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Improved modified Korteweg-de Vries (IMKdV) equation is shown to be non-integrable using Painlevé analysis. Exact travelling wave solutions are obtained using auto-Bäcklund transformation and Linearized transformation.

In this paper, we consider the (3 + 1)-dimensional time-fractional Schamel-Zakharov-Kuznetsov-Burgers (SZKB) equation. With the help of the Riemann-Liouville derivatives, the Lie point symmetries of the (3 + 1)-dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi-Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes.

A theoretical investigation of dust-acoustic solitary waves in one-dimensional, collisionless, and unmagnetized dusty plasma consisting of ion fluid, trapped as well as free electrons, and charge fluctuating immobile dust particles is considered. The nonlinear dynamics of dust ion-acoustic waves, whose phase speed is much smaller (larger) than the electron (ion) thermal speed, propagating in such a dusty plasma system is investigated. The reductive perturbation method is employed to reduce the basic set of fluid equations to the Schamel and Schamel-Korteweg-de Vries-Burger (S-KdVB) equations. The Schamel and Schamel-KdVB equations are shown to be non-integrable using Painlevé analysis. The Bäcklund transformations and some new exact solutions are formally derived. Finally, we discussed the results in this paper.

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In this paper, we constructed a traveling wave solutions expressed by three types of functions, which are hyperbolic, trigonometric, and rational functions. By using a fractional sub-equation method for some space-time fractional nonlinear partial differential equations (FNPDE), which are considered models for different phenomena in natural and social sciences fields like engineering, physics, geology, etc. This method is a very effective and easy to investigate exact traveling wave solutions to FNPDE with the aid of the modified Riemann–Liouville derivative.

The time-fractional Oskolkov-Benjamin-Bona-Mahony-Burgers (TF-OBBMB) equation is investigated in this paper. For this equation, we apply the Lie symmetry analysis to detect the symmetries and the vector fields provided using the definition of Riemann-Liouville (R-L) fractional derivatives. These symmetries allow us to construct the similarity reduction for the considered equation which converts it to a fractional ordinary differential (FOD) equation. Add to that, a set of solutions for the TF-OBBMB equation is obtained by the fractional sub-equation method. Also, we build a numerical solution by using the fractional Sumudu decomposition method in the sense of Caputo fractional derivatives accompanied by the absolute errors and the effect of the fractional-order α. Furthermore, we present a clear explanation for the physical meaning of both analytical and numerical solutions. Finally, we compute the conservation laws in detail in the light of the new conservation theorem.

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