On the integrability aspects of nonparaxial nonlinear Schrödinger equation and the dynamics of solitary waves
OOn the integrability aspects of nonparaxial nonlinear Schr ¨odingerequation and the dynamics of solitary waves
K. Tamilselvan a , T. Kanna a, ∗ , A. Govindarajan b a Nonlinear waves Research Lab, PG & Research Department of Physics, Bishop Heber college (A ffi liated toBharathidasan University), Tiruchirapppalli-620017, Tamil Nadu, India b Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, TamilNadu, India
Abstract
The integrability nature of a nonparaxial nonlinear Schr¨odinger (NNLS) equation, describing thepropagation of ultra-broad nonparaxial beams in a planar optical waveguide, is studied by em-ploying the Painlev´e singularity structure analysis. Our study shows that the NNLS equation failsto satisfy the Painlev´e test. Nevertheless, we construct one bright solitary wave solution for theNNLS equation by using the Hirota’s direct method. Also, we numerically demonstrate the stablepropagation of the obtained bright solitary waves even in the presence of an external perturbationin a form of white noise. We then numerically investigate the coherent interaction dynamics oftwo and three bright solitary waves. Our study reveals interesting energy switching among thecolliding solitary waves due to the nonparaxiality.
Keywords:
Bright solitary waves, Integrability, Painlev´e analysis, Hirota’s bilinearization method,Nonparaxial NLS, Solitary wave interaction
1. Introduction
The advent of the universal nonlinear Schr¨odinger (NLS) equation in nonlinear optics hasopened an avenue to explore various nonlinear waves like solitons, breathers, rogue waves, shockwaves, vortices and so on [1, 2, 3]. This ubiquitous model can be derived from the famousMaxwell’s equations by employing the so-called slowly varying envelope approximation (SVEA)alias paraxial approximation (PA), which is justified only if the scale of spatial and temporal vari-ations is larger than the optical wavelength and optical cycle, respectively. Under this approxima-tion, the second-order derivative of the normalized envelope field, with respect to its longitudinal(propagation) co-ordinate, can be ignored. From a practical point of view, the SVEA holds goodwhen the optical modes are propagating along (or at near-negligible angles with) the reference axis ∗ Corresponding author
Email addresses: [email protected] (K. Tamilselvan), [email protected] (T. Kanna), [email protected] (A. Govindarajan)
Preprint submitted to Phys. Lett. A July 22, 2020 a r X i v : . [ n li n . PS ] J u l ith its pulse / beam width being greater than the carrier wavelength. Though the NLS equationnaturally describes the pulse propagations in optical fibers [4] with such limitations, the pulses en-counter a catastrophic collapse when higher order traverse dimensions are included [5, 6]. It shouldbe noted that the inclusion of nonparaxiality (or spatial group velocity dispersion (S-GVD)) hasled to the stable propagation of localized pulses even in higher dimensional NLS equations [5].In addition to nonlinear optics, this S-GVD or nonparaxial e ff ect appears naturally in the dy-namics of exciton-polaritons in a waveguide of semiconductor material such as ZnCdSe / ZnSesuperlattice [7]. The underlying governing equation is the NLS equation with spatial dispersionterm. This system is formally equivalent to the nonparaxial NLS equation (also referred as nonlin-ear Helmholtz (NLH) equation) which is routinely used to study nonparaxial localized modes inoptical wave guides [8]. In the earlier work of Lax et al., [9], it was attempted to investigate thenonparaxial e ff ect by means of expanding field components as a power series in terms of a ratio ofthe beam diameter to the di ff raction length. Following this work, many studies have been carriedout to investigate the dynamics of nonparaxiality in various optical settings like nonparaxial ac-celerating beams [10], optical and plasmonic sub-wavelength nanostructures devices [11, 12, 13],and in the design of Fresnel type di ff ractive optical elements [14].Furthermore, the propagation of nonparaxial solitons has stimulated extensive studies in di ff er-ent nonlinear optical settings such as Kerr media [8], cubic-quintic media [15], power-law media[16], and saturable nonlinear media [17]. The soliton theory has also been formulated in theNLH equation with distinct nonlinearities based on relativistic and pseudo-relativistic framework[18, 19, 20]. The coupled version of the NLH equation has been studied to explore various kindsof nonlinear waves, including elliptic waves and solitary waves [21, 22]. Recently, the study ofnonparaxiality has been extended to the intriguing area of (cid:80)(cid:84) -symmetric optics [23]. Also, quiterecently, the present authors have done a systematic analysis of the modulational instability for thecubic-quintic NLH equation and reported various interesting chirped elliptical and solitary waveswith nontrivial chirping behavior [24].In nonlinear dynamical systems, the challenging problem is to identify new nonlinear inte-grable / nearly integrable models. This has an important consequence for exploring nontrivial lo-calized nonlinear waves with intriguing dynamical features in di ff erent physical media [2, 25, 26].Moreover, investigations of the integrability nature of dynamical systems have been extended tomultiple areas of physics, including fluid dynamics, nonlinear optics, Bose-Einstein condensates,bio-physics and so on. Specifically, one can verify the integrability nature of a nonlinear dynam-ical equation by using a powerful mathematical tool, namely, Painlev´e analysis [27, 28]. ThePainlev´e analysis is a potential tool among many integrability indicators such as the linear eigen-value problem, bilinear transformation, B¨acklund transformation, Lax-pair method, and inversescattering method [2, 29]. Through the Painlev´e analysis, the integrability nature has been testedfor various nonlinear models [30, 31, 32]. As mentioned earlier, the NNLS equation can serve asa fertile platform for studying dynamics of a wide range of physical systems. In a recent work,the symmetry reductions of the NNLS equation have been obtained by the Lie symmetry analysis[33]. However, the integrability nature of this NNLS equation is yet to be investigated. One of theobjectives of this paper is to inspect the integrability nature of the following dimensionless NNLS2quation. i ∂ Ψ ∂ z + κ ∂ Ψ ∂ z + ∂ Ψ ∂ t + γ | Ψ | Ψ = , (1)where Ψ is the normalized complex envelope field and normalized space z and retarded time t are expressed as Z / L D and T / T , in which the dispersion length L D is determined by T / | β | .The parameters β and T o account for group velocity dispersion (GVD) and input pulse width,respectively. The term κ refers to nonparaxial parameter which ranges from 10 − to 10 − with κ = / k L D (where k = π n /λ stands for wavenumber, in which n is refractive index) [18].Also, the term γ indicates the Kerr nonlinearity co-e ffi cient. In the limit, κ → ffi cient number of arbitrary functions without the movable critical singularity manifolds. In Sec.3, we obtain the bright solitary wave solution by employing the Hirota’s bilinearization method.Following that, the stability of the bright solitary wave solution in the presence of external pertur-bation is examined by numerical simulation in Sec. 4. In addition, the interaction of nonparaxialsolitary waves is analyzed by executing split step Fourier (SSF) method. Finally, we conclude ourfindings in Sec. 5.
2. Painlev´e Singularity Structure Analysis
In order to apply the Painlev´e singularity structure analysis to equation (1), we consider thedependent variable and its complex conjugate as Ψ = r , Ψ ∗ = s . Then the equation (1) and itscomplex conjugate equation can be rewritten as, ir z + κ r zz + r tt + γ ( r s ) = , (2a) − is z + κ s zz + s tt + γ ( s r ) = . (2b)The singularity structure analysis of the above equations (2a)-(2b) is carried out by seeking thefollowing generalized Laurent series expansion for the dependent variables in the neighbourhoodof the non-characteristic singular manifold φ ( z , t ) = φ z ( z , t ) (cid:44) φ t ( z , t ) (cid:44) r = φ α ∞ (cid:88) j = r j ( z , t ) φ j , r (cid:44) , (3a) s = φ β ∞ (cid:88) j = s j ( z , t ) φ j , s (cid:44) , (3b)where α and β are integers yet to be determined. Next, in order to analyze the leading ordersolution, we restrict the above series as, r = r φ α and s = s φ β . By using these relations in3quation (2) and balancing the most dominant terms, the unknown values α and β are determinedas, α = − , and β = − , accompanied by the following condition2 κφ z + φ t = − γ ( r s ) . (4)In equation (4), out of two functions r and s , one is arbitrary.Next, the resonances (powers at which arbitrary functions can enter into the Laurent series (3))are obtained by determing the values of j upon substitution of the following equations r = r φ − + · · · + r j φ j − , (5a) s = s φ − + · · · + s j φ j − (5b)into equations (2). By collecting the coe ffi cients of φ j − , we get (cid:32) ( j − j ) δ + γ ( r s ) γ r γ s ( j − j ) δ + γ ( r s ) (cid:33) (cid:32) r j s j (cid:33) = , (6)where δ = κφ z + φ t . By setting the above matrix determinant to be zero, we obtain a quarticequation for j as follows j − j + j + j = . (7)The roots of equation (7) are the resonance values and are found to be j = − , , ,
4, where theresonance value j = − φ ( z , t ). Except this, allother resonance values are positive as required by the Painlev´e test. The third step is to examine the existence of su ffi cient number of arbitrary functions at theseresonance values without introducing movable critical singular manifolds of the singularity struc-ture analysis. To this end, we expand the dependent variables as follows: r = r φ + r + r φ + r φ + r φ , (8a) s = s φ + s + s φ + s φ + s φ . (8b)Then, substituting the above equations (8) in equations (2) and collecting the co-e ffi cients at vari-ous orders of φ , one can study the arbitrariness of the singularity.First, collecting the terms at the order φ − which corresponds to the resonance value j =
0, weobtain 2 κφ z + φ t = − γ ( r s ) . (9)This equation is exactly the same as the leading order equation (4).4econd, collecting the coe ffi cients at the order φ − , we obtain the following equations whichare expressed in matrix form as (cid:32) − + κρ z ) γ r γ s − + κρ z ) (cid:33) (cid:32) r s (cid:33) = − (cid:32) ir ρ z + κ r ρ zz + κ r , z ρ z − is ρ z + κ s ρ zz + κ s , z ρ z (cid:33) . (10)Here, we have used the Kruskal ansatz of the form φ ( z , t ) = t + ρ ( z ), with ρ ( z ) being an arbitraryanalytic function to simplify the calculations and the r j and s j are functions of z only. We obtainthe following expressions for r and s from the above equation (10). r = + κρ z ) [( ir ρ z + κ r ρ zz + κ r , z ρ z )(2(1 + κρ z )) + γ r ( − is ρ z + κ s ρ zz + κ s , z ρ z )] , (11a) s = + κρ z ) [(2(1 + κρ z ))( − is ρ z + κ s ρ zz + κ s , z ρ z ) + γ s ( ir ρ z + κ r ρ zz + κ r , z ρ z )] . (11b)Thus, there is no arbitrary function at this order.Similarly, collecting the coe ffi cients at the order φ − , we obtain (cid:32) − + κρ z ) γ r γ s − + κρ z ) (cid:33) (cid:32) r s (cid:33) = − (cid:32) ir , z + κ r , zz + γ ( r s + r r s ) − is , z + κ s , zz + γ ( s r + s r s ) (cid:33) . (12)By solving the above set of algebraic equations, we find that r and s can be expressed interms of r and s as below r = + κρ z ) [2( ir , z + κ r , zz + γ ( r s + r r s ))(1 + κρ z ) + γ r ( − is , z + κ s , zz + γ ( s r + s r s ))] , (13a) s = + κρ z ) [2( − is , z + κ s , zz + γ ( s r + s r s ))(1 + κρ z ) + γ s ( ir , z + κ r , zz + γ ( r s + r r s ))] , (13b)where the expressions for r and s are as given in equations (11). The above equations (13)indicate that r and s are not arbitrary functions. Further, collecting the coe ffi cients at the order φ corresponding to the resonance value j =
3, we obtain, s r + r s = − γ r [ ir , z + κ r , zz + ir ρ z + κ r ρ zz + κ r , z ρ z + γ r ( r s + r s + r s ) + γ r r s ] , (14a) s r + r s = − γ s [ − is , z + κ s , zz − is ρ z + κ s ρ zz + κ s , z ρ z + γ s ( r s + r s + r s ) + γ s s r ] . (14b)5y carefully analyzing the right hand sides of expressions (14) by symbolic computation, wenote that they become non-identical except for the choice κ = , which corresponds to the resultof the standard integrable NLS equation. This clearly indicates the violation of arbitrariness forthe resonance j =
3, as there is no any arbitrary function. Hence the NNLS equation (1) fails tosatisfy the Painlev´e property at this stage.Finally, we move on to collect the coe ffi cients at the order φ and one obtains (cid:32) (1 + κρ z ) γ r (1 + κρ z ) γ r (cid:33) (cid:32) r s (cid:33) = A , (15)where the column matrix A is given by A = − i ( r , z + r ρ z ) + κ ( r , zz + r ρ zz + r , z ρ z ) + γ ( r s + r r s + r s r + r r s + r r s + r r s + r s ) − r (1 + κρ z ) ( i ( s , z + s ρ z ) + κ ( s , zz + s ρ zz + s , z ρ z ) + γ ( s r + s s r + s r s + s s r + s s r + s s r + s r )) As before, here also a rigorous analytical calculation involving symbolic computation showsthat the above two equations remain distinct as long as κ is non-zero. However, they becomeidentical for κ =
0, as expected. Thus, due to the failure of existence of su ffi cient number ofarbitrary functions (see equations (14) to (16)), we conclude that the NNLS equation (1) is notfree from movable critical singular manifolds. The above singularity structure analysis clearlyindicates that the NNLS equation (1) is not integrable in the Painlev´e sense.
3. Solitary wave solutions for the NNLS equation
As established in the previous section, the NNLS equation (1) fails to satisfy the Painlev´e testfor integrability. Hence, one has to consider quasi-analytical methods or numerical analysis tofind special solutions in the equation (1) [34, 35, 36]. However, we here attempt to find specialsolutions in equation (1) by using the well-known Hirota’s bilinearization method, in spite of theequation (1) being non-integrable. The NNLS equation (1) is expressed in a bilinear form byemploying the following transformation Ψ = g ( z , t ) f ( z , t ) , (16)where g and f are complex and real functions, respectively, and ∗ indicates complex conjugation.The resulting bilinear equations are( iD z + κ D z + D t )( g · f ) = , (17a)( κ D z + D t )( f · f ) = γ ( gg ∗ ) . (17b)6ne can obtain the single solitary wave solution by expression g = χ g , and f = + χ f in equa-tion (17), where χ is a formal expression parameter. Solving the resulting linear partial di ff erentialequations (17) at various orders of χ recursively, we obtain Ψ = ∆ e i η i sech (cid:18) η r + R (cid:19) . (18)Here η r = a r t + b r z , η i = a i t + b i z + θ, a i = (cid:114) − b i + κ ( b r − b i ) ± (cid:113) ( b r + b i )[1 + κ b i + κ ( b r + b i )] , a r = − b r (2 κ b i + a i , θ = tan − (cid:32) α i α r (cid:33) , R = (cid:115) γ ( αα ∗ )(8 κ b r + a r ) , (19a)where a r , a i , b r , b i , α r , and α i are real parameters. By direct substitution, we have also verifiedthat the solution (18) indeed satisfies the NNLS equation (1). This one bright solitary wave (18)is characterized by four arbitrary real parameters b r , b i , α r and α i . The amplitude and velocity ofone bright solitary wave (18) can be expressed as ∆ = α e R = b r (cid:113)(cid:104) κ a i + (1 + κ b i ) (cid:105) √ γ a i , and v = a i ( − κ b i − , (19b)respectively. Also, the phase part of the solitary wave is given by a i ( t + b i a i z ). Here, the amplitude,velocity and phase of the bright solitary wave are a ff ected significantly by the nonparaxial e ff ectdue to the explicit appearance of the nonparaxial parameter κ in their corresponding expressions.Note that, the solution (18) reduces to the standard NLS soliton in the paraxial limit (i.e., when κ → ffi cient number of parameters does not exist. This conclusion is in support of thePainlev´e analysis carried out in the previous section (2.1) showing the NLS system to be non-integrable.First, we show the propagation of the one bright solitary wave as in Fig. 1 which mimics thetypical soliton propagation in integrable systems. Then, in order to reveal the impact of nonparax-iality on the bright solitary wave, we display the intensity plots of the bright solitary wave fordi ff erent values of the nonparaxial parameter κ in Fig. 2. In the absence of the nonparaxial pa-rameter (i.e. κ = κ , the bright solitary wave undergoessignificant changes, not only in its amplitude and width but also in its central position. These aresignatures of the nonparaxiality [19, 20]. The influence of the nonparaxial parameter on physicalquantities such as amplitude and speed of the bright solitary wave is presented in Fig. 3. It clearlyshows that the increase in the nonparaxial parameter enhances the speed of the bright solitary7 igure 1: Propagation of the bright one solitary wave for the parametric choice b r = b i = α = κ = .
01, and γ = ff erent values of κ parameter. The parameters are assignedas b r = α = b i = . γ =
2, and z = wave. For the κ values lying in the window [-1,1], the amplitude decreases until κ becomes zeroand then it starts to increase.We have also investigated the stable dynamics of obtained bright solitary wave of the NNLSequation by employing the split-step Fourier method based on Feit-Flock algorithm [5]. To doso, we add a random uniform white noise as a perturbation at a rate of 10% in the initial solutionof bright solitary wave solution (18) [37]. Figure 4 demonstrates that the solitary pulse remainsstable for a long propagation distance which is quite larger for optical waveguides, without (seeFig. 4(a)) and with noise (see Fig. 4(b)). Hence the numerical evolution clearly demonstrates thatthe pulse is robust against small perturbations in the form of uniform white noise for the givensystem parameters.
4. Scattering dynamics of bright solitary waves in the NNLS system
Interaction of solitary waves is a key feature that determines their potential applications innonlinear optical systems. It is interesting to study the interaction between two solitons by launch-8 igure 3: Plot depicts the speed and amplitude of the bright solitary wave as a function of the nonparaxial parameter κ . The parameters are same as given in Fig. 2 (a) (b) Figure 4: Numerical evolution of stable bright solitary waves, in the absence of perturbation (a) and in the presenceof white noise 10% (b). The parameters are the same as in Fig. 1 ing the soliton pulses far enough from each other, at least with a separation distance around tentimes of their pulse-width [38, 39]. The implication of such criteria has really helped to overcomemultiple issues like pulse distortion, deteriorations of the data transmission and synchronization inthe high-bit-rate systems. In general, interaction of solitons can be classified into two main cate-gories as coherent and incoherent based on their relative phases [40, 41]. In practice, the coherenttype of interactions takes place when the interference e ff ects between the overlapping beams aretaken into account. It requires the medium to respond instantaneously. On the contrary, incoherentinteractions exist when the time response varies slower than the relative phase between solitons.Ultimately, solitons experience periodic collapse with neighboring solitons. It must be hence notedthat the incoherent interactions are undesirable in the practical viewpoint [42].Interaction of various types of solitons has been intensively discussed both from experimen-tal and theoretical aspects [1]. In particular, these studies considered interaction between soli-tons / solitary waves in the NLS and NLS-like equations [43, 44, 45, 46]. The multicomponentversions of these scalar NLS type equations support bright solitons with interesting shape chang-ing (energy sharing / switching) collisions [47, 48, 49, 50, 51]. These interesting energy sharing9a) φ = (b) φ = π/ (c) φ = π Figure 5: Interaction of dynamics of two bright solitary waves with parameters as κ = . γ = ∆ t =
1, and b r = b i = α = collisions find applications in the context of realizing gates based on soliton collisions [52, 53,54, 55, 56]. However, to date, the intriguing process of soliton interactions remains unexplored inthe context of nonparaxial regime except a work that showed a glimpse of the former [57]. Weare hence interested to study the interactions of bright solitons numerically. The split-step Fouriermethod based on Feit-Flock algorithm is adopted here to investigate the interaction between twotemporally separated bright solitons in the NNLS equation. To study the scattering dynamics ofbright solitons in the NNLS equation, we assume the following two temporal bright solitary pulseswith equal amplitudes ∆ ( ≈
1, in the normalized sense) Ψ (0 , t ) = Ψ (0 , t + ∆ t ) exp( i φ ) + Ψ (0 , t − ∆ t ) , (20)where Ψ (0 , t + ∆ t ) denotes the bright solitary wave solution given by equation (18) with ampli-tude ∆ ≈
1, and φ indicates an initial phase di ff erence between the two temporally solitary pulsesinitially separated by a distance ∆ t . For the simulations performed here, we choose the boundaryconditions to minimize the undesired e ff ects such as reflection of radiation at the boundaries ofthe computational window. In what follows, we present the coherent interactions of nonparaxialbright solitons with di ff erent parametric choices of obtained solutions and qualitatively discuss thephysics behind the interaction dynamics in detail.To start with, we consider the collision for the parametric choice κ = . ∆ t =
2, and varyphase from φ = π as presented in Fig. 5. For φ =
0, it exhibits an in-phase interaction dynam-ics and forms oscillating bound solitary waves as shown in Fig. 5(a). Note that, these localizedstructures maintain their velocity throughout the propagation and retain their shape throughout themedium. The scenario is changed for the choice of phase φ = π/ φ = π , the interacting pulses become unstableand dispersion radiations are created [see Fig. 5(c)]. Thus, when the pulses are separated by shortdistance, stable solitary waves are formed when their phases are correlated [37].10a) φ = (b) φ = π/ (c) φ = π Figure 6: The density plots of interaction of two bright solitary waves. The parameters are the same as in Fig. 5 except ∆ t = . Next, we increase the separation distance between the solitary waves and investigate the col-lision dynamics for in-phase, out-of-phase, and φ = π/ φ =
0. For φ = π/
2, there is a non-trivial energy switch-ing among the interacting solitary waves during the collision accompanied by bending (drifting)of the interacting solitary waves resulting an increase in their relative separation distance. Forthe choice, φ = π , also similar behavior takes place [see Figs, 6 (a)-(c)]. For larger separationdistance there is no passing through collision and we observe only parallel propagation of boundsolitons as noticed in Fig. 7(a). It is to be noted that in contrast to the standard NLS and Man-akov systems where the solitons exhibit conservation of energy during their collisions, the solitarywaves of the present system (1) do not preserve the total energy, rather it conserves the quantity c = (cid:82) + ∞−∞ (cid:16) | Ψ | − i κ (cid:16) Ψ ∗ ∂ Ψ ∂ z − Ψ ∗ ∂ Ψ ∗ ∂ z (cid:17)(cid:17) dt . This constant of motion c can be easily obtained byconsidering the NNLS equation and its complex conjugate equation and by using the asymptoticbehavior of bright solitary waves (cid:18) Ψ −−−−→ t →±∞ (cid:19) , with a simple algebra. Here the nonparaxial param-eter ( κ ) is responsible for the violation of the conservation of energy, which is also corroboratedthrough numerical simulations. In particular, as shown in Fig. 5(c), some amount of energy of thesolitary wave gets radiated which further proves that the norm of the solitary wave denoting thetotal intensity is non-conserved.Finally, it is interesting to reveal the scattering nature of three bright solitary waves by alteringthe phase of solitary waves from in-phase (zero) to out-of-phase ( π ) as shown in Fig. 8. In the caseof in-phase, three solitary waves are attracted to each other at z ∼ z denotes the propagationdistance of the medium), afterwards they get separated symmetrically to each other and also retaintheir shape after collision (see Fig. 8(a)). We also observe a non-trivial energy switching in thesecond (middle) solitary wave. By tuning the phase to π/
2, we identify an interesting interactiondynamics, where one solitary pulse (left side) is completely separated from other solitary wavesand is deviated away from the remaining solitary waves. The rest of the two solitary pulses ini-tially propagate within a very short separation distance and after the collision due to repulsionbetween the solitary pulses the separation distance between them is increased. In particular, twosolitary pulses have distinct intensity profiles, featuring an energy transfer from one solitary pulse11a) φ = Figure 7: The evolution of interaction of two bright solitary waves. The parameters are the same as in Fig. 5 except ∆ t = (a) φ = (b) φ = π/ (c) φ = π Figure 8: The density plots of interaction of three bright solitary pulses. The parameters are the same as in Fig. 5except ∆ t = to another one as presented in Fig. 8(b). A similar collision behavior with a significant energyswitching in the right most solitary wave (before collision) can be observed for the case φ = π with a slight modification as given in Fig. 8(c). Note that after collision, there is a correspondingdecrease in the intensities of other two solitary pulses.
5. Conclusion
To conclude, we have investigated the integrability aspects of the NNLS equation by employingthe Painlev´e singularity structure analysis. Based on this analysis, we have proven that the NNLSequation fails to satisfy the Painlev´e test as it is not free from the movable singularity at theresonance j =
3. Nevertheless, we have then constructed bright solitary wave for the NNLSequation by using the Hirota’s bilinearization method. We have demonstrated stable propagationover long distance even in the presence of external perturbation, which is seeded in the form of awhite noise, by employing the SSFM. The scattering dynamics of bright solitary waves has alsobeen investigated by numerical simulation for di ff erent values of separation distance and relativephase. This numerical study reveals that there is an energy / intensity switching among the collidingsolitons after collision, due to the nonparaxiality / spatial dispersion. Also, the collision leads to astronger repulsion between the solitons which results in an increase in the separation distancebetween the solitons after interaction. Ultimately, there is a significant deviation in the trajectory12f solitary wave. We anticipate that the results will shed light in the formation, propagation andcollision of solitary waves in nonparaxial nonlinear media. The energy switching phenomenonduring collision in the NNLS system can find applications in optical switching devices, beamsteering and in soliton collision based optical computing. Acknowledgement
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