10)
where β0 and χ0 are the parameters related to GVD and nonlinearity, σ1 > 0 corresponds to the DDF and γ0 denotes the constant net gain (>0) or loss (<0). On one hand, from Eq. (3) the effective propagation distance Z has a relation to the original χ2
propagation distance z with Z = − W β (σ +02σ +4γ ) {1 − exp[(σ1 + 2σ2 + 4γ0 )z ]}, which implies 0 0 1 2 0 that when σ1 + 2σ2 + 4γ0 > 0, one has Z > z and Z → ∞ with z → ∞; thus rogue waves are excited at z = −ln
χ02
Z0 W02 β0 (σ1 +2σ2 +4γ0 )+χ02
/(σ1 + 2σ2 + 4γ0 ) and then vanish quickly. When χ2
σ1 + 2σ2 + 4γ0 < 0, one has Z < z and Z → Zmax = − W0 β0 (σ2 +02σ1 +4γ0 ) with z → ∞. On the other hand, in the framework of Eq. (7), rogue waves reach their maximum amplitudes when Z = Z0 and then disappear, while in the framework of Eq. (1) the full excitation of rogue waves will be sustained for quite a long distance. Therefore, at first, if Zmax is slightly bigger than Z0 , the excitations of both the one-rogue wave and the two-rogue wave are postponed (cf. Fig. 1(b) and (d)), i.e. the complete rogue waves are not excited. When Zmax adds sequentially and is close to 3Z0 , the full rogue waves are excited fleetingly (cf. Fig. 1(a) and (c)). Here we set σ2 = −σ1 . Note that during the propagation, the values reached by the maximal amplitudes of rogue waves are (a) 6.3, (b) 6.8, (c) 7.0 and (d) 10.0 in Fig. 1. It is interesting that the typical maximal amplitude of the first-order rational solution of NLSE is three times larger than the surrounding waves and the typical maximal amplitude of the second-order rational solution is five times larger than the surrounding waves [35]. Obviously the values reached by the maximum amplitudes of the fast and postponed excited rogue waves are different from those of the standard NLSE [35]. Secondly, if Zmax = Z0 , the full excitations of both the one-rogue wave and the two-rogue wave can be maintained forever with self-similar propagating behaviors (cf. Fig. 2(a) and (c)), where their amplitudes and widths vary self-similarly after a short propagation distance from the initial condition. Finally, if Zmax < Z0 , the thresholds for exciting both the one-rogue wave and the tworogue wave are never reached and their excitations are restrained or even eliminated (cf. Fig. 2(b) and (d)), looking like a bright optical similariton and separated bright similariton pairs [23,32] with very small amplitudes propagating stably along the fiber on a non-zero background.
516
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
Fig. 1. ((a), (b)) Fast and postponed excitation of one-rogue waves. ((c), (d)) Fast and postponed excitation of two-rogue waves. The parameters are v = 0.2, W0 = 0.5, χ0 = 0.12, β0 = 0.1, γ0 = 0.001, Z0 = 8 with ((a) and (c)) σ1 = 0.035 and ((b) and (d)) σ1 = 0.065. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 6.3, (b) 6.8, (c) 7.0 and (d) 10.0.
4. Nonlinear tunneling effects for controllable rogue waves The concept of nonlinear tunneling follows directly from the nonlinear wave equation that leads to the associated nonlinear dispersion relation [38]. Historically, the study of soliton tunneling effects governed by the NLSE with variable coefficients began with the pioneering work of Serkin and Belyaeva [39]. They also investigated the nonlinear tunneling of optical solitons through strong nonlinear organic thin films and polymeric waveguides, which exhibit jumplike nonadiabatic fission reactions, eventually resulting in soliton ‘‘fission reactions’’ [40]. These results hint that the concept of soliton tunneling could inaugurate a new and exciting area in the application of optical solitons. Thus, recently, the tunneling effects of solitons governed by various NLSE were extensively investigated since being predicted as early as 1978, by Newell [38]. For instance, Yang et al. [41] confirmed the compression of the optical pulse by a nonlinear barrier. Wang et al. [42] studied the tunneling effects of spatial similaritons passing through the nonlinear barrier (or well). Dai et al. [43] discussed the tunneling effects of bright and dark similaritons governed by the generalized coupled NLSE for the birefringent fiber. Zhong and Belić [44] investigated the nonlinear tunneling effects of spatial solitons in the nonlinear and diffractive barriers (wells). More recently, enigmas of optical and matter–wave soliton nonlinear tunneling were uncovered in [45]. Moreover, it should be emphasized that the nonlinear soliton tunneling effect is a subject of constantly renewed interest. In femtosecond nonlinear fiber optics, the most intriguing enigma of optical solitons is connected with the so-called soliton spectral tunneling effect (SST). This effect is characterized in the spectral domain by the passage of a femtosecond soliton through a potential barrier-like spectral inhomogeneity of GVD, including a forbidden normal dispersion barrier (or well) [46,47]. Since this phenomenon shares many analogies with that of quantum mechanical tunneling through a potential barrier, it is generally referred to as the SST effect [46,47]. However, to our knowledge, the tunneling effects of self-similar rogue waves have been less investigated. To investigate the unique property of nonlinear tunneling effects for controllable rogue waves, we focus on rogue waves propagating through a nonlinear barrier (NB) or nonlinear well (NW) on an
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
517
Fig. 2. ((a), (b)) Sustainment and annihilation of self-similar one-rogue waves. ((c), (d)) Sustainment and annihilation of selfsimilar two-rogue waves. The parameters are the same as for Fig. 1 with ((a) and (c)) σ1 = 0.076 and ((b) and (d)) σ1 = 0.1. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 9.6, (b) 3.96, (c) 16.8 and (d) 4.4.
exponential background [42,43]:
β(z ) = β0 ,
χ (z ) = χ0 {re−gz + hsech2 [a(z − z0 )]},
γ (z ) = γ0 ,
(11)
where h denotes the height of the NB or NW, respectively. a is related to the NB or NW width, g is a decaying (g > 0) or increasing (g < 0) parameter, and z0 represents the longitudinal coordinate indicating the location of NB or NW. As β(z ) > 0, we assume that h > −1, where h > 0 indicates the NB and −1 < h < 0 represents the NW. We consider the case of function α(z ) being the constant α0 ; √ then Eq. (8) hints that the widths of self-similar rogue waves are given by W (z ) = W0 exp( 2β0 α0 z ). From Eq. (3), the effective propagation distance Z has a relation to the original propagation distance √ z with Z = √
Zmax =
2 β0
4α0 W02
2β0
4α0 W02
√ [1 − exp(− 2β0 α0 z )]; thus the maximum of effective propagation distance
for α0 > 0. Like in discussion above, when Zmax > Z0 , rogue waves are postponed
(cf. Fig. 3). If Zmax = Z0 , rogue waves can be sustained (cf. Fig. 4), while for Zmax < Z0 , rogue waves are restrained or even eliminated (cf. Fig. 6). As shown in Figs. 3(a) and (c), 4(a) and (c) and 5(a) and (c), it can be seen that when the postponed, sustained and annihilated rogue waves pass through the NB at z = z0 = 100, the pulses diminish in amplitude and form the channels; then they increase in amplitude and recover their original shapes. When they propagate through the NW at z = z0 = 100, the pulses are amplified and form peaks, and then attenuate and recover their original shapes, as shown in Figs. 3(b) and (d), 4(b) and (d) and 5(b) and (d), respectively. It was found that in certain circumstances, depending on the ratio of the amplitudes of the rogue waves to the barrier or well height, rogue waves can tunnel through the barrier or well in a
518
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
Fig. 3. Postponed one-rogue and two-rogue waves passing through ((a), (c)) the DB and ((b), (d)) the DW. The parameters are g = 0.058, r = 1, z0 = 100, α0 = 0.05 with (a) a = 0.6, h = 0.3 and (b) a = 2, h = −0.2. The other parameters are the same as for Fig. 1. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 2.9, (b) 3.6, (c) 4.1 and (d) 4.2.
lossless manner. Fig. 6 illustrates the evolution of self-similar rogue waves through the NB for the decaying or increasing parameter g. When g = 0.058 for the postponed rogue wave or g = 0.045 for the sustained rogue wave, the amplitudes of rogue waves are almost unchanged as we see from the comparison of the red line with the green line in Fig. 6. When g > 0.058 for the postponed rogue wave or g > 0.045 for the sustained rogue wave and g < 0.058 for the postponed rogue wave or g < 0.045 for the sustained rogue wave, the amplitudes gradually increase (compare the red line with the black line in Fig. 6) and decrease (compare the red line with the blue line in Fig. 6) along z, respectively. This implies that the parameter g can be used to control the amplitude of the rogue wave after it passes through the barrier. Analysis shows that rogue waves passing across the NW have the same properties. 5. Conclusions In summary, we have constructed the relation between the inhomogeneous NLSE and the standard NLSE via a similarity transformation. On the basis of this transformation, we analytically obtain the one-rogue wave and the two-rogue wave for the inhomogeneous NLSE. For certain parameter conditions, we discuss the propagation behaviors of controllable rogue waves under dispersion and nonlinearity management. In an exponentially dispersion-decreasing fiber, the postponement, annihilation and sustainment of self-similar rogue waves are modulated via the exponential parameter σ . Finally, we investigate nonlinear tunneling effects for self-similar rogue waves. Results show that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged or
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
519
Fig. 4. Sustained one-rogue waves passing through ((a), (c)) the DB and ((b), (d)) the DW. The parameters are the same as for Fig. 3 with g = 0.045 and α0 = 0.056. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 1.48, (b) 1.88, (c) 2.3 and (d) 2.9.
Fig. 5. Annihilated one-rogue waves passing through ((a), (c)) the DB and ((b), (d)) the DW. The parameters are the same as for Fig. 3 with g = 0.045 and α0 = 0.06. During the propagation, the values reached by the maximum amplitudes of the rogue waves are (a) 0.88, (b) 1.48, (c) 1.16 and (d) 1.78.
520
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521
a
b
Fig. 6. The sectional view before passing through the NB at z = 90 (dotted line) and after passing through the NB at z = 110 (solid line). (a) and (b) correspond to Figs. 3(a) and 4(a): the red line denotes the rogue wave before passing through the NB; green, black and blue denote the rogue wave after passing through the NB with g = 0.058, 0.063, 0.053 for the postponed one-rogue wave and g = 0.045, 0.05, 0.04 for the sustained one-rogue wave. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
decreasing amplitudes, through modulation of the ratio of the amplitudes of rogue waves to the barrier or well height. The results obtained in this paper supplement well our comprehension of rogue waves, that is, although they ‘‘appear from nowhere and disappear without a trace’’ [19], rogue waves can be controlled as discussed, in a similar way, in this paper. Moreover, these results may have potential values for the generation and sustainment of exceptionally high amplitude optical pulses—‘‘optical rogue waves’’ and their oceanic rogue wave counterparts. Of course, more practical implementation of these theoretical results might be an interesting task. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 11005092 and No. 11072219), the Program for Innovative Research Teams of Young Teachers (Grant No. 2009RC01) and the Scientific Research and Development Fund (Grant No. 2009FK42) of Zhejiang A&F University. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
W.J. Broad, Rogue Giants at Sea, The New York Times, 2006. P.A.E.M. Janssen, J. Phys. Oceanogr. 33 (2003) 863. N. Mori, P.A.E.M. Janssen, J. Phys. Oceanogr. 36 (2006) 1471. D.H. Peregrine, J. Aust. Math. Soc. Ser. B 25 (1983) 16. A.R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Method, Academic Press, San Diego, 2010. A.I. Dyachenko, V.E. Zakharov, JETP Lett. 81 (2005) 255. C. Kharif, E. Pelinovsky, A. Slyunyaev, Rogue Waves in the Ocean, Springer, Berlin, 2009. N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Rev. A 80 (2009) 043818. A. Chabchoub, N.P. Hoffmann, N. Akhmediev, Phys. Rev. Lett. 106 (2011) 204502. K. Hammani, B. Kibler, C. Finot, A. Picozzi, Phys. Lett. A 374 (2010) 3585. B. Kibler, K. Hammani, C. Michel, C. Finot, A. Picozzi, Phys. Lett. A 375 (2011) 3149. D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450 (2007) 1054. J. Kasparian, P. Béjot, J.-P. Wolf, J.M. Dudley, Opt. Express 17 (2009) 12070. N. Akhmediev, V.I. Korneev, Theor. Math. Phys. 69 (1986) 1089. J.M. Dudley, G. Genty, F. Dias, B. Kibler, N. Akhmediev, Opt. Express 17 (2009) 21497. B. Kibler, J. Fatome, C. Finot, et al., Nat. Phys. 6 (2010) 790. M. Erkintalo, G. Genty, J.M. Dudley, Opt. Lett. 34 (2009) 2468. D.R. Solli, B. Jalali, C. Ropers, Phys. Rev. Lett. 105 (2010) 233902. N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373 (2009) 2137. Y. Xia, W.P. Zhong, M. Belic, Acta Phys. Polon. B 42 (2011) 1881. V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. E 71 (2005) 056619.
C.-Q. Dai et al. / Annals of Physics 327 (2012) 512–521 [22] S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97 (2006) 013901; J.M. Dudley, C. Finot, D.J. Richardson, G. Millot, Nat. Phys. 3 (2007) 597. [23] C.Q. Dai, Y.Y. Wang, J.F. Zhang, Opt. Lett. 35 (2010) 1437; C.Q. Dai, D.S. Wang, L.L. Wang, J.F. Zhang, W.M. Liu, Ann. Phys. (NY) 326 (2011) 2356. [24] Q. Tian, Q. Yang, C.Q. Dai, J.F. Zhang, Opt. Commun. 284 (2011) 2222. [25] L. Li, Z.H. Li, S.Q. Li, G.S. Zhou, Opt. Commun. 234 (2004) 169. [26] R. Atre, P.K. Panigrahi, G.S. Agarwal, Phys. Rev. E 73 (2006) 056611. [27] V. Serkin, A. Hasegawa, T.S. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102. [28] W.P. Zhong, M.R. Belić, Y.Q. Lu, T.W. Huang, Phys. Rev. E 81 (2010) 016605. [29] W.J. Liu, B. Tian, T. Xu, K. Sun, Y. Jiang, Ann. Phys. (NY) 325 (2010) 1633. [30] X. Lü, B. Tian, T. Xu, K.J. Cai, W.J. Liu, Ann. Phys. (NY) 323 (2008) 2554. [31] Z.D. Li, Q.Y. Li, P.B. He, Z.G. Bai, Y.B. Sun, Ann. Phys. (NY) 322 (2007) 2945. [32] C.Q. Dai, X.G. Wang, J.F. Zhang, Ann. Phys. (NY) 326 (2011) 645; C.Q. Dai, R.P. Chen, G.Q. Zhou, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 145401. [33] J.F. Zhang, C.Q. Dai, Q. Yang, J.M. Zhu, Opt. Commun. 252 (2005) 408. [34] S.A. Ponomarenko, G.P. Agrawal, Opt. Express 15 (2007) 2963. [35] N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373 (2009) 675. [36] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601. [37] V.I. Kruglov, A.C. Peacock, J.D. Harvey, Phys. Rev. Lett. 90 (2003) 113902; L.H. Zhao, C.Q. Dai, Eur. Phys. J. D 58 (2010) 327. [38] A.C. Newell, J. Math. Phys. 19 (1978) 1126. [39] V.N. Serkin, T.L. Belyaeva, JETP Lett. 74 (2001) 573. [40] V.N. Serkin, V.M. Chapela, J. Percino, T.L. Belyaeva, Opt. Commun. 192 (2001) 237. [41] G.Y. Yang, R.Y. Hao, L. Li, Z.H. Li, G.S. Zhou, Opt. Commun. 260 (2006) 282. [42] J.F. Wang, L. Li, S.T. Jia, J. Opt. Soc. Am. B 25 (2008) 1254. [43] C.Q. Dai, Y.Y. Wang, J.F. Zhang, Opt. Express 18 (2010) 17548. [44] W.P. Zhong, M.R. Belić, Phys. Rev. E 81 (2010) 056604. [45] T.L. Belyaeva, V.N. Serkin, C. Hernandez-Tenorio, F. Garcia-Santibanez, J. Modern Opt. 57 (2010) 1087. [46] B. Kibler, P.-A. Lacourt, F. Courvoisier, J.M. Dudley, Electron. Lett. 43 (2007) 967. [47] F. Poletti, P. Horak, D.J. Richardson, IEEE Photon. Tech. Lett. 20 (2008) 1414.
521