From the point of view of three famous and important techniques we will achieve new expectations for the soliton configurations to the (2+1)-dimensional generalized nonlinear Schrödinger equation (GNLSE) with four waves mixing (FWM). The suggested model describes propagation of solitons in birefringent fiber. The FWM governed effectively the performance of the resultant soliton amplitudes. The three famous methods candidates for this purpose are the extended direct algebraic method (EDAM), the extended simple equation method(ESEM) and the solitary wave ansatz method (SWAM). The three techniques are implemented successively for the suggested model successfully. Surprise expectations for solitons via these three techniques to this model which weren’t achieved previously by any other authors who used other techniques have been demonstrated.

In this work, we will design unexpected configurations for the optical soliton propagation in lossy fiber system in presence the dispersion term solitons via two distinct and impressive techniques. The first one is the (G’/G)-expansion method, while the second is solitary wave ansatze method. The two methods are implemented in same vein and parallel. The obtained perceptions are new and weren’t achieved before. The comparison between our achieved visions and that achieved by other authors who used different schemas has been documented.

From point of view of two different schemas several new impressive lump solutions to (1+1)-dimensional Ito equation have been established. The first schema is the Paul-Painleve approach method (PPAM) which will be applied perfectly to extract multiple lump solutions of this model, while the second schema is the famous one of the ansatze method and has personal profile named the Ricatti-Bernolli Sub-ODE method. In related subject the numerical solutions corresponding to all lump solutions achieved via each method have been demonstrated individually in the framework of the variational iteration method (VIM).

In this article, the Paul-Painleve approach (PPA) which discovered recently and built on the balance role has been used perfectly to achieve new impressive solitary wave solutions to the nano-soliton of ionic waves (NSOIW) propagating along microtubules in living cells. In addition, the variational iteration method (VIM) has been applied in the same vein and parallel to establish the numerical solutions of this model.

In this article, imperssive exact solutions and hence effective regulations to the non-fractional order and the time-fractional order of the biological population models are achevied for the first time in the framwork of the Paul-Painlevé approach. When the variables appearing in the exact solutions take specific values, the solaitry wave solutions will be easily satisfied.The realized results prove the efficiency of this technique.

In this article, we employ the nonlinear complex Hirota-dynamical model which is one of the famous and important standards to the nonlinear Schrödinger equation in which the third derivative term represent the self-interaction in the high-frequency subsystem. Specially, in plasma this term is isomorphic to the so known self-focusing effect. The bright, dark and periodic optical soliton solutions to this equation will realized successfully for the first time in the framework of the solitary wave ansatz method. Furthermore, in this connection at the same time and parallel the extended simple equation method has been applied successfully to achieve new impressive solitary wave solutions to this model. A comparison between the obtained results and that satisfied in previous work has been established.