In this article, several kinds of wave solutions such as solitons, solitary waves and Jacobi elliptic function solutions in which many are novel of nonlinear fractional partial differential equations (NLFPDEs) in mathematical physics, has obtained with the aid of improved extended auxiliary equation method. The efficiency of the current technique is demonstrated by applications to three NLFPDEs, namely, (2+1)-dimensional space-time fractional Zoomeron equation, space-time fractional Benjamin–Bona–Mahony and space-time fractional modified third-order KdV equations. Several kinds of exact solutions are constructed which have key applications in applied sciences. The geometrical shapes for some of the obtained results are depicted for various choices of the free parameters that appear in the results. The resulting numerous solutions will also be helpful in applied mechanics and various fields of sciences. This powerful technique can be applied to other wave models that can arise in mathematical physics.

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The unstable nonlinear Schrödinger equations (UNLSEs) are universal equations of the class of nonlinear integrable systems, which reveal the temporal changing of disruption in slightly stable and unstable media. In current paper, an improved auxiliary equation technique is proposed to obtain the wave results of UNLSE and modified UNLSE. Numerous varieties of results are generated in the mode of some special Jacobi elliptic functions and trigonometric and hyperbolic functions, many of which are distinctive and have significant applications such as pulse propagation in optical fibers. The exact soliton solutions also give information on the soliton interaction in unstable media. Furthermore, with the assistance of the suitable parameter values, various kinds of structures such as bright-dark, multi-wave structures, breather and kink-type solitons, and several periodic solitary waves are depicted that aid in the understanding of the physical interpretation of unstable nonlinear models. The various constructed solutions demonstrate the effectiveness of the suggested approach, which proves that the current technique may be applied to other nonlinear physical problems encountered in mathematical physics.

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The Kaup–Newell equation is used to model sub-picoseconds pulses that travel throughout optical fibers. The fractional-order perturbed Kaup–Newell model, which represents extensive waves parallel to the field of magnetic, is examined. In this paper, two analytical techniques named, improved F-expansion and generalized exp[Formula: see text]-expansion techniques, are employed and new analytical solutions in generalized forms like bright solitons, dark solitons, multi-peak solitons, peakon solitons, periodic solitons and further wave results are assembled. These soliton solutions and other waves findings have important applications in applied sciences. The configurations of some solutions are shown in the form of graphs through assigning precise values to parameters, and their dynamics are described. The illustrated novel structures of some solutions also assist engineers and scientists in better grasping the physical phenomena of this fractional model. A comparison analysis has been given to explain the originality of the current findings compared to the previously achieved results. The results of computer simulations show that the procedures described are effective, simple, and efficient.