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Graphical abstract Interactions between solitons can be controlled through adjusting the phase shift of solitons. Highlights • Effects of the reciprocal of group velocity on solitons were discussed. • Energy exchange of solitons occured during the phase shift. • Solitons in a bound state were amplified or attenuated. • Parabolic soliton interactions were analysed to decrease the interactions. • Parabolic solitons can be reduced to dromion-like structures. Abstract In nonlinear optics, the soliton transmission in different forms can be described with the use of nonlinear Schrödinger (NLS) equations. Here, the soliton transmission is investigated by solving the NLS equation with the reciprocal of the group velocity β 1 (z) , the group velocity dispersion coefficient β 2 (z) and nonlinear coefficient γ (z). Two-soliton solutions for the NLS equation are obtained through the Hirota method. According to the solutions obtained, β 1 (z) and γ (z) with different function forms are taken to study the characteristics of solitons. The effect of the phase shift on the soliton interaction is discussed, and the non-oscillating soliton amplification, which is transmitted in a bound state, is explored. Parabolic solitons with oscillations are analysed. Moreover, parabolic solitons can be reduced to dromion-like structures. Results indicate that the transmission of solitons can be adjusted with the group velocity dispersion and Kerr nonlinearity coefficients. The phase shift, amplification, oscillation and attenuation of solitons can also be controlled by other related parameters. This work accomplishes the theoretical study of transmission characteristics of optical solitons in spatially dependent inhomogeneous optical fibres. The conclusions of this research have theoretical guidance for the research of optical amplifier, all-optical switches and mode-locked lasers. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method based on Adomian decomposition method is applied for solving Burgers’ equation with Dirichlet and Neumann boundary conditions. This mathematical-physical model describes the propagation of waves in nonlinear optical media. Finally, some illustrative examples are presented to show the efficiency of the proposed method and the results obtained by this method are compared by the exact solution. [ABSTRACT FROM AUTHOR]

In this work, we utilize the improved modified extended tanh method to get optical solitons and other exact solutions for (3+1) dimensional Sasa–Satsuma model with third order dispersion that describe the dynamics of ultrashort pulses in the optical fibers. Many exact solutions are raised with the aid of the proposed method. These solutions include bright solitons, dark solitons, singular solitons, periodic wave solutions, hyperbolic wave solutions, Weierstrass elliptic solutions and Jacobi elliptic solutions. [ABSTRACT FROM AUTHOR]

This study discusses the soliton and other solutions for a weakly nonlocal Schrödinger equation of the fifth order involving nonlinearity of the parabolic law and an external potential. We use solitary wave ansatz of different forms to obtain bright, dark, singular, periodic soliton and exponential solutions. The results obtained can be useful in understanding and interpreting the wave dynamics of the proposed model. [ABSTRACT FROM AUTHOR]

The search for soliton solutions of nonlocal Schrödinger equations in the presence of nonlinear effects has received special interest in the last few decades. In the present paper, a weakly nonlocal Schrödinger equation (WNSE) involving the parabolic law nonlinearity is considered and as a success, its soliton solutions including bright and dark solitons are constructed. Such a key goal is formally carried out using a series of efficient approaches such as Kudryashov and exponential methods (Their new versions). The results presented in this paper are of a significant role to describe the propagation of soliton waves in the weakly nonlocal parabolic law medium. [ABSTRACT FROM AUTHOR]

This paper examines new travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) model with Kerr nonlinearity using Bäcklund transformation method based on Riccati equation, Kudryashov method and a new auxiliary ordinary differential equation (ODE). The three methods are adequately utilised, and some new rational-type hyperbolic and trigonometric function solutions are derived in different shapes for the aforementioned model. We confirm that our methods are more efficient than the other methods and it might be used in many other such types of nonlinear equations arising in the basic fabric of communications network technology and nonlinear optics. [ABSTRACT FROM AUTHOR]

This work presents novel perception into the optical solitons of the cubic-quartic nonlinear Schrödinger's equation (CQ-NLSE) by introducing the cubic-quintic-septic-nonic (CQSN) form of self-phase modulation (SPM). The proposed model has a significant role in physics and engineering and it is considered the equation of pulses dissemination in optical communication. To extract the optical solitons such as the bright, kink, periodic and singular solitons, the Kudryashov's method and extended hyperbolic function method (EHFM) are applied for the first time. The results show that these methods are suitable, productive and competent to find the solutions for such kind of nonlinear evolution equations (NLEEs) in optical fibers and nonlinear sciences. Moreover, for physical illustration of obtained solutions, some 3d, contour and 2d graphs are plotted which shed light on the differences between the solutions obtained through Kudryashov's method and extended hyperbolic function method (EHFM). Furthermore, comparison section is also given to prove the novelty of the work over the reported results. [ABSTRACT FROM AUTHOR]

This paper studies the resonant optical solitons that comes with anti-cubic nonlinearity in presence of Hamiltonian perturbation terms. The extended trial function method is employed to retrieve cnoidal waves, bright and singular soliton solutions as well as periodic singular solutions to the model. [ABSTRACT FROM AUTHOR]

This paper obtains dispersive dark and singular optical solitons, governed by Schrödinger–Hirota equation, in optical fibers. The integration algorithm is the modified simple equation method. Both Kerr law and power laws of nonlinearity are considered. [ABSTRACT FROM AUTHOR]

This paper obtains exact bright, dark and singular dispersive optical soliton solutions to DWDM system that is studied with spatio-temporal dispersion in addition to group-velocity dispersion. The governing model is Schrödinger–Hirota equation. The method of undetermined coefficients is employed to secure these soliton solutions that appear with necessary constraints for their existence. [ABSTRACT FROM AUTHOR]

This paper applied modified simple equation method to extract soliton solutions to birefringent fibers, in presence of perturbation terms. Both Kerr law and parabolic law of nonlinearity are studied. Dark and singular optical soliton solutions are listed for these nonlinearities. The existence of these solitons is guaranteed with appropriate constraints that are also presented. [ABSTRACT FROM AUTHOR]

The new (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation, which was introduced by Wazwaz and El-Tantawy in 2016, has been studied analytically. There are two tools of integration are employed to solve this mathematical–physical model. First, We reported the solitary wave solutions including the dark and singular solitons using the generalized Kudryashov method. Then we obtained the hyperbolic traveling wave solutions, trigonometric traveling wave solutions and rational function solutions with the help of the G ′/ G -expansion method, these solutions can be reduced solitary wave solutions and periodic wave solutions under some particular cases. [ABSTRACT FROM AUTHOR]

In nonlinear optics, the soliton transmission in different forms can be described with the use of nonlinear Schrödinger (NLS) equations. Here, the soliton transmission is investigated by solving the NLS equation with the reciprocal of the group velocity β 1 ( z ) , the group velocity dispersion coefficient β 2 ( z ) and nonlinear coefficient γ ( z ) . Two-soliton solutions for the NLS equation are obtained through the Hirota method. According to the solutions obtained, β 1 ( z ) and γ ( z ) with different function forms are taken to study the characteristics of solitons. The effect of the phase shift on the soliton interaction is discussed, and the non-oscillating soliton amplification, which is transmitted in a bound state, is explored. Parabolic solitons with oscillations are analysed. Moreover, parabolic solitons can be reduced to dromion-like structures. Results indicate that the transmission of solitons can be adjusted with the group velocity dispersion and Kerr nonlinearity coefficients. The phase shift, amplification, oscillation and attenuation of solitons can also be controlled by other related parameters. This work accomplishes the theoretical study of transmission characteristics of optical solitons in spatially dependent inhomogeneous optical fibres. The conclusions of this research have theoretical guidance for the research of optical amplifier, all-optical switches and mode-locked lasers.