Nondegenerate solitons have been demonstrated in recent years that they potentially allow a further increase of information transmission rates in optical communication due to their stable double-hump structures. In this paper, the femtosecond nondegenerate solitons in optical fibers which are described by the coupled higher-order nonlinear Schrödinger equations, are investigated. Analytical nondegenerate soliton solutions are constructed by the developing Hirota method. And the constraints for stable double-hump structures are put forward. Furthermore, soliton molecules and asymmetric solitons are also found as the new types of nondegenerate solitons. The interactions between two nondegenerate solitons are studied in their propagation.

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With the aid of the Painlevé analysis, we obtain residual symmetries for a new (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the (3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.

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The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method. The auto-Bäcklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries. Furthermore, the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.