Optical Solitons for the Perturbed Nonlinear Schrödinger Equation with Kerr Law and Non-Kerr Law Nonlinearity

1 Introduction

In nonlinear optics, the nonlinear Schrödinger equation (NLSE) [1], [2], [3], [4], [5], [6] can be used to describe the propagation of an optical pulse in nonlinear media including Kerr law and non-Kerr law media [7], [8], [9], [10]. Several advances have been made in the area of the non-Kerr law media, such as power law, parabolic law, dual power law, and so on, which have attracted much attention.

The analytical solutions of the Schrödinger equations may give more insight into the described optical pulse transmission. So, an important issue with Schrödinger equations is to find their new analytical solutions and study the characteristics of the solutions. There are many relative methods, such as the modified mapping method and the extended mapping method, the modified trigonometric function series method, the bifurcation method and qualitative theory of dynamical systems, the modified (GG′)-expansion method, and the dynamical systems approach, to study Schrödinger equations [11], [12], [13], [14], [15]. One type of the most significant solutions comprise the optical solitons, which hold their shapes in the process of transmission and collisions. Because of the perfect balance between dispersion or (and) diffraction and nonlinear effect in a nonlinear optical fibre, optical solitons are often utilised in optical fibre communication areas.

This paper is devoted to studying the NLSE with perturbation [3], [11]

where x represents the non-dimensional distance along the fibre, t represents time in dimensionless form, and a and b are real-valued constants. The dependent variable q(x, t) is a complex-valued function that represents the wave profile. α is the inter-modal dispersion, λ represents the coefficient of self-steepening for short pulses (typically ≤100 fs), and f is the higher order dispersion coefficient. The parameter m is a nonlinearity parameter [16]. F(|q|²) is a real-valued function which represents the type of nonlinear media. When the right-hand side of the equation is zero, (1) is the simplified dimensionless form of the NLSE. It is a nonlinear partial differential equation that is not integrable, in general. The non-integrability is not necessarily related to the nonlinear terms in it.

Equation (1) is widely used in optics as a good model for optical pulse propagation in nonlinear fibres. In this paper, we consider four types of nonlinearity, namely Kerr law, power law, parabolic law, and dual power law. It is obvious that Kerr law and parabolic law are special cases of the other two laws. The three different effective integration schemes are the tanh function method, the Kudryashov method, and the trial solution method [17], [18], [19]. To the best of our knowledge, these methods have not been employed in solving the NLSE with perturbation.

Let

where u(s) represents the shape of the pulse, s=x−vt, and ϕ=−kx+ωt+θ. The function ϕ(x, t) is the phase component of the soliton, k is the soliton frequency, while ω is the wave number, ϕ is the phase constant, and v is the velocity of the soliton.

Substituting (2) into (1), and then equating the real and imaginary parts to zero, we obtain two equations. The imaginary part gives

Here, ((2m+1)λ+2mf) must be equal zero (when m≠0). The real part gives

Equation (3) gives the velocity of the wave, while (4) can be integrated to compute the soliton profile provided the functional is known. NLSE with perturbation will be considered for the following four forms of nonlinearity, as discussed in the following two sections.

2 Power Law

Power law nonlinearity is encountered in various materials including semiconductors [9]. This law is also applicable to higher order photon processes that dominate at different intensities. However, the power law solves the problem of small-K condensation in weak turbulence theory in nonlinear plasmas. We can see this law as a generalisation of the Kerr law nonlinearity.

For power law nonlinearity

Substituting (5) into (4) gives

where the parameter n dictates the power law nonlinearity. It is necessary to take 0<n<2 because of stability issues.

2.1 Application of the Kudryashov Method

In this subsection, we will apply the Kudryashov method for the NLSE with perturbation with power law nonlinearity. Balancing u″ with u²ⁿ⁺¹ to obtain closed-form solutions gives N=1n. Using the transformation

(7)u=v1n,

(6) becomes

(8)an(1n−1)v′2+anvv″−(w+αk+ak2)v2 −λkv2+2mn+bv4=0.

Case 1:

When mn=12. In this case, (8) takes the form

(9)an(1n−1)v′2+anvv″−(w+αk+ak2)v2−λkv3+bv4=0.

Balancing vv″ with v⁴ in (9), we find that N=1. The solution of (9) is of the form

(10)v(ξ)=A0+A1Q(ξ),

where A_i (i=0, 1) are some constants to be determined later, and Q(ξ) satisfies

(11)Qξ=μQ−Q2.

Substituting (10) into (9) with (11), collecting the coefficients of Q, and solving the resulting system, we have

Case 1.1:

(12)A0=−μA1, A1≠0, w=aμ2−ak2n2−αkn2n2,

where μ is an arbitrary constant.

Consequently, (6) has the following exact soliton solutions:

(13)u(x, t)=(−μA1−A1μ(1±tanh(±μ(x+(2ak+α)t)2))−2)1n,

and

(14)u(x, t)=(−μA1−A1μ(1±coth(±μ(x+(2ak+α)t)2))−2)1n,

where μ is an arbitrary constant.

Figure 1 shows the kink soliton with power law nonlinearity. The parameter values are A₁=μ=a=k =α=n=1.

Figure 1:

Optical soliton with power law nonlinearity.

Case 1.2:

(15)A0=0, A1≠0, w=k(−A1λμ2+A1λk2n2−anμ−2aμ)μ(2+n),

where μ is an arbitrary constant.

Consequently, we obtain the exact solutions of (6) as follows:

(16)u(x, t)=(−A1μ(1±tanh(±μ(x+(2ak+α)t)2))−2)1n,

and

(17)u(x, t)=(−A1μ(1±coth(±μ(x+(2ak+α)t)2))−2)1n,

where μ is an arbitrary constant.

3 Dual Power Law

This model is used to describe the saturation of the nonlinear refractive index and solitons in photovoltaic-photo-refractive materials such as LiNbO₃. For dual power law nonlinearity

Substituting (34) into (4) gives

3.2 Application of the Trial Solution Method

In this subsection, we will apply the trial solution method to handle the NLSE with perturbation with dual power law nonlinearity.

Case 1:

Equation (38) has the trial solution as follows:

(56)(v′)2=A0+A1v+A2v2+A3v3+A4v4.

Thus, we have

(57)v″=A12+A2v+3A32v2+2A4v3.

Inserting (56) and (57) into (38), equating all coefficients of v, and solving the resulting system, we get

(58){A0=A1=0,A2=4n2(n+1)(2n+1)a(2an2k2+3ank2+2kn2α+ak2+3knα+2wn2+kα+w+3nw),A3=4n2(n+1)(2n+1)a(2knλ+kλ−b−2bn),A4=4n2(n+1)(2n+1)a(−lbn−lb),

where w is an arbitrary constant.

Then (56) becomes

(59)(v′)2=4n2(n+1)(2n+1)a(2an2k2+3ank2+2kn2α+ak2+3knα+2wn2+kα+w+3nw)v2+(2knλ+kλ−b−2bn)v3+ (−lbn−lb)v4).

Equation (35) has the following exact soliton solutions:

(60)u(x, t)=(−A2A3sech2(A2(x+t(2ak+α))2)A32−A2A4(1±tanh(A2(x+t(2ak+α))2)))1n,

which are valid for

(61)A2>0,

and

(62)u(x, t)=(−A2sec2(−A2(x+t(2ak+α))2)A3±−A2A4(1±tan(−A2(x+t(2ak+α))2)))1n,

which are valid for

(63)A2<0, A4>0.

Figure 2 shows the periodic solution with dual power law nonlinearity. The parameter values are A₂=A₄=2, A₃=a=k=α=n=1.

Figure 2:

Optical soliton with dual power law nonlinearity.

Case 2:

Equation (47) has the trial solution as follows:

(64)(v′)2=A0+A1v+A2v2+A3v3+A4v4.

Thus, we have

(65)v″=A12+A2v+3A32v2+2A4v3.

Inserting (64) and (65) into (47), equating all coefficients of v, and solving the resulting system, we get

(66){A0=A1=0,A2=4n2(n+1)(2n+1)a(2an2k2+3ank2+2kn2α+ak2+3knα+2wn2+kα+w+3nw),A3=4n2(n+1)(2n+1)a(b+2bn),A4=4n2(n+1)(2n+1)a(lbn+lb−knλ−kλ),

where w is an arbitrary constant.

Then (64) becomes

(67)(v′)2=4n2(n+1)(2n+1)a(2an2k2+3ank2+2kn2α+ak2+3knα+2wn2+kα+w+3nw)v2+(b+2bn)v3+(lbn+lb−knλ−kλ)v4).

Equation (35) has the following exact soliton solutions:

(68)u(x, t)=(−A2A3sech2(A2(x+t(2ak+α))2)A32−A2A4(1±tanh(A2(x+t(2ak+α))2)))1n,

which are valid for

(69)A2>0,

and

(70)u(x, t)=(−A2sec2(−A2(x+t(2ak+α))2)A3±−A2A4(1±tan(−A2(x+t(2ak+α))2)))1n

which are valid for

(71)A2<0, A4>0.

The special case with n=1 for dual power law nonlinearity reduces to the parabolic law nonlinearity. Dual power law nonlinearity and parabolic law nonlinearity of the solution process are similar, so we omit the latter here.

4 Discussion and Conclusions

The dynamics of soliton propagation through optical fibres for the perturbed NLSE with two forms of nonlinear optical fibers was studied in detail in this paper. They are the power law nonlinearity and the dual power law nonlinearity. For the first case, the Kudryashov method was applied, which led to new kink solitons. The trial solution method and the tanh function method for this law gave new periodic solutions. For dual power law, only two algorithms were applied, which were the Kudryashov method and the trial solution method, and both these also retrieved new kink solitons and new periodic solutions. In using these methods, the parameter relationships for m and n were required to allow integrability of the perturbed NLSE. These results, which have not been reported previously, are immensely helpful for additional investigation of the perturbed NLSE, which is under way. However, we are considering whether there is still an effective method to investigate the perturbed NLSE with nonlinear optical fibres for future study. These results will be expected to eventually change the future of optical fibre communication technology.