The present paper studies the unstable nonlinear Schrödinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schrödinger equation and its modified form are analytically solved using two efficient distinct techniques, known as the modified Kudraysov method and the sine-Gordon expansion approach. As a result, a wide range of new exact traveling wave solutions for the unstable nonlinear Schrödinger equation and its modified form are formally obtained.

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Controlling of a rotational inverted pendulum is considered as a challenging problem, mainly due to the system’s inherent nonlinear and unstable dynamics. In fact, the goal of this control is to maintain the pendulum vertically upward regardless of external disturbances. This paper aims to optimally design a model reference adaptive proportional integral derivative (PID) control for rotary inverted pendulum system based on a novel hybrid particle swarm optimization algorithm, combining sine cosine algorithm and levy flight distribution. Evaluation of the performance quality of the proposed adaptive controller is accomplished based on the stabilization and tracking control of rotary inverted pendulum system. In addition, two other PID controllers are designed to get a better understanding of the performance and robustness of the proposed controller. To make a complete comparison, the performance of the hybrid particle swarm optimization algorithm is examined against two other optimization techniques known as simple particle swarm optimization and whale optimization algorithm. Finally, the obtained simulation results demonstrate that the proposed optimal adaptive controller is superior to the other controllers, especially in terms of the transient response characteristics and the magnitude of control output signal.

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This work aimed to improve the physicochemical properties of a low carbon steel C15 by a nitriding treatment in a salt bath at 580• C. The micrographs of treated sample show that the nitriding treatment causes significant structural changes, it is allowed to delineate the nitriding layer. The X-ray diffraction and the Raman spectroscopy of the treated samples permitted to identify different nitrides (Fe2−3N, Fe4N) formed. The obtained hardness profile determines the nitriding depth. Potentiodynamic curves show that the corrosion current density of treated C15 is the 10 times lower than untreated specimen. The values of impedance parameters obtained after nitriding treatment indicate that the resistance values and exponent n increase, whereas the capacitance decreases. We concluded that the nitriding treatment is an effective method for improvement of the corrosion behavior and surface properties of the low alloy steel C15.

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