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We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)-dimensional cubic-quintic nonlinear Schrödinger equation with spatial distributed coefficients. For restrictive parameters, these periodic wave solutions acquire the form of localized spatial solitons. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and gain (or loss). We then demonstrate the nonlinear tunneling effects and controllable compression technique of three-dimensional bright and dark solitons when they pass unchanged through the potential barriers and wells affected by special choices of the diffraction and/or the nonlinearity parameters. Direct numerical simulation has been performed to show the stable propagation of bright soliton with 5% white noise perturbation.

In this paper, we obtain the exact bright and dark soliton solutions for the nonlinear Schrödinger equation (NLSE) which describes the propagation of femtosecond light pulses in optical fibers in the presence of self-steepening and a self-frequency shift terms. The solitary wave ansatz method is used to carry out the derivations of the solitons. The parametric conditions for the formation of soliton pulses are determined. Using the one-soliton solution, a number of conserved quantities have been calculated for Hirota and Sasa–Satsuma cases and finally, we have constructed some periodic wave solutions by reducing the higher order nonlinear Schrödinger equation (HNLS) to quartic anharmonic oscillator equation. The obtained exact solutions may be useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a HNLS model equation.

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