Peregrine Rogue Wave dynamics in the continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity

Samit Kumar Gupta and Amarendra K. Sarma* *[email protected]

Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781 039, Assam, India.

Abstract - In this work, we have studied the peregrine rogue wave dynamics, with a solitons on finite background (SFB) ansatz, in the recently proposed (Phys. Rev. Lett. 110 (2013) 064105) continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity. We have found that the continuous nonlinear Schrödinger system with PT- symmetric nonlinearity also admits Peregrine Soliton solution. Motivated by the fact that Peregrine solitons are regarded as prototypical solutions of rogue waves, we have studied Peregrine rogue wave dynamics in the c-PTNLSE model. Upon numerical computation, we observe the appearance of low-intense Kuznetsov-Ma (KM) soliton trains in the absence of transverse shift (unbroken PT-symmetry) and well-localized high-intense Peregrine Rogue waves in the presence of transverse shift (broken PT-symmetry) in a definite parametric regime.

nonlinearity |휓|2휓 is replaced by its PT-symmetric ∗ 1. Introduction form: 휓(푥, 푧)휓 (−푥, 푧)휓(푥, 푧). The study also reveals that this equation is fully integrable since it possesses linear Lax pairs and The idea of parity-time (PT)-symmetry in the context of a wide an infinite number of conserved quantities. In another work, class of non-Hermitian Hamiltonians counter-intuitively having Christodoulides et al. have studied dynamical behaviors of entirely real eigen-spectra once the system obeys parity-time continuous and discrete Schrödinger systems exhibiting parity- (PT)-symmetry was proposed in the path-breaking work of time (PT) invariant nonlinearity [32]. The study reveals that the Bender and Boettcher [1]. Since then, it has witnessed extensive system yields simultaneous bright and dark soliton solutions and research activities both in theoretical and experimental domains that the shift in the transverse co-ordinate ‘x’ results in the PT- [2-34]. A necessary condition for the Hamiltonian 퐻 = symmetry breaking. This way the wave dynamics of the solitons 1 푑2 undergoes instability. It is worthwhile to mention that the idea of − + 푉(푥), where we have 푉(푥) = 푉 (푥) + 푖 푉 (푥), to be 2 푑푥2 푅 퐼 parity-time symmetry and the transverse shift, in the context of PT-symmetric is that the complex potential 푉(푥) and its real and Rogue waves, have also been explored by a group of researchers imaginary parts 푉푅(푥) and 푉퐼(푥) respectively obey the following [33,34]. Other works in this connection include dark and anti- ∗ relationships: 푉 (−푥) = 푉(푥) 푖. 푒. 푉푅(−푥) = 푉푅(푥) and dark soliton interactions in the nonlocal nonlinear Schrödinger 푉퐼(−푥) = −푉퐼(푥). In this case, the spectrum of the system is equation with self-induced parity-time-symmetric potential [35], real and the system is said to be in unbroken PT-phase regime, periodic and hyperbolic soliton solutions in nonlocal PT- whereas, when the imaginary part of the potential 푉퐼(푥) goes symmetric equations [36]. On a different side, as a limiting case beyond a critical value, the spectrum becomes complex or of a wide class of solutions to the nonlinear Schrödinger imaginary and the system is said to be in broken PT-phase. Of equations, Peregrine solitons (PSs) being localized both in late, it has been found that optics can provide the platform for evolution and transverse variables draws fundamental investigating PT-symmetry related ideas due to the fact that the importance [37]. It is also regarded as the limiting case of the paraxial equation of diffraction in optics is isomorphic to the transverse co-ordinate periodic Akhmediev breather [37-39] or Schrödinger equation in quantum mechanics [2-7] allowing the longitudinal co-ordinate periodic Kuznetsov-Ma (KM) observation of PT-symmetry in optical waveguide structures and breather [38, 40]. In spite of being theoretically predicted in lattices [8]. In the numerous studies and investigations, it has 1983 by H. Peregrine [41], it was not until 2010 by Kibler et al. been found that the idea of PT-symmetric optics can result in [40] that the Peregrine soliton has been experimentally novel optical components and applications [9-19]. On the other demonstrated in nonlinear fiber optics. After that, Peregrine front, nonlinear Schrödinger systems with PT-symmetric linear soliton has been studied in numerous contexts such as: Peregrine 1 2 solitons in a multi-component plasma with negative ions [42], potentials (푖휓푧 + 휓푥푥 + 푉(푥)휓 + |휓| 휓 = 0) have been 2 Peregrine soliton generation and breakup in the standard studied quite rigorously in the recent past [20-30]. In a nonlinear telecommunications fiber [43], interaction of Peregrine solitons Schrödinger system, Kerr nonlinearity can dynamically create an [44], breather-like solitons extracted from the Peregrine rogue effective linear potential which in general may not be PT- wave [45], Peregrine solitons and algebraic soliton pairs in Kerr symmetric. In consequence, if the even symmetry of the real part media [46] and so on. Often, the Peregrine soliton is seen as a of the effective potential gets broken, the system can observe PT- rogue-wave prototype [47] for its close resemblance to the symmetry breaking instability in its wave evolution dynamics. rogue-wave dynamics. It is interesting to note that “Rogue In a recent study, Ablowitz and Musslimani [31] have waves” or “freak waves” which are large amplitude “waves considered an alternative class of highly nonlocal nonlinear appearing from nowhere and disappearing without a trace” Schrödinger equation where the standard third-order (WANDT) find its roots in hydro-dynamical systems [48, 49],

nonlinear fiber optics [42], Bose-Einstein condensates [50] etc.

Optical rogue waves have been studied in singly resonant

parametric oscillators [51], mode-locked lasers [52], in optically injected lasers [53]. In this work, we have considered a Peregrine soliton ansatz to the focusing nonlinear Schrödinger equation with anomalous dispersion with parity-time (PT) symmetric nonlinearity. We intend to study the dynamics of the c-PTNLSE under Peregrine soliton or more generally solitons on finite background (SFB) excitation both in unbroken and broken PT- phases. Since the Peregrine solitons have been seen as a prototype of rogue waves, we study the dynamics of initial PS solution in the c-PTNLSE model and see if it can yield Peregrine rogue (PR) wave in the broken PT-phase. This work may encourage the nonlinear physics community to investigate many other nonlinear systems [54-57] starting with quadratic to power law media with such novel type of nonlinearity. 2. Theoretical model In this work, as stated in the previous section, we are considering the nonlocal nonlinear Schrodinger equation, in normalized units, where the standard third order nonlinearity | (x,z) |2  (x,z) is replaced with its PT symmetric counterpart  (x,z) *(x,z) (x,z) to obtain: 1 ( ) ∗( ) ( ) (1) 푖휓푧 + 2휓푥푥 + 휓 푥, 푧 휓 −푥, 푧 휓 푥, 푧 = 0 Here  (x,z) is the dimensionless field. 푥 and 푧 are normalized distance and time. It is straightforward to show that Eq. (1) possesses the following solitons on finite background (SFB) solutions: (1−4 푎)cosh(푏푧)+ 2푎 cos(훺 푥)+푖 푏 sinh (푏푧) 휓(푥, 푧) = [ √ ] (2) √2푎 cos(훺 푥)−cosh (푏푧) where, Ω is the dimensionless spatial modulation frequency, 푎 = 1/2(1 − 훺2/4) with 0 <푎< 1/2 determines the frequencies experiencing gain and 푏 = √{8푎(1 − 2푎)} is the instability growth parameter. For 푎 → 1/2, the above solution reduces to the rational soliton form i.e. Peregrine soliton: 4(1+2푖푧) 휓(푥, 푧) = [1 − ] 푒푖푧 (3) 1+4푥2+4푧2

3. Numerical simulations and analysis Fig. 1: (Color online) Left hand panel: Evolutions of optical In order to analyze the continuous PTNLSE numerically, in the intensity in the z-x plane. Right hand panel: corresponding 3D context of SFB solutions, we have taken the following initial density plots. (a) 푎=0.20, (b) 푎=0.30, (c) 푎=0.45, (d) 푎=0.47. (1−4 푎)+ 2푎 cos(훺 푥) 휀 = 0.0. excitation from Eq. (2): 휓(푥, 0) = [ √ ]. Eq. √2푎 cos(훺 푥)−1 (1) is solved numerically by using the so-called split-operator method. For a detailed discussion of the method readers are The spatial periodicity of KM solitons keeps on increasing with referred to Ref. [58]. In a recent work [32] it is shown that increase in the modulation parameter ‘푎’ and results in the introduction of the shift, 휀, in the transverse co-ordinate, 푥, may Peregrine solitons as 푎 → 1/2, as could be seen from Fig. 1 (c)- give rise to instability in the wave dynamics. This happens (d). In passing, it is worth noting that the standard NLSE because the self-induced potential 푉 (푥, 푧) starts to have supports Akhmediev breather solutions, which become more imaginary contribution, due to transverse shift, which once and more localized both in time and space co-ordinates with the goes beyond a critical value results in instability of the wave increase of the modulation parameter 'a ' [40]. dynamics. This motivates us to consider various cases with regard to broken and unbroken PT-phases. With a small but nonzero transverse shift. Fig. 2 elucidates the Without any transverse shift. We first consider the case without effect of shift in the transverse co-ordinate ‘x’ on the evolution transverse shift. Fig. 1 exhibits the dynamical evolutions of the of the optical field dynamics. From Fig. 2, we can infer that intensity of the optical field under the SFB excitation. As can with the increase in the transverse shift parameter, the optical be observed from Fig.1 (a)-(b), we find existence of Kuznetsov- field dynamics tend to be more localized in the z-x plane. The Ma (KM) solitons periodic in the evolution variable ‘z’. KM role of the transverse shift is two-fold: first, in triggering the soliton is dependent on the modulation parameter 푎. The spatial PT-instability, and second, ensuring the nonlinear interactions periodicity of KM solitons keeps on increasing with increase in between localized modes. That is why, in Fig. 2 (a) and (b), the modulation parameter ‘푎’. although we see presence of a dominant single second-order kind of PS with a wing-like structure, in Fig. 2 (c), we find

appearance of another second-order type of PS in the CW Fig. 2: (Color online) Left hand panel: Evolutions of optical background. It is worthwhile to note that a strict mathematical intensity in the z-x plane. Right hand panel: corresponding 3D definition of second order PS soliton for the continuous density plots. (a)ε = 2 × 10-6, (b)ε = 3 × 10-6, (c)ε = 4 × Schrodinger equation considered in this work, unlike some 10-6, (d)ε = 4.53 × 10-6. Other parameter value: 푎 = 0.47. other nonlinear systems [34], is not available. Our assumption of second order PS is based on plots resulting from numerical It is worthwhile to mention here that in a recent work in the simulations. More interestingly, in Fig. 2 (d), we finally context of PT-symmetric coupled waveguides, occurrence of observe one PS in the CW background. This is simply because two KM solitons trains hinged upon a second-order Peregrine in Fig. 2 (d), further increase of the transverse shift acts as a soliton creating a wing-like shape has been reported [38]. Our perturbation which gets amplified via modulation instability study on c-PTNLSE model also reveals existence of similar and through nonlinear wave-mixing processes the multiple PS wing-like KM soliton trains hinged upon a second-order profiles interact with each other to transfer energy from one to Peregrine soliton as is evident from Fig. 2 (a)-(b). the other resulting into giant Peregrine rogue wave. One The fact that inclusion of a nonzero shift in the transverse co- crucial point to mention here is that the occurrence of the PRW ordinate results in the instability in the wave dynamics, as depends on very definite parameter values, which is quite suggested in [32], could be clearly seen from Fig.3. This evident as far as the rarity of the rogue wave phenomenon is instability can be attributed to the induction of an imaginary concerned. In the corresponding density plot of Fig. 2(d), we part in the self-induced potential, which goes beyond a critical also observe occurrence of certain radiation states alongside the value. In Fig. 3 (b) and its corresponding density plot shows the Peregrine rogue wave profile. onset of instability in the optical field dynamics in presence of the transverse shift. The intensity of the optical field keeps on increasing with the normalized propagation distance.

Fig. 3: (Color online) Left hand panel: Evolutions of optical intensity in the z-x plane. Right hand panel: corresponding 3D density plots. (a)ε = 0.0, (b)ε = 3 × 10-3. Other parameter value: a = 0.30

Without any transverse shift, in Fig. 4(a) we observe two lines of peaks of KM soliton train along the z-axis, existing at different locations. These two lines of peaks of KM soliton

trains are hinged upon a second order Peregrine soliton. Fig. 4(c) depicts the corresponding power along the propagation distance. Now, quite interestingly, we find that for the same set of parameter values but with a nonzero shift in the transverse co-ordinate, the system supports robust (with respect to the transverse shift parameter in the range 휀 = [4.5,4.605] × 10−6) Peregrine soliton solution with enhanced intensity, strongly localized in the z-x plane, as is clear from Fig. 4 (b) and (d). This enhancement in the intensity of the optical field is owing to the broken PT-symmetry of the self-induced potential. Due to the prototypical analogy between Peregrine soliton and

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