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Dynamical properties, modulation instability analysis and chaotic behaviors to the nonlinear coupled Schrödinger equation in fiber Bragg gratings

    https://doi.org/10.1142/S0217984923502391Cited by:29 (Source: Crossref)

    Abstract

    The nonlinear coupled Schrödinger equation in fiber Bragg gratings is studied in this paper. The existence of soliton solutions and periodic solutions are proved by qualitative analysis, and exact solutions are given, as well as the parameter condition of each solution is described. Then the modulation instability (MI) analysis is carried out and the linear stability criterion is given. In particular, external perturbation terms are introduced to prove that the equation exists chaotic behaviors.

    1. Introduction

    In the real world, there are many models using data1,2,3,4,5,6 for model analysis. However, general mathematical models are always necessary due to this aspect. Some mathematical models are used to get some more general results. The nonlinear partial differential equations model (NPDE) is a very important model that can be used to analyze problems in a common sense. NPDE can be used to describe nonlinear physical phenomena, such as mathematics,7 engineering,8 electrical science9 and others. There are many NPDEs that have been extensively studied, such as generalized (2+1)-dimensional shallow water wave equation,10 the (3+1)-dimensional Burger system (3DBS) equation,11 Schrödinger equation12,13,14 and so many others.15,16 In this paper, we consider the nonlinear coupled Schrödinger equation in fiber Bragg gratings as17

    iqt+ia1rxxx+b1rxxxx+(c1|q|2+d1|r|2)q+(ξ1|q|4+η1|q|2|r|2+ξ1|r|4)q+(l1|q|6+m1|q|4|r|2+n1|q|2|r|4+p1|r|6)q+iα1qx+β1r+σ1qr2=i[γ1(|q|2q)x+θ1(|q|2)xq+μ1|q|2qx],(1)
    and
    irt+ia2qxxx+b2qxxxx+(c2|r|2+d2|q|2)r+(ξ2|r|4+η2|r|2|q|2+ξ2|q|4)r+(l2|r|6+m2|r|4|q|2+n2|r|2|q|4+p2|q|6)r+β2q+σ2rq2=i[γ2(|r|2r)x+θ2(|r|2)xr+μ2|r|2rx],(2)
    where q(x,t) and r(x,t) are forward propagation and backward propagation wave profile, respectively, aj(j=1,2) and bj are third-order and fourth-order dispersion, cj, ζj, lj are the coefficients of self-phase modulation terms, dj, ηj, ζj, mj, nj, pj are the cross-phase modulation terms, αj is the inter-modal dispersion, βj is the detuning parameter, σj is the four-wave-mixing effect, γj is self-steepening, θj and μj are nonlinear dispersion coefficients.17

    Fiber Bragg grating has a large group velocity dispersion, by selecting characteristics such as frequency and nonlinearity of gratings, it can be applied in many fields, such as filter design18,19 and pulse compression technology20 and fiber Bragg grating can form optical solitons in a very short distance, and optical soliton plays an important role in the field of fiber optical communications.21,22 In particular, Eqs. (1) and (2) are a dual extension to the Kerr type of nonlinearity and are occasionally referred to as cubic–quintic–septic nonlinear form, having an important application in compensation for the dispersion velocity of one of the two sources of stable soliton transport over intercontinental distances.17 As a result, this equation has attracted the attention of many researchers in recent years. Zayed studied the nonlinear polynomial law and obtained the bright and singular soliton solution of the equation.17 Kamel constructed periodic and solitary wave solutions to this equation, representing the propagation of different waveforms in fiber Bragg gratings.23 The propagation of Bragg solitons is shown by studying the nonlinear pulse propagation of out-of-band frequency through Bragg grating when the grating is transmitted but highly dispersive.24 However, they didn’t conduct the qualitative analysis, and didn’t find all the traveling wave solutions.

    Many famous methods are used to deal with NDPE, such as Darboux transform,25 sinh-Gordon equation expansion method,26 Bäcklund transformation,27 tanh method28 and the complete discrimination system for polynomial method (CDSPM),29,30,31,32 Among the above method, CDSPM is widely used,33,34,35 not only can it obtain the traveling wave solutions of many important NPDE, but it also can prove the existence of periodic and soliton solutions by qualitative analysis of equations. Kai uses this method to prove the existence of periodic soliton of Degasperis–Procesi model and Kudryashov equation.36,37 Hu uses this method for qualitative analysis of a generalized Kudryashov equation.38 Therefore, in this paper, this method is used to qualitatively analyze the equation to give the dynamical properties, proving that periodic solutions and soliton solutions exist, and construct all traveling wave solutions to verify it. In particular, the modulation instability (MI) of Eqs. (1) and (2) is analyzed, and the chaotic behaviors of the equation are observed by introducing perturbation terms.

    2. Integral Form of Eqs. (1) and (2)

    By taking the following transformation

    q(x,t)=ϕ1(ξ)exp[iψ(x,t)],r(x,t)=ϕ2(ξ)exp[iψ(x,t)],ξ=k0xv0t,ψ(x,t)=k0t+w0t+θ0,(3)
    where v0, θ0, k0, w0 are nonzero constants, which are soliton velocity, phase constant, soliton frequency and wave number, respectively. ψ(x,t) represents the phase component of soliton, and ϕj(ξ)(j=1,2) stands for the pulse shape.

    When

    ϕ2(ξ)=Πϕ1(ξ),(Π0,1),(4)
    then substituting Eqs. (3) and (4) into Eqs. (1) and (2) yields14
    b1Πϕ1+3(a1k2b1k2)Πϕ1+[(β1a1k3+b1k4)Π +α1kω]ϕ1+[c1(μ1+γ1)k]+(σ1+d1)Π2]ϕ31 +(ζ1Π4+η1Π2+ξ1)ϕ51+(l1+m1Π2+n1Π4+p1Π6)ϕ71=0,(5)
    and
    b2ϕ1+3(a2k2b2k2)ϕ1+[β2a2k3+b2k4+(α2kω)Π]ϕ1+[c2(μ2+γ2)k]Π3+(σ2+d2)Πϕ31+(ξ2Π5+η2Π3+ζ2Π)ϕ51+(l2Π7+m2Π5+n2Π3+p2Π)ϕ71=0.(6)
    Thus we can get
    b1Π=b2,(a1k2b1k2)Π=a2k2b2k2,(β1a1k3+b1k4)Π+α1kω=β2a2k3+b2k4+(α2kω)Π,c1(μ1+γ1)k+(σ1+d1)Π2=[c2(μ2+γ2)k]Π3+(σ2+d2)Π,ζ1Π4+η1Π2+ξ1=ξ2Π5+η2Π3+ζ2Π,l1+m1Π2+n1Π4+p1Π6=l2Π7+m2Π5+n2Π3+p2Π.(7)
    We can see from the equations above that we only need to deal with Eq. (6), and then ϕ1 shall be replaced by ϕ. Then Eq. (6) becomes
    b2ϕ+3g1ϕ+g2ϕ+g3ϕ3+g4ϕ5+g5ϕ7=0,(8)
    where
    g1=a2k2b2k2,g2=[β2a2k3+b2k4+(α2kω)Π),g3=[c2(μ2+γ2)k]Π3+(σ2+d2)Π,g4=(ξ2Pi5+η2Π3+ζ2Π),g5=(l2Π7+m2Π5+n2Π3+p2Π).(9)
    By introducing the following trial equation39,40
    (ϕ)2=f5ϕ5+f4ϕ4+f3ϕ3+f2ϕ2+f1ϕ+f0,(10)
    we have
    ϕ=5f5ϕ42+2f4ϕ3+3f3ϕ22+f2ϕ+f12,(11)
    and
    ϕ=30f5ϕ2+24f4ϕ+3f3)ϕ2+(10f5ϕ3+12f4ϕ2+3f3ϕ+f2).(12)
    Substituting (10)–(12) into (8) yields
    ϕ7:55b2f25+g5=0,ϕ6:104b2f4f5=0,ϕ5:(111f3f52+48f24)b2+g4=0,ϕ4:(85f2f52+51f3f4)b2+15g1f52=0,ϕ3:(35f1f5+38f2f4+15f232)+6g1f4+g3=0,ϕ2:(30f1f4+30f0f5+15f2f32)b2+9g1f32=0,ϕ1:(9f1f32+f22+2f0f4)b2+3g1f2+g2=0,ϕ0:(f1f22+3f0+f3)b2+3g1f12=0,(13)
    then we have
    f5=g555b2,f4=0,f3=2g4111g555b2,f2=3g117b2,f1=2g3+3300g2412321g5,f0=666g1g3+666g1g317b2300g2437g5g162912g4b2,(14)
    thus (8) becomes
    (ϕ)2=f5ϕ5+f3ϕ3+f2ϕ2+f1ϕ+f0.(15)
    Equation (15) is analyzed in the following.

    3. Dynamical Properties of Eq. (15)

    Dynamical system of (15) is as follows :

    {ϕ=v,v=5f52(ϕ4+h2ϕ2+h1ϕ+h0),(16)
    where h2=3f35f5, h1=2f25f5 and h0=f15f5. And the Hamiltonian is
    H(u,v)=v2(f5ϕ5+f3ϕ3+f2ϕ2+f1ϕ+f0),(17)
    which satisfies
    Hv=ϕ,Hϕ=v.(18)
    The potential energy can be given by
    F(ϕ)=f5ϕ5+f3ϕ3+f2ϕ2+f1ϕ+f0.(19)
    The roots of F(ϕ) are discussed to analyze the dynamic properties of Eq. (15)
    F(ϕ)=5f52(ϕ4+h2ϕ2+h1ϕ+h0).(20)
    We introduce the complete discrimination system for polynomial
    N1=4,N2=h2,N3=2h32+8h2h09h21,N4=h32h21+4h42h0+36h2h21h032h22h2027h414+64h30,F2=9h2232h2h0,(21)
    and nine cases that will be discussed here.

    Case 1.N4=0, N3>0, N2>0, then F(ϕ) is given by

    F(ϕ)=5f52(ϕα1)2(ϕα2)(ϕα3),(α2>α3).(22)
    When f5<0, (α1,0),(α3,0) are saddle points and (α2,0) is a center and if f5>0, (α3,0) is a center and (α1,0),(α2,0) are saddle points. For example, when f5=±1, h2=±1, h1=0, h0=±1 we have α1=0, α2=1, α3=1, see Fig. 1(b).

    Fig. 1.

    Fig. 1. (Color online) Dynamic system (16): (a) f5=1, h2=1, h1=1, h0=0; (b) f5=1, h2=1, h1=0, h0=1.

    Trajectory I implies that periodic solutions exist. Trajectories II and III indicate that bell-shaped soliton solutions exist.41

    Case 2.N4=0, N3=0, N2>0, E2=0, we have

    F(ϕ)=5f52(ϕα1)3(ϕα2).(23)
    When f5<0, (α2,0) is a saddle point and (α1,0) is a center in Fig. 2(a), and when f5>0, (α1,0) is a saddle point, (α2,0) is a center in Fig. 2(b). When f5=1, h2=±6, h1=±8 and h0=±3, we can get α1=1, α2=3.

    Fig. 2.

    Fig. 2. (Color online) Dynamic system (16): (a) f5=1, h2=6, h1=8, h0=3; (b) f5=1, h2=6, h1=8, h0=3.

    The trajectories I and II in Fig. 2(a) imply that the periodic solutions exist, the trajectory III indicates that the soliton solutions exist. However, in Fig. 2(b) trajectory I implies that the periodic solutions exist, and trajectories II and III imply that the bell-shaped soliton solutions exist.

    Case 3.N1>0, N2>0, N3>0, we can get

    F(ϕ)=5f52(ϕα1)(ϕα2)(ϕα3)(ϕα4).(24)
    When f5<0, (α1,0) and (α4,0) are saddle points, (α2,0) and (α3,0) are center points. For instance, f5=±1, and h2=±5, h1=0, h0=±4, and then we can get α1=1, α2=1, α3=2, α4=2.

    The specific phase portrait is shown in Fig. 3, Trajectories I and III imply that periodic solutions exist, and trajectories II and IIII indicate that bell-shaped soliton solutions exist.

    Fig. 3.

    Fig. 3. (Color online) Dynamic system (16): (a) f5=1, h2=5, h1=0, h0=4; (b) f5=1, h2=5, h1=0, h0=4.

    Case 4.N4<0, N3N20, we can get

    F(ϕ)=5f52(ϕα1)(ϕα2)((ϕl1)2+β21).(25)
    The dynamic system has a center point (α1,0) and has a saddle point (α2,0) when f5<0. For instance, f5=±1 and h2=0, h1=0, h0=±1, and then we can get α1=1, α2=1, l1=0, β1=1, phase portrait is shown in Fig. 4.

    Fig. 4.

    Fig. 4. (Color online) Dynamic system (16): (a) f5=1, h2=0, h1=0, h0=1; (b) f5=1, h2=0, h1=0, h0=1.

    Trajectory I implies that periodic solution exists. Trajectories II and III imply that soliton solution exists.

    Here are other five cases where there are no soliton solutions and periodic solutions, it does not help our discussion, we omit specific discussion.

    4. Exact Solutions of Eq. (15)

    Transform (15) as

    τ=(f5)15ϕ,ξ=(f5)15ξ1,(26)
    and rewrite Eq. (16) into the following form :
    (τ)2=τ5+k3τ3+k2τ2+k1τ+k0,(27)
    where k3=f3f355, k2=f2f255, k1=f1f155, and k0=f0. We can get
    F(τ)=τ5+k3τ3+k2τ2+k1τ+k0,(28)
    thus Eq. (33) is converted into the following integral form :
    ±(ξξ0)=dττ5+k3τ3+k2τ2+k1τ+k0,.(29)
    By introducing the following complete discrimination system for polynomial
    N2=k3, N3=40k1k312k3345k22,N4=12k43k14k33k22+117k22k1k388k21k2340k2k0k2327k42300k2k1k0+160k31, N5=1600k2k0k313750k3k2k30+2000k3k20k214k33k22k21+16k33k32k016k33k32k0+900k1k20k33+825k23k22k20+144k3k22k31+2250k1k22k20+16k43k31+108k53k20128k41k2327k21k42+108k0k52256k51+3125k40+560k0k2k21k2372k1k0k2k43630k1k3k0k32,E2=160k21k33+900k22k2148k53k1+60k1k23k22+1500k0k1k2k3+16k22k431100k2k0k23+625k20k233375k0k32,F2=3k228k1k3,(30)
    there are 12 cases to discuss.

    (I) When N5=0, N4=0, N3>0, E20, we can get

    F(τ)=(τo1)2(τo2)2(τo3),(31)
    where o1, o2, o3 are real numbers, and o1o2o3, while τ>o3 the corresponding solutions can be given by
    ±o1o22(ξξ0)=o3o1arctanτo3o3o1o3o2arctanτo3o3o2,(32)
    while o3>o1, o3>o2,
    ±o1o22(ξξ0)=o3o1arctanτo3o3o11o2o3ln|τo3o2o3τo3+o2o3|,(33)
    while o3>o1, o3<o2, k3=2, k2=0, k1=0, k0=1, we can get the corresponding 2D graph in Fig. 5. When the solution reaches a certain value, derivative takes a finite transition and it gradually trends to 0 when ξ trends to infinitely. Obviously, this solution is a peak on the bright soliton solution.
    ±o1o22(ξξ0)=o3o2arctanτo3o3o2+12o1o3ln|τo3o1o3τo3+o1o3|,(34)
    while o3>o2, o3<o1,
    ±o1o22(ξξ0)=12o1o3ln|τo3o1o3τo3+o1o3|12o2o3ln|τo3o1o3τo3+o1o3|,(35)
    while o3<o1, o3<o2.

    Fig. 5.

    Fig. 5. (Color online) Peakon bright soliton solution when k3=2, k2=0, k1=0, k0=1.

    (II) When N5=0N4=0N3=0, N20, F20, we can get

    F(τ)=(τo1)3(τo2)2,(36)
    where o1, o2 are real numbers and o1o2. While τ>o1 we can get
    ±o1o22(ξξ0)=12τo1o1o2arctanτo1o1o2,o1>o2,(37)
    or
    ±o1o22(ξξ0)=12τo112o2o1ln|τo1o2o1τo1+o2o1|,o1<o2.(38)

    (III) When N5=0, N4=0, N3=0 and N20, F=0

    F(τ)=(τo1)4(τo2),(39)
    where o1, o2, are real numbers and o1o2. While τ>o1, we can get the corresponding solutions as follows :
    ±o1o22(ξξ0)=τo22(τo1)12o1o2arctanτo2o2o1,o1<o2,(40)
    or
    ±o1o22(ξξ0)=τo22(τo1)14o1o2ln|τo2o1o2τo2+o1o2|,o1>o2.(41)

    (IV) When N5=0, N4=0, N3=0, N2=0, F(τ) is given by could be shown as

    F(τ)=(τo)5,(42)
    where o is a real number. While τ>o, we can get the following solutions :
    ±(ξξ0)=23(τo)23.(43)

    (V) When N5=0, N4=0, N3<0, E20, we have

    F(τ)=(τo)(τ2+k1τ+k0)2,(44)
    where o is real number and k214k0<0, while τ>o we can get
    ±(ξξ0)=2χ4k0k21(cosφarctan22χsinφτoτoχ2)+sinφ2ln|τoχ22χcosφτoτoχ2+2χcosφτo|,(45)
    where
    χ=(o2+k1o+k0)14,φ=12arctan4k0k212ok1.(46)
    When k3=0, k2=5, k1=415, k0=92, we obtain dark soliton solution in Fig. 6.

    Fig. 6.

    Fig. 6. (Color online) Dark soliton solution when k3=0, k2=5, k1=415, k0=92.

    (VI) When N5=0 and N4>0, we get

    F(τ)=(τo)2(τι1)(τι2)(τι3),(47)
    where o, ι1, ι2, ι3 are real numbers and ι1>ι2>ι3. Then we can get
    ±(ξξ0)=2(oι2)ι2ι3×(F(φ,m)ι1ι2ι1oπ(φ,ι1ι2ι1o)),(48)
    where oι1, oι2, oι3, and
    F(φ,m)=φ011msin2φdφ,π=φ01(1+nsin2φ)1msin2φdφ.(49)

    (VII) When N5=0, N4=0, N3<0, we have

    F(τ)=(τo)3((τl)2+y2),(50)
    where o, y, l, are real numbers. While τ>o, if ol+y then we take the following solution:
    ±(ξξ0)=tanθ+cotθ2(ytanθlo)ysin32θF(φ,m)ytanθ+ycotθycotθ+l+o×(tanθ+lo(ycotθ+lo)sinθ1m2sin2θ+F(φ,m)Γ(φ,m)),(51)
    if o=l+y we can get
    ±(ξξ0)=sin32θ4h30(1marcsin(msinφ)F(φ,m)),(52)
    where tan2θ=yol, and m=sinθ, 0<θ<π2
    Γ(φ,m)=φ01m2sin2φdφ.(53)

    (VIII) When N5>0, N4>0, N3>0, N2>0 we obtain

    F(u)=(τo1)(τo2)(τι1)(τι2)(τι3)(54)
    use hyper-elliptic function or a hyper-elliptic integral, such as
    ±(ξξ0)=1(τo1)(τo2)(τι1)(τι2)(τι3)dτ.(55)

    (IX) When N5=0, N4=0, N3>0, E2=0, we obtain

    F(τ)=(τo1)3(τo2)(τo3),(56)
    where o1, o2, o3, are real numbers. While τ>o1>o2>o3 we can get
    ±o1o22(ξξ0)=1o1o3Γ(arcsino1o3τo3,o2o1o1o3τo2(τo3)(τo1)),(57)
    while τ>o2>o1>o3 and so on, the corresponding solutions can be given in a similar way, which we omit for simplicity.

    (X) When N5=0, N4<0, we get

    F(τ)=(τo1)2(τo2)((τn)2+y2),(58)
    where o1, n, y, are real numbers, while τ>o2, if o1nytanθ and o1nycotθ then we take the following solution:
    ±(ξξ0)=tanθ+cotθ2(ytanθno1)ysin32θF(φ,m)ytanθ+ycotθycotθ+n+o1×(tanθ+no1(ycotθ+no1)sinθ1m2sin2θ+F(φ,m)Γ(φ,m)),(59)
    where tan2θ=yo1n, and m=sinθ, 0<θ<π2
    ±(ξξ0)=sin32θ4h03F(φ,m)11m2×ln1m2sin2φ+1m2sinφcosφ,(60)
    where tanθ=yo1o2, m=sinθ, 0<θ<π2.

    (XI) When N5>0, N40, N30, N20, we have

    F(τ)=(τo)((τn)2+y2)((τn1)2+y12),(61)
    then hyper-elliptic functions or hyper-elliptic integral may be used to define the relevant solutions, we can get
    ±(ξξ0)=1(τo)((τn)2+y2)(τn1)2+y12dτ.(62)

    (XII) When N5<0, we have

    F(τ)=(τo1)(τo2)(τo3)((τn)2+y2),(63)
    then hyper-elliptic functions or hyper-elliptic integral may be used to define the relevant solutions, we can get
    ±(ξξ0)=1(τo1)(τo2)(τo3)(τn)2+y2dτ.(64)
    Through the results given above, all traveling wave solutions are found, among which the rational type solution, elliptic function solutions are new solutions.

    5. Stability Analysis

    MI analysis is of great significance for nonlinear equations42,43,44,45. In this section, due to ϕ2(ξ)=Πϕ1(ξ), we can get

    r(x,t)=Πq(x,t),(65)
    thus we only need to analyze Eq. (1). Rewrite it as
    iqt+Aqxxx+Bqxxxx+C|q|2q+D|q|4q+E|q|6q+iα1qx+FqΠ+σ1qPi2q2=i[γ1(|q|2q)x+θ1(|q|2)xq+μ1|q2qx|],(66)
    where
    A=ia1Π,B=b1Π,C=c1+d1Π2,D=ξ1+Π2η1+Π4ζ1,E=l1+m1Π2+n1Π4+p1Π6,F=β1Π.(67)
    The steady-state perturbation of the original equation is
    q(x,t)=(ϵ(x,t)+λ)ei(F+Cλ+Dλ2+Eλ3)t,ϵλ,(68)
    where λei(F+Cλ+Dλ2+Eλ3)t is the steady state solution. Substituting (73) into (71) and linearizing yield
    iϵt+iA3ϵx3+B4ϵx4+Cλ(ϵ+ϵ)+2Dλ2(ϵ+ϵ)+3Eλ3(ϵ+ϵ)=iγ1λ(2ϵx)+θ1λϵx+ϵx+μ1λϵx,(69)
    where ∗ represents the complex conjugate. Assuming the solution for (74) is36,46,47
    ϵ(x,t)=f1ei(lxtω)+f2ei(lxtω),(70)
    where l stands for the normalized wave number and ω for the frequency of perturbation. Substituting (75) into (74), the following results can be obtained :
    ω=±lB2l4+2BTl2+(γ1λ+θ1λ)2Al3Sl,(71)
    where S=(α1+μ1λ+2γ1λ+θ1λ), T=Cλ+Dλ2+Eλ3. If ω has the real part, the steady state is stable under small perturbation. Conversely, if ω has the imaginary part, the steady state becomes unstable as the disturbance grows exponentially. So the condition of unstable steady state is given by
    B2l4+2BTl2+(γ1λ+θ1λ)2<0.(72)
    Let us show the unstable and stable MI zones in Figs. 7(a) and 7(b) by considering B=2, T=1, a=0.5, b=0.3. 0>l>1 is the unstable zone in Fig. 7(a) and the area of 1>l>0 is the unstable zone in Fig. 7(b).

    Fig. 7.

    Fig. 7. (Color online) MI zones when B=2, T=1, a=0.5, b=0.3.

    In this case the MI gains spectrum G(λ), which can be expressed as

    G(λ)=2Im(lB2l4+2BTl2+(γ1λ+θ1λ)2).(73)

    6. Chaotic Studies of Eq. (15)

    Chaotic phenomena can be used to describe many complex physical phenomena, such as analysis of neural network and optical fiber communications.48,49 Therefore, we study the chaotic phenomena of the equation. We add the perturbed term ξD(η) to the dynamic system (15) to get the following perturbed system :

    dϕdξ=v,dvdξ=(a4ϕ4+a2ϕ2+a1ϕ+a0)+ξD(η),(74)
    where a4=5f52, a2=5f5h22, a1=5f5h12, a0=5f5h02, ξ denotes coefficient of perturbation term, D(η) is the perturbed function. When D(η)=l(0.035η), g(x) is Gaussian function, which is expressed as follows :
    g(x)=1σ2πe12(xμσ)2.(75)
    If μ=0, σ=1, a4=20, a2=10, a1=20, a0=10, ξ=200, we get the corresponding phase diagram in Figs. 8 and 9, we find that the system (74) has chaotic properties. The largest Lyapunov exponents (LLEs) verify this conclusion. From Figs. 9(a)–(d) we can see the relationship between parameters a4, a2, a1 and a0 and LLE, and it is easy to find that those parameters have effects on chaotic behavior of the perturbed system.38,50,51 Moreover, from Figs. 8(a) and 8(b), we can find that the trajectories of the perturbed system (74) intersect, so the solution is not unique. In addition, different perturbed systems produce different chaotic behaviors, which will be confirmed later.

    Fig. 8.

    Fig. 8. (Color online) The phase graph of the perturbed system with Gaussian distribution when a4=20, a2=10, a1=20a0=10, g(ξ)=200(0.035ξ).

    Fig. 9.

    Fig. 9. (Color online) The perturbed system (15): (a) LLE for a4; (b) LLE for a2; (c) LLE for a1; (d) LLE for a0.

    We study another perturbation function: the cosine function. If D(η)=0.5cos(0.5η), a4=5, a2=5, a1=10, a0=5, ξ=0.5. The corresponding phase diagram in Fig. 10 and the relationship between LLE and parameters are shown in Figs. 11(a)–11(d). The analysis of LLE is similar to the Gaussian perturbation, it is clear that the parameters a4, a2, a1 and a0 have effects on chaotic behaviors of the perturbed system. It is easy to find that there is chaotic behaviors in this case and that the solution is not unique, and the phase portraits are different from the Gaussian perturbation, which confirms the above conclusion. The chaotic behaviors to this equation may have many potential physical applications which can provide a new direction for future study.

    Fig. 10.

    Fig. 10. (Color online) The phase graph of the perturbed system with Gaussian distribution when a4=5, a2=5, a1=10, a0=5, D(η)=0.5cos(0.5η).

    Fig. 11.

    Fig. 11. (Color online) The perturbed system (15): (a) The LLE for a4; (b) The LLE for a2; (c) The LLE for a1; (d) The LLE for a0.

    7. Conclusion

    The purpose of this study is to solve the nonlinear coupled Schrödinger equation in fiber Bragg gratings by CDSPM. The method used in this paper is simple but effective. We can obtain the existence of solitons and periodic solutions by qualitative analysis through its dynamical system without constructing the exact solutions. We also give concrete solutions to verify our solutions by conducting the quantitative analysis and all traveling wave solutions are given, and some new solutions of the equation are shown, namely the rational type solution, elliptic function solutions, which makes our results more complete. In particular, the MI of the original equation is analyzed. The linear stability is presented, and the critical condition for stability is provided. Finally, the chaotic behaviors of the equation under different perturbation terms are shown. As far as we know, these findings to this equation are given first and a new research direction for the equation is provided.

    ORCID

    Yue Kai  https://orcid.org/0000-0002-2935-8634

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