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Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

By Mati Youssoufa, Ousmanou Dafounansou, Camus Gaston Latchio Tiofack and Alidou Mohamadou

Submitted: August 30th 2021Reviewed: September 10th 2021Published: October 18th 2021

DOI: 10.5772/intechopen.100396

Downloaded: 47

Abstract

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

Keywords

  • nano-optical fibers
  • perturbed nonlinear Schrödinger equation
  • auxiliary equation method
  • exact traveling wave solutions
  • modulational instability
  • planar dynamic system
  • chaotic motions

1. Introduction

The wave process is innumerable in nature. Such familiar examples include water waves, plasma waves, and optical waves and are governed by nonlinear partial differential equations. The study of nonlinear evolution equations helps a lot in understanding certain interesting physical properties posed by themselves, in several physical systems. Recently, an important amount of studies has been related to nonlinear systems having multidegrees of freedom: The well-known nonlinear Schrödinger (NLS) equation is a particular example. The idea behind the NLS equation was originated from the work of Erwin Schrödinger, an Autrichian physician, in 1926 [1]. This equation governs weakly nonlinear and dispersive wave packets in one-dimensional (1D) physical systems. It was first derived, in a general setting, by Benney and Newell in 1967 [2]. Also, it was derived in the study of modulational stability of deep-water waves by Zakharov in 1968 [3]. Afterward, Hasegawa and Tappert (1973) showed that the same equation governs light-pulse propagation in optical fibers [4]. For instance, the cubic-NLS has been widely used to model the propagation of light pulse in material’s systems involving third-order susceptibility χ3[5, 6, 7]. In the same context, the nonlinear interaction between the high-frequency Langmuir waves and the ion-acoustic waves by ponderomotive forces [8, 9] in a region of reduced plasma density, and the nonlinear interaction between Langmuir waves and electrons, were described by the “parabolic law nonlinearity” (cubic-quintic CQ) that existing in nonlinear media such as the CdSxSe1xdoped glass [10, 11], the poly-toluene sulfonate (PTS) crystals, special semiconductor waveguides (e.g., AlGaAs, CdS) [12]. Furthermore, Serkin et al. [13], Dai et al. [14], and others have devoted pioneering works in order to analyze the dynamical propagation of light pulse in CQ-nonlinear media, by considering the CQ-NLS equation.

Generally, the NLS-type models are important classes of nonlinear evolution equations that play a crucial role in the study of nonlinear dynamical problems in several areas of nonlinear sciences such as nonlinear optics, plasmas and Bose-Einstein condensates, and nano-optical fibers among others [15, 16, 17, 18]. Although these equations explain the pulse dynamics in optical fibers [19, 20, 21, 22], some of these nonlinear models are non-integrable. In this context, various computational and analytical methods have been proposed and used in the past few decades, to examine many classes of Schrödinger equation [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Nonetheless, these investigations reveal that the dynamic of solutions in non-integrable systems can be important and more complex.

Our study will be focused on a nano-optical fiber-system, described by the following extended perturbed NLS equation (integrable equation named as Biswas-Arshed model), involving power-law nonlinearity and higher-order dispersions [20, 21, 22]:

z+a1ψxx+a2ψxt+b1ψ2nψ+b2ψ4nψiαψx+γψ2nψx+σψ2nxψ+δψ4nψx+λψ4nxψ+θψ2nψx=0,E1

where the complex-valued function ψzxis designated for waveform, which depends on the temporal variable zand the spatial variable x; a1and a2are, respectively, the group velocity dispersion (GVD) and spatiotemporal dispersion coefficients (STD). b1and b2correspond to the coefficient of power-law nonlinearity; αaccounts for the inter-modal dispersion. γand δaccount for the self-steepening perturbation terms, while σ, λ, and θprovide the effect of nonlinear dispersion coefficient. Finally, ndenotes the strength of the power-law nonlinearity.

This model is relevant to some applications in which higher-order nonlinearities are important and describe the dynamics of solitary-wave propagation through optical fibers and other forms of waveguides, and contains, under different circumstances, several integrable NLS-types such as the Hirota equation [43], the Sasa Satsuma model [44], Gerdjikov-Ivanov equation, Lakshmanan-Porsezian-Daniel model, Schrödinger-Hirota equation, and a variety of other such models. More specially, Eq. (1) with b2=σ=δ=λ=θ=0and 0<n<2was used to study chaotic motions for the perturbed NLS equation with the power-law nonlinearity based on the equilibrium points by Yin et al. [20] and was also considered by Savescu et al. [45] to analyze nonlinear dynamical problems in the nano-optical fibers. Here, we study the model Eq. (1) with arbitrary parameters that are valid for several types of highly nonlinear mediums and give rise to some new results. For this purpose, we would like to obtain the exact solutions of Eq. (1) by using the auxiliary equation method [46, 47, 48] and the bifurcation theory of planar dynamical systems [49, 50].

The auxiliary equation method is a powerful solution method for the computation of exact traveling wave and soliton solutions. It is a one of the most direct and effective algebraic methods for finding exact solutions of nonlinear partial differential equations. This method is applicable to a large class of equations and does not need therefore to make strong assumptions about the nonlinear equations, as compared to the well-known inverse scattering transform, which uses powerful analytical methods and therefore makes strong assumptions.

The bifurcation theory of planar dynamical systems plays a crucial role in the study of the evolution of higher-order nonlinear equations. The bifurcation analysis can be used to obtain chaotic motions for Eq. (1) based on the equilibrium points.

We will discuss model Eq. (1) and explore the dynamics of traveling wave solutions by employing the auxiliary equation method. In addition, using the linear stability analysis formulation, we will analyze and report the typical outcomes of the nonlinear development of the modulational instability (MI). Finally, we will point out the development of the chaotic motions for systems described by Eq. (1) through the bifurcation theory.

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2. Exact solutions

2.1 The auxiliary equation method

In order to obtain the exact analytic traveling wave solutions of Eq. (1), we can employ the auxiliary equation method by considering the following transformation:

ψzx=Uζeζ,ζ=k1xk2z.E2

Here, k1and k2are real constants, Uζdenotes the amplitude and ϕζcharacterizes the phase component of the soliton.

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

b1U2n+1+b2+k1ϕζU4n+1+k1γ+θϕζU2n+1+k2+αk1ϕζUk1a1k1a2k2ϕζ2U+Uζζ=0,E3

and

k1δ+4nδ+λUζU4nk2+αk1Uζk1γ+θ+2nγ+σUζU2n+k1a1k1a2k22ϕζUζ+ϕζζU=0.E4

We set:

ϕζ=p1p2U2n,ϕζζ=2np2UζU2n1.E5

The substitution of Eq. (5) into Eq. (4) gives

p1=k2+αk12k1a1k1a2k2,p2=γ+θ+2nγ+σ2n+1a1k1a2k2,E6

under the restraint relation

λ=4n+1δ.E7

Plugging Eq. (5) into Eq. (3) with respect to Eqs. (6) and (7), we get

k1a1k1a2k2Uζζ+p1k2+αk1p12k1a1k1a2k2U+b1+k1p1γ+θ+p2k2+αk12p1p2k1a1k1a2k2U2n+1.+b2+k1p1+k1p2γ+θp22k1a1k1a2k2U4n+1+k1p2U6n+1=0.E8

The substitution of

Uζ=V12nζ,E9

in Eq. (8) yields:

k1a1k1a2k22nVVζζ+12nVζ2+4n2k1p2V5
+4n2b2+k1p1+k1p2γ+θp22k1a1k1a2k2V4
+4n2b1+k1p1γ+θ+p2k2+αk12p1p2k1a1k1a2k2V3
+4n2p1k2+αk1p12k1a1k1a2k2V2=0.E10

We consider the trial equation as [46, 47]:

Vζ2=FV=l=0NμlVl,E11

where μll=01Nare constants to be determined according to the balance principle. The previous Eq. (11) can be rewritten by the integral form

±ζζ0=dVFV.E12

Balancing VVζζand V5in Eq. (10), we get N=5. Using the solution procedure of the trial equation method [46, 47], a system of algebraic equations is obtained (see Appendix) and the resolution of this obtained system yields the following:

μ0=0,μ1=0,μ2=4n2p12p1k2+αk1k1a1k1a2k2,μ3=4n2n+12p1p2b1+k1p1γ+θ+p2k2+αk1k1a1k1a2k2,μ4=4n22n+1p22b2+k1p1+k1p2γ+θk1a1k1a2k2,μ5=4n2p2a1k1a2k23n+1.E13

Now, from Eqs. (11) and (12), we can write

±ζζ0=dVVμ2+μ3V+μ4V2+μ5V3.E14

The integral Eq. (14) admits many types of solutions that can be listed in accordance with [51, 52, 53].

Since μ2=4n2p12p1k2+αk1k1a1k1a2k2=n2k2+αk12k12a1k1a2k22<0, and in order to highlight our analysis, we consider the following parametric setting γ=σ=θ. After these considerations, the phase component of the soliton can be written as follows:

ϕzx=k2+αk12k1a1k1a2k2k1xk2z+c,E15

where cis a real constant number.

  1. If 4μ2μ4μ32=0and μ4>0, we have the following exact traveling solution.

    ψazx=μ34μ41±cotanhμ3μ44μ4k1xk2z12nezx,E16

under the constraint μ4>0.

  • If μ324μ2μ4>0and μ2<0we get a singular periodic solution

  • ψbzx=2μ2μ3±μ324μ2μ4sinμ2k1xk2z12nezx.E17

    2.2 Stability analysis method: Modulational instability (MI) of the continuous wave (CW) background

    Modulational instability (MI) is a fundamental and ubiquitous phenomenon originating from the interplay between nonlinear self-interaction of wave fields and linear dispersion or diffraction. This process appears in most nonlinear systems [42, 52, 53, 54, 55, 56, 57]. Unlike the well-known pulse kinds, the solitons are relatively stable, even in a perturbed environment.

    In this section, we investigate the stability of the previous solutions that are sitting on a CW background, which may be subject to MI. To do so, we apply the standard linear stability analysis [52, 53, 56] on a generic CW (steady-state solution)

    ψ0zx=P0eic1x+c2z,E18

    where P0, c1, and c2are real constants.

    Putting Eq. (18) into Eq. (1), we get:

    c2=11+a2c1αa1c1c1+b1+γ+θc1P02n+b2+δc1P04n.E19

    Adding infinitesimal perturbation field υon CW solutions by introducing the following expansion

    ψzx=P0+υzxeic1x+c2z,E20

    one can find the linearized equation satisfied by the complex perturbation υas:

    ir1υz+ir2υx+a1υxx+a1υxz+r3υ+υ+ir4υx+υx=0,E21

    where

    r1=1+a2c1,r2=2a1c1+a2c2αγ+θP02nδP04n,r3=nP02nb1+γ+θc1+2b2+δc1P02n,r4=nP02nγ+σ+2δ+λP02n.E22

    The solution of Eq. (21) is given by collecting the Fourier modes as

    υzx=υ+eiKzΩx+υeiKzΩx,E23

    where Ωaccounts for the wavenumber and Krepresents the frequency of perturbation. υ+and υare much less than the background amplitude P0(υzxP0). In this case, the instability of the steady state (CW) is achieved by the exponential growth of the perturbed field.

    Deputing the expression of perturbed nonlinear background Eq. (23) into Eq. (21), we obtain after linearization, a system of homogeneous equations satisfied by υ+and υ:

    a2Ωr1K+r2+r4Ω+r3a1Ω2r3+Ωr4r3Ωr4a2Ω+r1Kr2+r4Ω+r3a1Ω2υ+υ=00.E24

    This set has a nontrivial solution only when the previous 2 × 2 determinant matrix vanishes. By requiring the determinant of the associated matrix to be zero, we get the dispersion relation:

    a22Ω2r12K2+2Ωa2r3a1Ω2+r1r2+r4K+r3a1Ω22Ω2r2+r42=0.E25

    In order to observe MI, one of the two roots of the previous dispersion relation should possess a negative imaginary part, which corresponds to an exponential growth of the perturbation amplitude. So, the MI is measured by power gain, and it is defined at any pump frequency as [52, 53]:

    gΩ=2ImKmax,E26

    where the factor 2 converts gΩto power gain, and ImKmaxdenotes the imaginary part of the polynomial root with the largest value Kmax.

    Figure 1 depicts the MI gain spectra as a function of the modulation frequency (Ω) and second-order dispersion (a1), for a fixed value of the initial power (P0=10kW), the other parameter values being a2=0.005, b1=b2=α=0.2, c1=2, γ=θ=σ=1, δ=0.5. Firstly, we consider the strength of the power-law nonlinearity n=1, which yields to symmetrical sidelobes of instability around the zero-perturbation frequency Ω=0in Figure 1(a). In this map, the width and magnitude of the two sidelobes remain constant in the normal group velocity dispersion (a1<0), while they increase in the anomalous dispersion regime (a1>0). For the nonlinearity power index n=2, we obtain in Figure 1(b), two similar sidelobes due to MI, which stand symmetrically around the line Ω=0. In this case, the magnitude and width of the sidelobes remain constant in the normal as well as the anomalous dispersion regimes.

    Figure 1.

    MI gain spectragΩversus frequency shiftΩand second-order dispersiona1for parameter values:a2=0.005,b1=b2=α=0.2,c1=2,γ=θ=σ=1,δ=0.5,P0=10: (a)n=1; (b)n=2.

    Figure 2 exhibits the enlarged MI gain spectra in 2D-plot at four power levels (P0=10kW;20kW;30kW;50kW) with the same values of parameters as in Figure 1. The MI gain profile, indicated by colored solid curves, is constitutive of two gain bands in the Stokes frequency shift region (Ω<0) and in the anti-Stokes frequency shift region (Ω>0). We can see that the MI gain exists only within a limited range of frequency Ω<100and the maximum gain increases with the increasing input power P0.

    Figure 2.

    2D plot showing the variation of the MI gain spectragΩversus frequencyΩat a four-power levelP0[for yellow solid line(P0=10kW), red solid line(P0=20kW), blue solid line(P0=30kW), and green solid line(P0=50kW)], with the same parameter values as inFigure 1.

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    3. Planar dynamical system and Hamiltonian: Phase portraits

    In this section, we transform Eq. (8) to a dynamical system by introducing new variables Xand Y, in order to investigate the equilibrium points, the periodic, quasiperiodic, and chaotic motions of systems in the presence of an external periodic perturbation, viathe bifurcation method [49, 50, 58].

    3.1 Formation of a dynamical system

    Now, we rewrite Eq. (8) as

    Uζζ=AU+BU2n+1+CU4n+1+DU6n+1,E27

    where

    A=p12p1k2+αk1k1a1k1a2k2=k2+αk124k12a1k1a2k22<0,B=2p1p2b1+k1p1γ+θ+p2k2+αk1k1a1k1a2k2,C=p22b2+k1p1+k1p2γ+θk1a1k1a2k2,D=k1p2k1a1k1a2k2.E28

    By setting

    X=U,Y=Uζ,E29

    we can rewrite Eq. (27) as a planar dynamical system

    Xζ=Y,Yζ=fX=AX+BX2n+1+CX4n+1+DX6n+1.E30

    The Hamiltonian of the dynamical system Eq. (30) is defined as

    HXY=12Y2A2X2B2n+1X2n+2C22n+1X4n+2D23n+1X6n+2,E31

    and satisfy to

    dH=HζXζ+HζYζ=0.E32

    This result implies that the Hamiltonian is a constant of motion [i.e., HXYCst] and the system Eq. (30) is an integrable Hamiltonian system.

    3.2 Chaotic motion analysis

    3.2.1 Equilibrium state derivation

    Using the bifurcation analysis and qualitative theory, we analyze equilibrium points for system Eq. (30). We consider the following Jacobian matrix of system (30) at the equilibrium points XkYk:

    J=XζXXζYYζXYζX=01A+2n+1BXk2n+4n+1CXk4n+6n+1DXk6n0,E33

    where Xkare the zeros (solutions) of fX, and the determinant of Jis expressed by MdetJas

    M=A2n+1BXk2n4n+1CXk4n6n+1DXk6n.E34

    We research equilibrium points that satisfy Xζ=Yζ=fX=0, and we find

    Y=0,XA+BX2n+CX4n+DX6n=0.E35

    It is obvious to notice that TraceJ=0, and through the bifurcation theory [49, 50, 58], we know that, the solution XjYjof Eq. (35) is a:

    • center point, if M>0;

    • saddle point, if M<0;

    • degenerate point, if M=0.

    From Eq. (35), firstly, we have one equilibrium point XY=00for the dynamical system Eq. (30), which is a center point, and hence stable.

    • If we consider A=0, we have

      Y=0,X2n+1B+CX2n+DX4n=0,E36

    and by setting =C24BD, we can discuss the following situations:

    • If <0, Eq. (36) has only one real root, which indicates that the dynamical system Eq. (30) has one equilibrium point XY=00, which is a center point.

    • If =0and C2D>0, Eq. (36) has three real roots, which indicates that the dynamical system Eq. (30) has three equilibrium points: XY00;C2D2n0;C2D2n0. The first equilibrium point 00is a center point; the second C2D2n0and third C2D2n0points are also center points if M>0. Else, if M<0, they are saddle points and hence unstable.

    • If >0, C<0, and D<0, Eq. (36) has just one root; we find one equilibrium point 00,which indicates a stable center point.

  • For A0, we get

    Y=0,A+BX2n+CX4n+DX6n=0.E37

    We consider an evident root of the polynomial equation A+BX2n+CX4n+DX6n=0, as X0[i.e., A+BX02n+CX04n+DX06n=0] and setting =C+DX02n24DB+X02nC+DX02n. After this consideration, we can discuss the equilibrium points for the dynamical system Eq. (30):

    • If <0, we get a first equilibrium point as X00, which is a center point.

    • If =0and C+DX02n2D>0, there are three equilibrium points: one center point X00, and two saddle points C+DX02n2D2n0and C+DX02n2D2n0for M<0. In contrast for M>0, the three equilibrium points are center points.

    • If >0, C>0, and D>0, the dynamical system Eq. (30) has just equilibrium point X00, which indicates a stable center point.

  • Figure 3 shows the phase portrait of the dynamical system (30) for a1=1.4, a2=1, b1=b2=α=2, γ=3, θ=0.0005, σ=4, k1=0.51, k2=3.5, and n=1. We observe one limit cycle about the origin 00. This implies that the waves are stable, and there are no noises to disturb them.

    Figure 3.

    Phase portrait of the system(30)for parameter values:a1=1.4,a2=1,b1=b2=α=2,γ=3,θ=0.0005,σ=4,k1=0.51,k2=3.5, andn=1.

    The periodicity of Xand Y, based on system (30) with the same values of parameters as in Figure 3, is shown in Figure 4.

    Figure 4.

    Periodicity ofXandYbased on systemEq. (30), with the same values of parameters asFigure 3.

    3.2.2 Quasiperiodic and chaotic motions of the perturbed system

    In this section, we will study the quasiperiodic and chaotic motions for Eq. (1) under the external perturbation. As in the previous process, we find the following perturbed dynamical system:

    Xζ=Y,Yζ=gX=AX+BX2n+1+CX4n+1+DX6n+1Rζk1a1k1a2k2,E38

    where X,Y,A,B,C, and Dare given by Eqs. (28) and (29); Rζ=R0cosωζis an external periodic perturbation, R0is a strength of the external perturbation, and ωis the frequency. The difference between the system (30) and the system (38) is that only external periodic perturbation is added with the system (38).

    In Figure 5, we have presented a phase portrait of the perturbed system (38) under the conditions of parameter values as those in Figure 3, except those fixed E0=0.01and ω=1. For the same parameter values as Figure 5, we plotted in Figure 6 the quasiperiodicity of Xand Yversus ζ. From this plot, it is obvious to notice that the perturbed system (38) has quasiperiodic motion even with the consideration of the external periodic perturbation.

    Figure 5.

    Phase portrait of the perturbed system(38)for the same parameter values as those inFigure 3, under external perturbationRζ=E0cosωζ, whereE0=0.01andω=1.

    Figure 6.

    Variation ofXandYwith respect toζof the perturbed system(38), for the same values of parameters as inFigure 5.

    If we increase the strength of the periodic perturbation by considering E0=0.01, the other parameter values remain as in Figure 5, the perturbed system (38) shows quasiperiodic route to chaos as it is shown in Figure 7. In this case, the solutions ignore the periodic motions and represent random sequences of uncorrelated oscillations (see Figures 7 and 8).

    Figure 7.

    Phase portrait of the perturbed system(38)for the same parameter values as those inFigure 5withE0=0.1.

    Figure 8.

    Quasi-periodicity ofXandYbased on system(38), for same values of parameters asFigure 7.

    From the above observations, it is straightforward to notice that the strength of the periodic perturbation significantly enhances the development of the quasiperiodic motion of the perturbed system (38) and quasiperiodic route to chaotic motion of the system (38). Thus, the perturbed NLS Eq. (1) with the power-law nonlinearity in a nano-optical fiber not only has solitonic and periodic wave solutions but could also possess quasiperiodic and chaotic motions.

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    4. Conclusion

    In this chapter, we have investigated the perturbed nonlinear Schrodinger equation involving power-law nonlinearity and higher-order dispersions. We have constructed exact traveling wave solutions of the model by means of the well-known auxiliary equation method. We showed the existence of a family of traveling wave solutions and we have reported the parametric conditions on the physical parameters for the existence of these propagating solutions. Moreover, by employing Stuart and DiPrima’s stability analysis method, a dispersion relation for the MI gain has been obtained. The outcomes of the instability development depend upon the nonlinearity, the power levels, and the dispersion parameters; the instability region increases regardless of the dispersion regime. The results may find straightforward applications in nonlinear optics, particularly in fiber-optical communication. Afterward, equivalent two-dimensional planar dynamic system and Hamiltonian have been derived and equilibrium points of the corresponding system have been gotten through the bifurcation theory. In addition, we have addressed the periodic, quasiperiodic, and chaotic behaviors of the traveling waves considering an external periodic perturbation. It has been observed that the perturbed system shows quasiperiodic route to chaos as a result of the strength of the periodic perturbation enhancement.

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    Conflict of interest

    The authors declare no conflict of interest.

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    The system of algebraic equations obtained by balancing VVζζand V5in Eq. (10) is as follows:

    • V5coeff:

      k1a1k1a2k23n+1μ5+4n2k1p2=0,

  • V4coeff:

    4n2b2+k1p1+k1p2γ+θp22k1a1k1a2k2
    +k1a1k1a2k22n+1μ4=0,

  • V3coeff:

    4n2b1+k1p1γ+θ+p2k2+αk12p1p2k1a1k1a2k2
    +k1a1k1a2k2n+1μ3=0,

  • V2coeff:

    4n2p1k2+αk1p12k1a1k1a2k2
    +k1a1k1a2k2μ2=0,

  • V1coeff:

    k1a1k1a2k21nμ1=0,

  • V0coeff:

    k1a1k1a2k212nμ0=0.

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    Mati Youssoufa, Ousmanou Dafounansou, Camus Gaston Latchio Tiofack and Alidou Mohamadou (October 18th 2021). Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions [Online First], IntechOpen, DOI: 10.5772/intechopen.100396. Available from:

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