Modulation stability analysis of exact multidimensional solutions to the generalized nonlinear Schrödinger equation and the Gross-Pitaevskii equation using a variational approach.

We analyze the modulation stability of spatiotemporal solitary and traveling wave solutions to the multidimensional nonlinear Schrödinger equation and the Gross-Pitaevskii equation with variable coefficients that were obtained using Jacobi elliptic functions. For all the solutions we obtain either unconditional stability, or a conditional stability that can be furnished through the use of dispersion management.

Analytical chirped solutions to the (3 + 1)-dimensional Gross-Pitaevskii equation for various diffraction and potential functions.

Analytical solutions to the (3 + 1)-dimensional Gross-Pitaevskii equation in the presence of chirp and for different diffraction and potential functions are found. We utilize a method we formulated to solve the Riccati equation for the chirp function that arises when the F-expansion technique and the homogeneous balance principle are applied to the Gross-Pitaevskii equation. Three specific examples of physical interest are considered in some detail.

Analytical traveling-wave and solitary solutions to the generalized Gross-Pitaevskii equation with sinusoidal time-varying diffraction and potential.

We determine analytical extended traveling-wave and spatiotemporal solitary solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation with time-dependent coefficients, for the sinusoidally time-varying diffraction and quadratic potential strength. A number of periodic and localized solutions are obtained whose intensity does not decrease in time in the absence of externally induced gain or loss. Stability analysis of our solitary solutions is carried out, to display their modulational stability.

Exact traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Schrödinger equation with polynomial nonlinearity of arbitrary order.

We obtain exact traveling wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with variable coefficients and polynomial Kerr nonlinearity of an arbitrarily high order. Exact solutions, given in terms of Jacobi elliptic functions, are presented for the special cases of cubic-quintic and septic models. We demonstrate that the widely used method for finding exact solutions in terms of Jacobi elliptic functions is not applicable to the nonlinear Schrödinger equation with saturable nonlinearity.

Spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation.

Exact extended traveling wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Gross-Pitaevskii equation with time-dependent coefficients are obtained. The case with constant diffraction and parabolic potential strength, but with variable gain, is discussed in some detail. It is found that gain in the system is necessary for the appearance of stable solitons.

Exact spatiotemporal wave and soliton solutions to the generalized (3+1)-dimensional Schrödinger equation for both normal and anomalous dispersion.

We obtain exact extended traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equations for both the normal and the anomalous dispersion.