1 Introduction

As the solution of the nonlinear evolution equations(NLEE), soliton has caused many interests, since it plays an important role in some fields of science and engineering such as the nonlinear optics, Bose-Einstein condensates, fluid mechanics and plasma physics.^[1–5] Soliton is self-localized, robust, long-lived, and preserves the identity, when it travels in a nonlinear, dispersive medium.^[6–7]

N-soliton solutions can be derived with the Hirota method and Darboux transformation, and they can be expressed as an N-th order polynomial in the N exponentials.^[8–10] However, it is difficult to verify such N-soliton solutions, but the solutions in the Wronskian-form are relatively easier to be differentiate.^[8–9,11] Therefore, the Wronskian solutions can be verified by virtue of the direct substitution into bilinear forms of the NLEE.^[11–12]

It has developed the classical soliton concept for the nonlinear and dispersive systems that have been autonomous, which means that the time is the only factor that plays the role of the independent variable, and the solitary waves in nonautonomous nonlinear and dispersive systems can propagate in the form of nonautonomous solitons.^[13–15] Under investigation in this paper is the following generalized nonautonomous nonlinear equation,^[16–18]

which describes the ultrashort optical pulse propagating in the nonlinear inhomogeneous fiber, where t and x are the retarded time and normalized distance, respectively, ψ(x,t) is the complex envelope of the electrical field in a comoving frame, the coefficients σ₁(x,t), σ₂(x,t) and σ₃(x,t) are the first-, second- and third-order dispersion parameters, respectively, ρ(x,t) is the Kerr nonlinear parameter, r(x,t) is the time-delaying effect parameter and v(x,t) is an inhomogeneous parameter related to the phase modulation and gain (loss).^[16] Darboux transformation and soliton solutions for Eq. (1) have been investigated analytically.^[16] Rogue-wave solutions for Eq. (1) have been derived via the generalized Darboux transformation.^[17] Bright one- and two-soliton solutions for Eq. (1) have been studied.^[18]

However, the N-soliton solutions for Eq. (1) with the Wronskian technique have not yet been reported yet, and the purpose of the present work will be to investigate the N-soliton solutions for Eq. (1). In Sec. 2, we will derive the explicit N-soliton solutions for Eq. (1) in the form of the double-Wronskian determinant. In Sec. 3, based on those solutions, the three-soliton solutions will be analyzed graphically.

2 Wronskian Form of the N-Soliton Solutions

Bilinear forms for Eq. (1) have been derived in Ref. [18]:

via the dependent variable transformation

under the constraints^[16]

where g(x,t) is a complex function of x and t, and f(x,t) is a real one, while α(x), β(x), γ(x), μ(x), and δ(x) being the functions of x, D_x and D_t being the bilinear differential operators.^[8]

The double-Wronskian determinant is defined as

with

where φ_n and ϕ_n (n = 1, 2, …, N + M) are all the complex functions of x and t, T represents the vector transpose, and N and M are both the integers. For convenience, W^N,M will be written as .

Considering the Lax pair of Eq. (1), we assume that

where

with

where ξ_j are all the complex constants. Then, we derive

Based on the Wronskian technique, it can be proved that g and f defined in the double-Wronskian-form (6) indeed satisfy the Bilinear forms (2). Firstly, the derivatives of g and f with respect to x and t can be given as follows:

Based on the double-Wronskian determinant identities in the Appendix A, substituting the derivatives of g and f into the Bilinear forms (2), we derive

where the bold type denotes the contributions from the second half of the determinant.

Up to now, we have verified that g and f defined in the doubele-Wronskian-form (6) indeed satisfy the Bilinear forms (2). Therefore, N-soliton solutions for Eq. (1) can be expressed as

3 Discussions and Conclusion

For N = 1, the bright one-soliton solution is given as

For N = 2, the two-soliton solutions are shown as

with

Similarly, we can obtain the expressions of the three- and four-soliton solutions, even the N-soliton solutions for Eq. (1).

Since the propagation of the one solitons and collisions between the two solitons have been investigated in Ref. [18], we will study the collisions among the three solitons in this section. Also, influences of σ₁(x,t), σ₂(x,t), σ₃(x,t) and v(x,t) on the collisions are graphically discussed.

In Fig. 1(a), we observe the head-on collision among the three solitons, where the solitons keep their shapes unchanged before and after the collision, which means the collision is elastc. When σ₁(x,t) is chosen as the function of x, head-on collision also occurs in Fig. 1(b), and amplitudes of the solitons do not alter, while the speed of one of the solitons changing, compared with Fig. 1(a). In Fig. 1(c), σ₂(x,t) is found to affect the amplitudes and speeds of the two of the solitons. Figure 2(a) reveals that the collision features of the solitons alter with σ₃(x,t) = −1.8x. As shown in Fig. 2(b), traveling directions of the three solitons are observed to be parallel.

In conclusion, a generalized nonautonomous nonlinear equation (i.e., Eq. (1)), which describes the ultrashort optical pulse propagating in the nonlinear inhomogeneous fiber, has been investigated analytically in this paper. Via the Wronskian technique, the bright N-soliton solutions in the double-Wronskian-form for Eq. (1) have been constructed under the Constraints (4). In this paper, we have only studied the collisions among the three solitons, as the properties of the one and two solitons have been investigated. Also, influences of σ₁(x,t), σ₂(x,t), σ₃(x,t) and v(x,t) on the collisions have been graphically analyzed. Our study may be helpful for investigating the soliton propagation and collision in the nonlinear inhomogeneous fiber theoretically and experimentally. Furthermore, higher-order effects such as the third-order dispersion and nonlinearity could lead to state transitions between breather and soliton,^[19–21] which will be the focus of our future research.