Wave profile analysis of a couple of(3+1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach

The(3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation,dynamics of shallow water wave,plasma fluid model,chemical kinematics,chemical engineering,geochemistry,and many other topics.In this article,we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it.As a result,wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations,containing kink,breather soliton,bell-shaped soliton,and periodic waves.To extract the soliton solutions of the high-dimensional nonlinear evolution equations,a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly.The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations.This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension(HNEEs).Closed-form solutions are carefully illustrated and discussed through diagrams.

Aluminum Expansion Approaches The Inflection Point,With The Core Of A New Round Of Overcapacity Cutting As Self-Built Power Plants

Shandong Development and Reform Commission recently announced in a documen that based on the enterprise self-inspection and provincial inspection results,it had ordered Shandong Weiqiao Pioneering Group and Xinfa Group to close 3.21 million tons o illegal aluminum production capacity by the end of July.Besides Shandong,other provinces and regions with high aluminum

The Approach of Compensation of Air Refractive Index in Thermal Expansion Coefficients Measurement Based on Laser Feedback Interferometry

We present the thermal expansion coefficient （TEC） measurement technology of compensating for the effect of variations in the refractive index based on a Nd： YA G laser feedback system, the beam frequency is shifted by a pair of aeousto-optic modulators and then the heterodyne phase measurement technique is used. The sample measured is placed in a muffle furnace with two coaxial holes opened on the opposite furnace walls. The measurement beams hit perpendicularly and coaxially on each surface of the sample. The reference beams hit on the reference mirror and the high-refiectivity mirror, respectively. By the heterodyne configuration and computing, the influences of the vibration, distortion of the sample supporter and the effect of variations in the refractive index are measured and largely minimized. For validation, the TECs of aluminum samples are determined in the temperature range of 29-748K, confirming not only the precision within 5 × 10-7 K-1 and the accuracy within 0.4% from 298K to 448K but also the high sensitivity non-contact measurement of the lower reflectivity surface induced by the sample oxidization from 448 K to 748 K.

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

Finite symmetry transformation group and localized structures of the (2+1)-dimensional coupled Burgers equation

In this paper, the finite symmetry transformation group of the （2＋1）-dimensional coupled Burgers equation is studied by the modified direct method, and with the help of the truncated Painleve′ expansion approach, some special localized structures for the （2＋1）-dimensional coupled Burgers equation are obtained, in particular, the dromion-like and solitoff-like structures.

New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrdinger Equations

The present paper studies the unstable nonlinear Schr¨odinger equations, describing the time evolution of disturbances in marginally stable or unstable media. More precisely, the unstable nonlinear Schr¨odinger 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 Schr¨odinger equation and its modified form are formally obtained.

Optimal control of thermoelastic beam vibrations by piezoelectric actuation

This paper presents the vibrations suppression of a thermoelastic beam subject to sudden heat input by a single piezoelectric actuator. An optimization problem is formulated as the minimization of a quadratic functional in terms of displacement and velocity at a given time and with the least control effort. The solution method is based on a combination of modal expansion and variational approaches. The modal expansion approach is used to convert the optimal control of distributed parameter system into the optimal control of lumped parameter system. By utilizing the variational approach, an explicit optimal control law is derived and the determination of the corresponding displacement and velocity is reduced to solving a set of ordinary differential equations. Numerical results are presented to demonstrate the effectiveness and the applicability of the proposed method.

A variety of novel closed-form soliton solutions to the family of Boussinesq-like equations with different types

This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion.The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary,kink,anti-kink,combo,and singular-periodic wave solutions.The attained solutions are expressed by the trigonometric and hyperbolic functions including tan,sec,cot,csc,tanh,sech,coth,csch,and of their combination.In addition,the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions.Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family.For the bright wave solutions in terms of Atangana’s conformable derivative,the amplitudes of the bright wave gradually decrease,but the smoothness increases when the fractional parametersαandβincrease.On the other hand,the periodicities of periodic waves increase.The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases.The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.