Open access peer-reviewed chapter

Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium

Written By

Mati Youssoufa, Ousmanou Dafounansou and Alidou Mohamadou

Submitted: 03 February 2020 Reviewed: 13 May 2020 Published: 19 June 2020

DOI: 10.5772/intechopen.92819

From the Edited Volume

Nonlinear Optics - From Solitons to Similaritons

Edited by İlkay Bakırtaş and Nalan Antar

Chapter metrics overview

782 Chapter Downloads

View Full Metrics


In this chapter, evolution of light beams in a cubic-quintic-septic-nonical medium is investigated. As the model equation, an extended form of the well-known nonlinear Schrödinger (NLS) equation is taken into account. By the use of a special ansatz, exact analytical solutions describing bright/dark and kink solitons are constructed. The existence of the wave solutions is discussed in a parameter regime. Moreover, the stability properties of the obtained solutions are investigated, and by employing Stuart and DiPrima’s stability analysis method, an analytical expression for the modulational stability is found.


  • higher-order nonlinear Schrödinger equation
  • spatial solitons
  • stability analysis method
  • modulational instability
  • optical fibers

1. Introduction

The study of spatial solitons in the field of fiber-optical communication has attracted considerable interest in recent years. In a uniform nonlinear fiber, soliton can propagate over relatively long distance without any considerable attenuation. The formation of optical solitons in optical fibers results from an exact balancing between the diffraction and/or group velocity dispersion (GVD) and the self-phase modulation (SPM). The theorical prediction of a train of soliton pulses from a continuous-wave (CW) light in optical fibers was first suggested by Hasegawa and Tappert [1, 2] and first experimentally demonstrated by Mollenauer et al. [3] in single-mode fibers in the case of negative GVD, in liquid CS2 by Barthelemy et al. in 1985 [4]. In nonlinear optic, optical solitons are localized electromagnetic waves that transmit in nonlinear Kerr or non-Kerr media with dispersion or (and) diffraction without any change in shapes. In nonlinear media, the dynamics of spatial optical solitons is governed by the well-known nonlinear Schrödinger (NLS) equation. Depending on the signs of GVD, the NLS equation admits two distinct types of soliton, namely, bright and dark solitons. The bright soliton exists in the regime of anomalous GVD, and the dark soliton arises in the regime of normal GVD. The physics governing the soliton differs depending on whether one considers a bright or a dark soliton and accordingly features distinct applications [5, 6, 7, 8]. The unique property of optical solitons, either bright or dark, is their particle-like behavior in interaction [9].

In addition to fundamental bright and dark solitons, various other forms and shapes of solitary waves can appear in nonlinear media. Kink solitons, for example, are an important class of solitons which may propagate in nonlinear media exhibiting higher-order effects such as third-order dispersion, self-steepening, higher-order nonlinearity, and intrapulse stimulated Raman scattering. In the setting of nonlinear optics, a kink soliton represents a shock front that propagates undistorted inside the dispersive nonlinear medium [10]. This type of solitons has been studied extensively, both analytically and numerically [11, 12, 13]. These spatial soliton solutions can maintain their overall shapes but allow their widths and amplitudes and the pulse center to change according to the management of the system’s parameters, such as the dispersion, nonlinearity, gain, and so on [14].

The cubic nonlinear Schrödinger equation (CNLSE) has been widely used to model the propagation of light pulse in material’s systems involving third-order susceptibility χ3, though, for moderate pulse intensity, the higher-order nonlinearities are related to higher-order nonlinear susceptibilities (nonlinear responses) of a material. For example, the cubic-quintic-nonlinear Schrödinger equation (CQNLSE) models materials with fifth-order susceptibility χ5. This kind of nonlinearity (cubic-quintic CQ) is named as parabolic law nonlinearity and existing in nonlinear media such as the p-toluene sulfonate (PTS) crystals. The parabolic law can closely describe the nonlinear interaction between the high-frequency Langmuir waves and the ion acoustic waves by ponderomotive forces [15, 16], in a region of reduced plasma density, and the nonlinear interaction between Langmuir waves and electrons. In addition, CQ was experimentally proposed as an empirical description of special semiconductor (e.g., AlGaAs, CdS, etc.) waveguides and semiconductor-doped glasses, particularly for the CdSxSe1x-doped glass, which exhibit a significant fifth-order susceptibilities χ5 as experimentally reported earlier [17, 18]. Moreover, using high laser intensity, the saturation of nonlinearity has been established experimentally in many materials such as nonlinear organic polymers, semiconductor-doped glasses, and so on, which have the property that their absorption coefficient decreases [19]. More generally, a self-defocusing χ5 usually accounts for the saturation of χ3.

In recent years, many influential works have devoted to construct exact analytical solutions of CQNLSE, such as the pioneering work of Serkin et al. [20]. In particular, Dai et al. [21, 22, 23, 24, 25] obtained exact self-similar solutions (similaritons), their nonlinear tunneling effects of the generalized CQNLSE, and their higher-dimensional forms with spatially inhomogeneous group velocity dispersion, cubic-quintic nonlinearity, and amplification or attenuation.

Since the measurement of third-, fifth-, and seventh-order nonlinearities of silver nanoplatelet colloids using a femtosecond laser [26], an extension of nonlinear Schrödinger equation including the cubic-quintic-septic nonlinearity was used to model the propagation of spatial solitons. In [27], for example, the authors performed numerical calculations based on higher-order nonlinearity parameters including seventh-order susceptibility χ7 (a chalcogenide glass is an example). This seeds several motivations to discover new features of solitons with combined effects of higher-order nonlinear parameters. In this regard, Houria et al. [28] constructed dark spatial solitary waves in a cubic-quintic-septic-nonlinear medium, with a profile in a functional form given in terms of “sech23”. They have also investigated chirped solitary pulses for a derivative nonical-NLS equation on a CW background [29]. It is obvious to notice that the contributions of the higher-order nonlinearities can give way to generate stable solitons in homogeneous isotropic media and influence many aspects of filamentation in gases and condensed matters [30, 31, 32, 33].

Recently, the study of modulational instability (MI) in non-Kerr media has receiving particular attention. MI is a fundamental and ubiquitous process that appears in most nonlinear systems [6, 9, 34, 35, 36, 37]. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. During this process, small perturbations upon a uniform intensity beam grow exponentially due to the interplay between nonlinearity and dispersion or diffraction. As a result, under specific conditions, a CW light often breaks up into trains of ultrashort solitons like pulses [9]. To date, there has not been any report of MI in the cubic-quintic-septic-nonical-nonlinear Schrödinger equation (CQSNNLSE).

Our study will be focused on the analysis of solitary wave’s solutions of systems described by the higher-order NLSE named CQSNNLSE. We will discuss the model with higher-order nonlinearities and explore the dynamics of bright, dark, and kink soliton solutions. Finally, the linear stability analysis of the MI is formulated, and the analytical expression of the gain of MI is obtained. Moreover, the typical outcomes of the nonlinear development of the MI are reported.


2. Model equation

The dynamics of (1 + 1)-dimensional (one spatial and one temporal variables) spatial optical solitons is the well-known nonlinear Schrödinger equation. If we consider the higher-order effects, an extended model is required, and the propagation of optical pulses through the highly nonlinear waveguides can be described by the CQSNNLSE:


where Ezt is the slowly varying envelope of the electric field, the subscripts z and t are the spatial and temporal partial derivatives in the frame moving with the pulsed solutions, α1 is the parameter of diffraction or dispersion, and α2, α3, α4, and α5 are the cubic, quintic, septic, and nonical nonlinear terms, respectively. This model is relevant to some applications in which higher-order nonlinearities are important.

For example, Eq. (1) with α1=12, α4=1, and α5=0 was used to study numerically the stability conditions of one-dimensional spatial solitons [38]. Recently, Eq. (1) with α5=0 was analyzed for systems that are valid for several types of septic nonlinear materials [28]. Here, we consider arbitrary parameters αjj=12345 for the sake of a general analysis that is valid for several types of nonical media.

To obtain the exact analytic optical solitary-wave solutions of Eq. (1), we can employ the following transformation:


Here, θζ is a real function and β is a real constant to be determined.

Upon substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts, one obtains


Eq. (4) represents the evolution of an anharmonic oscillator with an effective potential energy V [28] defined by


Integrating Eq. (4) yields




and ξ is the constant of integration, which can represent the energy of the anharmonic oscillator [39].

In order to get the exact soliton solutions, we first rewrite Eq. (6) in a simplified form by using transformation:


By substituting Eq. (8) into Eq. (6), we obtain a new auxiliary equation possessing a sixth-degree nonlinear term:


To solve Eq. (9), we will employ three types of localized solutions named bright, dark, and kink solitons. In the following, we solve Eq. (9) by using appropriate ansatz and obtain alternative types of solitary-wave solutions on a CW background and investigate parameter domains in which these optical spatial solitary waves exist.


3. Exact solitary-wave solutions

In this section, we find bright-, dark-, and kink-solitary-wave localized solutions of Eq. (9), by using a special ansatz:

3.1 Bright solitary-wave solutions

The bright solitary solutions of Eq. (9) have the form:


where Ab, Nb, and αb are real constants which represent wave parameters (Ab and αb related to the amplitude and pulse width of the bright wave profiles, respectively) to be determined by the physical coefficients of the model.

Substituting the ansatz Eq. (10) into Eq. (9), we obtain the unknown parameters Ab, Nb, αb, and energy ξ:


with parametric conditions


Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1) are of the form:


3.2 Dark solitary-wave solutions

The dark solitary solutions of Eq. (9) take the form [40]:


Here Nd is a real constant supposed to be positive. Real parameters Ad and αd are related to the amplitude and pulse width of the dark wave profiles, respectively.

By substituting the ansatz Eq. (14) into Eq. (9), we get the unknown parameters Ad, αd, and energy ξ:


subject to the parametric conditions


The exact dark solitary-wave solutions on a CW background of Eq. (1) are of the form:


3.3 Kink solitary-wave solutions

The kink solitary solutions of Eq. (9) are in the following form:


where Ak and αk are real parameters related to the amplitude and pulse width of the kink wave profiles, respectively.

Substituting Eq. (18) into Eq. (9), we get


under the parametric conditions


Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1) are of the form:


The previous three exact solitary-wave solutions (13), (17), and (21) exist for the governing nonical-NLS model due to a balance among diffraction (or dispersion) and competing cubic-quintic-septic-nonical nonlinearities. For better insight, we plot in Figure 1 the intensity profile on top of the related first two exact solution solitons named bright and dark, corresponding to the CQNLS models (with α4=0,α5=0) that is available in the current literature. As we can see from Eq. (21), the kink solitons exist only if a50, consequently α50; thus, we cannot plot the corresponding CQNLS kink solution.

Figure 1.

Intensity |Ej(z, t)|2 distribution of the (a) bright and (b) dark solitons given by Eqs. (14) and (17), respectively, with the parameter values corresponding to CQNLS models as α1=0.5, α2=0, α3=1, α4=0, α5=0, k=1, and ω=1.


4. Modulational instability of the CW background

One of the essential aspects of solitary waves is their stability on propagation, in particular their ability to propagate in a perturbed environment over an appreciable distance [41]. Unlike the conventional pulses of different forms, the solitons are relatively stable, even in an environment subjected to external perturbations.

The previous three exact solitary-wave solutions given by the expressions (13), (17), and (21) are sitting on a CW background, which may be subject to MI. If this phenomenon occurs, then the CW background will be quickly destroyed, which will inevitably cause the destruction of the soliton. It is therefore of paramount importance to verify whether the condition of the existence of the soliton can be compatible with the condition of the stability of the CW background. Since MI properties can be used to understand the different excitation patterns on a CW in nonlinear systems, in this section, we perform the standard linear stability analysis [9, 34] on a generic CW:


in the system modeled by Eq. (1), where ϕnl=P0α2+α3P0+α4P02+α5P03z is the nonlinear phase shift induced by self-phase modulation and non-Kerr quintic-septic-nonical nonlinear terms, P0 being the initial power inside a medium exhibiting optical nonlinearities up to the ninth order. A perturbed nonlinear background plane-wave field for the CQSNNLSE (Eq. (1)) can be written as


where azt is a small perturbation field which is given by collecting the Fourier modes as


a+ and a are much less than the background amplitude P0, and Ω represents the perturbed frequency. Here, the complex field |azt|P0. Thus, if the perturbed field grows exponentially, the steady state (CW) becomes unstable. Inserting the expression of a perturbed nonlinear background Eq. (23) into Eq. (1), with respect to Eq. (24), we obtain after linearization the following dispersion relation:


where Ωc=1α12α2P0+4α3P02+6α4P03+8α5P04 and sgnα1=±1 depending on the sign of α1sgnα1=+1forα1>0andsgnα1=1forα1<0. The dispersion relation (25) shows that the steady-state stability depends critically on whether the light experiences normal or anomalous GVD inside the fiber. In the case of normal GVD (α1<0), the wave number K is real for all Ω, and the steady state is stable against small perturbations. By contrast, in the case of anomalous GVD (α1>0), K becomes imaginary for Ω<Ωc, and the perturbation azt grows exponentially with z. As a result, the CW solution E0 is inherently unstable for α1>0. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. Similar instabilities occur in many other nonlinear systems and are often called self-pulsing instabilities [9, 42, 43, 44, 45]. Then, by setting sgnα1=1, one can obtain the MI gain G=2ImK, where the factor 2 convert G to power gain. The gain exists only if for |Ω|<Ωc and is given by


The gain attains its peak values when the modulated frequency reaches its optimum value, i.e., its optimum modulation frequency (OMF). The OMF corresponding to the gain spectrum (26) is given by


and the peak value given by


In Figure 2, we have shown the variation of OMF, computed from Eq. (27) as a function of the GVD parameter (α1). The parameter values we have used are given as [34]

Figure 2.

Variation of optimum modulation frequency Ωop as a function of second-order dispersion α1.


We can observe that the OMF increases (respectively decreases) with the increasing α1<0 (respectively with the increasing α1>0).

Figure 3 shows the variation of MI gain as a function of the nonic nonlinearity α5. The MI gain increases with the decreasing nonic nonlinearity. In Figure 4, as the input power increases, the maximum gain also increases.

Figure 3.

Variation of the MI gain G as a function of the nonic nonlinearity α5, with the same parameter values as in Figure 2.

Figure 4.

Variation of the MI gain Gkm1 as a function of frequency ΩHz, at a four-power level P0 for an optical fiber. The other parameters are α5=0.5ps2/km,α2=2736W1/km,α3=2.63W2/km,α4=9.12×104W3/km,α5=0.5W4/km.

The MI gain spectrum in Figure 5 is a constitutive of two symmetrical sidebands which stand symmetrically along the line Ω=0. The maximum gain is nil at the zero perturbation frequency Ω=0; thus, there is no instability at the zero perturbation frequency.

Figure 5.

Variation of the MI gain Gkm1 as a function of frequency Ω and the GVD α1. The other parameters are P0=15W,α5=0.5ps2/km,α2=2736W1/km,α3=2.63W2/km,α4=9.12×104W3/km,α5=5W4/km.


5. Conclusion

In this chapter, we have investigated the higher-order nonlinear Schrödinger equation involving nonlinearity up to the ninth order. We have constructed exact solutions of this equation by means of a special ansatz. We showed the existence of a family of solitonic solutions: bright, dark, and kink solitons. The conditions on the physical parameters for the existence of this propagating envelope have also been reported. These conditions show a subtle balance among the diffraction or dispersion, Kerr nonlinearity, and quintic-septic-nonical non-Kerr nonlinearities, which has a profound implication to control the wave dynamics. Moreover, by employing Stuart and DiPrima’s stability analysis method, an analytical expression for the MI gain has been obtained. The outcomes of the instability development depend on the nonlinearity and dispersion (or diffraction) parameters. Results may find straightforward applications in nonlinear optics, particularly in fiber-optical communication.


Conflict of interest

The authors declare no conflict of interest.


  1. 1. Hasegawa A, Tappert FD. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Applied Physics Letters. 1973;23:142
  2. 2. Hasegawa A, Tappert FD. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Applied Physics Letters. 1973;23:171
  3. 3. Mollenauer LF, Stolen RH, Gordon JP. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Physical Review Letters. 1980;45:1095
  4. 4. Barthelemy A, Maneuf S, Froehly C. Soliton propagation and self-confinement of laser-beams by Kerr optical nonlinearity. Optics Communication. 1985;55:201
  5. 5. Hasegawa A, Kodama Y. Solitons in Optical Communications. Oxford: Oxford University Press; 1995
  6. 6. Agrawal GP. Nonlinear Fiber Optics. New York: Academic Press; 2013
  7. 7. Abdullaev F, Darmanyan S, Khabibullaev P. Optical Solitons. Berlin: Springer-Verlag; 1991
  8. 8. Kivshar YS, Luther-Davies B. Dark optical solitons: Physics and applications. Physics Reports. 1998;298:81
  9. 9. Kivshar YS, Agrawal GP. Optical Solitons: “From Fibers to Photonic Crystal”. San Diego: Academic Press; 2003
  10. 10. Agrawal GP, Headley C III. Kink solitons and optical shocks in dispersive nonlinear media. Physical Review A. 1992;46:1573
  11. 11. Raju TS, Panigrahi PK. Self-similar propagation in a graded-index nonlinear-fiber amplifier with an external source. Physical Review A. 2010;81:043820
  12. 12. Goyal A, Gupta R, Kumar CN, Raju TS. Chirped femtosecond solitons and double-kink solitons in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift. Physical Review A. 2011;84:063830
  13. 13. Porubov AV, Andrievsky BR. Kink and solitary waves may propagate together. Physical Review E. 2012;85:046604
  14. 14. Agrawal GP. Optical Solitons, Autosolitons, and Similaritons. NY: Institute of Optics, University of Rochester; 2008
  15. 15. Zhou Q, Liu L, Zhang H, Wei C, Lu J, Yu H, et al. Analytical study of Thirring optical solitons with parabolic law nonlinearity and spatio-temporal dispersion. The European Physical Journal - Plus. 2015;130:138
  16. 16. Jiang Q, Su Y, Nie H, Ma Z, Li Y. New type gray spatial solitons in two-photon photorefractive media with both the linear and quadratic electro-optic effects. Journal of Nonlinear Optical Physics & Materials. 2017;26(1):1750006 (9 pp)
  17. 17. Topkara E, Milovic D, Sarma AK, Zerrad E, Biswas A. Optical solitons with non-Kerr law nonlinearity and inter-modal dispersion with time-dependent coefficients. Communications in Nonlinear Science and Numerical Simulation. 2010;15:2320-2330
  18. 18. Jovanoski Z, Roland DR. Variational analysis of solitary waves in a homogeneous cubic-quintic nonlinear medium. Journal of Modern Optics. 2001;48:1179
  19. 19. Stegeman GI, Stolen RH. Waveguides and fibers for nonlinear optics. Journal of the Optical Society of America B: Optical Physics. 1989;6:652
  20. 20. Serkin VN, Belyaeva TL, Alexandrov IV, Melchor GM. Optical pulse and beam propagation III. In: Band YB, editor. SPIE Proceedings. Vol. 4271. Bellingham: SPIE; 2001. p. 292
  21. 21. Dai CQ, Wang YY, Zhang JF. Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation. Optics Letters. 2010;35:1437
  22. 22. Dai CQ, Zhang JF. Exact spatial similaritons and rogons in 2D graded-index waveguides. Optics Letters. 2010;35:2651
  23. 23. Dai CQ, Zhu SQ, Wang LL, Zhang JF. Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. Europhysics Letters. 2010;92:24005
  24. 24. Dai CQ, Wang XG, Zhang JF. Nonautonomous spatiotemporal localized structures in the inhomogeneous optical fibers: interaction and control. Annals of Physics (NY). 2011;326:645
  25. 25. Dai CQ, Yang Q, He JD, Wang YY. Nonlinear tunneling effect in the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation with variable coefficients. The European Physical Journal. 2011;D63:141
  26. 26. Jayabalan J, Singh A, Chari R, Khan S, Srivastava H, Oak SM. Transient absorption and higher-order nonlinearities in silver nanoplatelets. Applied Physics Letters. 2009;94:181902
  27. 27. Reyna AS, Jorge KC, de Araújo CB. Two-dimensional solitons in a quintic-septimal medium. Physical Review A. 2014;90:063835
  28. 28. Triki H, Porsezian K, Dinda PT, Grelu P. Dark spatial solitary waves in a cubic-quintic-septimal nonlinear medium. Physical Review A. 2017;95:023837
  29. 29. Triki H, Porsezian K, Choudhuri A, Tchofo Dinda P. Chirped solitary pulses for a nonic nonlinear Schrödinger equation on a continuous-wave background. Physical Review A. 2016;93:063810
  30. 30. Boyd RW, Lukishova SG, Shen YR, editors. Self-focusing: “Past and present (fundamentals and prospects)”. In: Topics in Applied Physics. Vol. 114. Berlin: Springer; 2009
  31. 31. Zeng J, Malomed BA. Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity. Physics Review. 2012;E86:036607
  32. 32. Couairon A, Mysyrowicz A. Femtosecond filamentation in transparent media. Physics Reports. 2007;441:47
  33. 33. Liu W, Petit S, Becker A, Akцozbek N, Bowden CM, Chin SL. Intensity clamping of a femtosecond laser pulse in condensed matter. Optics Communication. 2002;202:189
  34. 34. Mohamadou A, Latchio Tiofack CG, Kofané TC. Wave train generation of solitons in systems with higher-order nonlinearities. Physical Review E. 2010;82:016601
  35. 35. Shukla PK, Rasmussen JJ. Modulational instability of short pulses in long optical fibers. Optics Letters. 1986;11:171
  36. 36. Potasek MJ. Modulation instability in an extended nonlinear Schrödinger equation. Optics Letters. 1987;12:921
  37. 37. Porsezian K, Nithyanandan K, Vasantha Jayakantha Raja R, Shukla PK. Modulational instability at the proximity of zero dispersion wavelength in the relaxing saturable nonlinear system. Journal of the Optical Society of America. 2012;B29:2803
  38. 38. Reyna AS, Malomed BA, de Araújo CB. Stability conditions for one-dimensional optical solitons in cubic-quintic septimal media. Physical Review A. 2015;92:033810
  39. 39. Palacios SL, Guinea A, Fernández-Díaz JM, Crespo RD. Dark solitary waves in the nonlinear Schrodinger equation with third order dispersion, self-steepening, and self-frequency shift. Physics Review. 1999;E60:R45
  40. 40. Youssoufa M, Dafounansou O, Mohamadou A. W-shaped, dark and grey solitary waves in the nonlinear Schrödinger equation competing dual power-law nonlinear terms and potentials modulated in time and space. Journal of Modern Optics. 2019;66(5):530-540
  41. 41. Tang XY, Shukla PK. Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrodinger equation with an external potential. Physical Review A. 2007;76:013612
  42. 42. Boyd RW, Raymer MG, Narducci LM, editors. Optical Instabilities. London: Cambridge University Press; 1986
  43. 43. Arecchi FT, Harrison RG, editors. Instabilities and Chaos in Quantum Optics. Berlin: Springer-Verlag; 1987
  44. 44. Weiss CO, Vilaseca R. Dynamics of Lasers. New York: Weinheim; 1991
  45. 45. van Tartwijk GHM, Agrawal GP. Progress in Quantum Electronics. 1998;22:43

Written By

Mati Youssoufa, Ousmanou Dafounansou and Alidou Mohamadou

Submitted: 03 February 2020 Reviewed: 13 May 2020 Published: 19 June 2020