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In this paper, the exact solutions of generalized nonlinear Schrödinger (GNLS) equation are obtained by using Darboux transformation(DT). We derive some expressions of the 1-solitons, 2-solitons and n-soliton solutions of the GNLS equation via constructing special Lax pairs. And we choose different seed solutions and solve the GNLS equation to obtain the soliton solutions, breather solutions and rational wave solutions. Based on these obtained solutions, we consider the elastic interactions and dynamics between two solitons. [ABSTRACT FROM AUTHOR]

We investigate a variable-coefficient fractional nonlinear Schrödinger(vc-FNLS) equation with Wadati potential and PT-symmetric potential. We find the Lévy index can be used to transition from a breather wave to a soliton as the fractional order derivative is increasing. The influences of fractional α and β on the breather, dark and bright solitons of space–time vc-FNLS equation are analyzed in detail. Some different propagation dynamics are generated via the different parameters α and β. We find that the value of α decreases, the number of oscillations or singularities increases for small time values and decreases for large time values. And the stability of fundamental soliton and asymmetric soliton is explored via the linear stability analysis and direct propagations. Interestingly stability of the soliton is also explored against collisions with Wadati potential and PT-symmetric potential. To additionally explore robustness of the solitons in the vc-FNLS model, it is also relevant to show that they keep its integrity against "bombardment" by impinging waves. These states indicate that the interaction acts both repulsively and attractively, and the analysis can be extended for vc-FNLS equations with more than two different fractional-diffraction terms. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials. • We investigate a variable-coefficient fractional nonlinear Schrödinger equation. • The fractional order Lévy index can be used to transition from a breather wave to a solito. • Some stabilities and interactions of the novel fractional solutions are considered via numerical simulation. • The novel features of interaction dynamics are different from these usual features of solitons. [ABSTRACT FROM AUTHOR]

Some bright wave solutions of the coupled derivative nonlinear Schrödinger(CDNLS) equations are derived with Riemann–Hilbert(RH) method. We observe some novel non-autonomous bright soliton structures by choosing the different nonlinear perturbation function g (t). It is shown that there are some possibilities to manipulate the interactions of non-autonomous wave solution through choosing nonlinear and gain/loss functions. Some dynamic behaviors of interactions between two bright solitons have been asymptotically analyzed, the two parabolic-type bright solitons propagate with the opposite directions after the interaction. And some interactions of the linear-, parabolic- and periodic-type bright two-solitons are elastic. A general approach is presented to manipulate soliton waves, which has the potential application for the optical self-routing, non-Kerr media and Bose–Einstein condensates(BEC). [ABSTRACT FROM AUTHOR]

A high-order dispersive cubic-quintic Gross–Pitaevskii (HDCQGP) equation (a generalized variable coefficients nonlinear Schrödinger equation with the third and fourth-order and the cubic-quintic nonlinear terms) is considered, and is transformed into a standard cubic-quintic nonlinear Schrödinger equation (NLSE). By using the generalized tanh-function method, we study exact solutions of the HDCQGP equation with time-modulated potential and nonlinearity. In particular, based on the similarity transformation, we report several families of non-autonomous wave solutions of the HDCQGP equation with snaking behaviors and different amplitude surfaces. At last, we consider the numerical simulation of two solitons collision for the NLSE with different parameters. These results may raise the possibility of relative experiments and potential applications. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we construct explicit rogue wave solutions and dark-bright solutions for the inhomogeneous coupled nonlinear Schrödinger equation with variable coefficients by means of similarity transformations. Some explicit solutions to the coupled nonlinear Schrödinger equation with potentials and nonlinearities depending on both time and spatial coordinates are considered. We present the general approach and use it to calculate rogue wave solutions and dark-bright solutions for nonlinearities and potentials of physical interest in applications to Bose–Einstein condensates and nonlinear optics. [Copyright &y& Elsevier]

Abstract: In this paper, the rational solutions with arbitrary constants of a Hirota equation are presented employing the Darboux transformations. Then, the similarity transformation allows us to relate certain class of multi-rogue wave solutions of the higher-order nonlinear Schrödinger equation (NLSE) to the solutions of (1+1)-dimensional integrable Hirota equation. The multi-rogue wave solutions with arbitrary constants are considered. The obtained rogue wave solutions can be used to describe the possible formation mechanisms for optical and Bose–Einstein condensates. [Copyright &y& Elsevier]

Until now, the soliton solutions of nonlocal equations have also been put forward for the corresponding local equations. However, a natural motivation of our work is to find some soliton solutions to the mixed local–nonlocal equations that have not been reported for the associated nonlocal-type equations. A novel mixed local–nonlocal nonlinear Schrödinger (MLN-NLS) equation is presented, then we investigate the Hirota's bilinearization transformation and obtain the one- and two-soliton solutions of the MLN-NLS equation. Based on the obtained solutions, the propagation and interaction structures are shown, the broken and unbroken soliton solutions are derived with Hirota's transformation. We find that the MLN-NLS equation has some novel results, which is unlike the local or nonlocal cases. And these results seem to be the characteristic features of the mixed local–nonlocal nonlinear system in the present work. [ABSTRACT FROM AUTHOR]

We develop a nonstandard bilinearization procedure to generate more general bright soliton, breather soliton and quasi-rogue wave solutions for the PT -symmetric coupled nonlocal nonlinear Schrödinger (CNNLS) equations. We achieve some novel solutions by bilinearizing both the CNNLS equations and their associated auxiliary equations in a novel way. The soliton equations are written into bilinear operator forms by means of auxiliary equations, then the 1-soliton solution and 2-soliton solution of PT -symmetric CNNLS equations are constructed. Some novel interaction properties of the PT -symmetric two-soliton solutions are derived, which can present the potential applications to the soliton wave phenomena in nonlocal wave models. [ABSTRACT FROM AUTHOR]