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A generalized variable-coefficient Gardner equation with an external-force term is investigated which can model the propagation and interaction of some nonlinear waves in a fluid or plasma. Via the Hirota bilinear method, we have obtained some bilinear forms using a dependent variable transformation under certain constraints. The bilinear Bäcklund transformations, Lax pair and Lax-type Bäcklund transformations have also been constructed via the bilinear forms. By virtue of the extended variable-coefficient homogeneous balance method and truncated Painlevé expansion, we have obtained the same Bäcklund transformations and three different kinds of the analytical solutions. Additionally, profiles of some types of the obtained solutions are illustrated graphically. Graphic analysis on those waves shows that: (i) For the soliton-like and rational waves, the characteristic lines and velocities are related to the dispersive, dissipative, perturbed and external-force coefficients; the backgrounds are related to the perturbed and external-force coefficients; (ii) For the periodic waves, the velocities are related to the dispersive, dissipative, perturbed and external-force coefficients; the amplitudes are related to the dispersive, perturbed and external-force coefficients, while the backgrounds are related to the perturbed and external-force coefficients.

Acoustic waves on a crystal lattice, long internal waves in a density-stratified ocean, ion acoustic waves in a plasma, and shallow-water waves with weakly non-linear restoring forces are all represented mathematically by the KdV equation. Its importance and wide range of applications have led to the development and analysis of multiple solutions in the scientific community. Beside those in this article we prove the existence of superposed solutions of KdV equation. Some theorems and corollary on the existence of superposed and superposed-type solutions for KdV equations with variable coefficients are presented in this article. The six sets of superposed solutions to the variable coefficient KdV equation are obtained by using the corollary and theorem on the existence of superposed solutions. It was demonstrated that superposed solutions of the KdV problem with variable coefficients can be constructed by combining two elementary solutions that contain reciprocal Jacobi elliptic functions. Additionally, we present a few theorems and corollaries about the existence of superposed-type solutions for this equation in the literature. The most significant and fascinating of them all is the splitting technique theorem. We obtained many superposed-type solutions of KdV equations with variable coefficients in terms of the Jacobi elliptic function by using the splitting technique. It is additionally confirmed that the generalised Miura transformation is a sub-case of the splitting procedure. This represents an additional modification to the generalised Miura transformation. These theorems explain why a number of seemingly bizarre superposition-type solutions to a number of newly published nonlinear equations have appeared. It is further demonstrated that the solutions produced by the generalised Miura transformation are specific examples of solutions obtained through the application of the splitting technique. Plots in two dimensions, three dimensions, contour, and density have all been used to illustrate the features of the derived solutions.

Plasmas and fluids are of current interest, supporting a variety of wave phenomena. Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a generalized (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation for the nonlinear phenomena in a plasma or fluid. Based on the existing bilinear form, N -soliton solutions in the Gramian are derived, where N  = 1, 2, 3…. With N  = 3, three-soliton solutions are constructed. Fission and fusion for the three solitons are presented. Effects of the variable coefficients, i.e. h ( t ), l ( t ), q ( t ), n ( t ) and m ( t ), on the soliton fission and fusion are revealed: soliton velocity is related to h ( t ), l ( t ), q ( t ), n ( t ) and m ( t ), while the soliton amplitude cannot be affected by them, where t is the scaled temporal coordinate, h ( t ), l ( t ) and q ( t ) give the perturbed effects, and m ( t ) and n ( t ), respectively, stand for the disturbed wave velocities along two transverse spatial coordinates. We show the three parallel solitons with the same direction.

The variable coefficient nonlinear Schrödinger equation has a wide range of applications in various research fields. This work focuses on the wave propagation based on the variable coefficient nonlinear Schrödinger equation and the variable coefficient fractional order nonlinear Schrödinger equation. Due to the great challenge of accurately solving such problems, this work considers numerical simulation research on this type of problem. We innovatively consider using a mesoscopic numerical method, the lattice Boltzmann method, to study this type of problem, constructing lattice Boltzmann models for these two types of equations, and conducting numerical simulations of wave propagation. Error analysis was conducted on the model, and the convergence of the model was numerical validated. By comparing it with other classic schemes, the effectiveness of the model has been verified. The results indicate that lattice Boltzmann method has demonstrated advantages in both computational accuracy and time consumption. This study has positive significance for the fields of applied mathematics, nonlinear optics, and computational fluid dynamics.

Friction coefficient depends on various factors or surface characteristics during tribological testing, and this friction coefficient can be modified by altering the properties of one of the two contacting surfaces. It is crucial to monitor the friction coefficient continuously, not only at the conclusion of the test. This research examined the evolution of friction coefficient of silicon nitride (Si 3 N 4 ) coating and H13 steel over different sliding distances (250, 500, 750, 1000 m). The study assessed surface wear and oxidation through three-dimensional profilometry and SEM/EDX. The findings indicated a reduction in friction coefficient by 22%, a decrease in wear rate by 88%, and a reduction in wear volume by 87% when comparing the silicon nitride coated steel to the uncoated steel. Furthermore, the changes in friction coefficient provided insights into the timing of the complete fracture of the hard coating. Graphical abstract

In this paper we use simulations to examine how endogeneity biases the results reported by ordinary least squares (OLS) regression. In addition, we examine how instrumental variable techniques help to alleviate such bias. Our results demonstrate severe bias even at low levels of endogeneity. Our results also illustrate how instrumental variables produce unbiased coefficient estimates, but instrumental variables are associated with extremely low levels of statistical power. Finally, our simulations highlight how stronger instruments improve statistical power and that endogenous instruments can report results that are inferior to those reported by OLS regression. Based on our results, we provide a series of recommendations for scholars dealing with endogeneity.

In this paper the classification of single traveling wave solutions of (1+1)\\((1+1)\\) dimensional Gardner equation with variable coefficients is obtained by applying the complete discrimination system to the polynomial and trial equation methods. In particular, the corresponding solutions for the concrete parameters are constructed to show that each solution in the classification can be realized. Moreover, numerical simulations shown in the paper could help us better understand the nature of each solution.

We consider estimation of and inference about coefficients on endogenous variables in a linear instrumental variables model where the number of instruments and exogenous control variables are each allowed to be larger than the sample size. We work within an approximately sparse framework that maintains that the signal available in the instruments and control variables may be effectively captured by a small number of the available variables. We provide a LASSO-based method for this setting which provides uniformly valid inference about the coefficients on endogenous variables. We illustrate the method through an application to demand estimation.

Studied in this paper is a ( 3 + 1 )-dimensional nonlinear Schrödinger equation with the group velocity dispersion, fiber gain-or-loss and nonlinearity coefficient functions, which describes the evolution of a slowly varying wave packet envelope in the inhomogeneous optical fiber. With the Hirota method and symbolic computation, the bilinear form and dark multi-soliton solutions under certain variable-coefficient constraint are derived. Interactions between the different-type dark two solitons have been asymptotically analyzed and presented. Both velocities and amplitudes of the two linear-type dark solitons do not change before and after the interaction. The two parabolic-type dark solitons propagating with the opposite directions both change their directions after the interaction. Interaction between the two periodic-type dark solitons is also presented. Interactions between the linear-, parabolic- and periodic-type dark two solitons are elastic.

The double-disc opener of maize precision seeder is an important component which affects sowing quality. After the double-disc opening operation, there will be many unfavorable phenomena such as a W-shaped bottom with pointed ridge, returning soil to furrow, loose and rough furrow sidewall, and large soil blocks in the furrow bottom. These phenomena often cause the problems of poor sowing depth consistency and seed spacing uniformity. In order to solve the above problems, the furrow compaction device with opener was designed to compact and reshape the original seed furrow, eventually forming a smooth and flat V-shaped seed furrow. Through theoretical calculations and kinematic analysis, the main structural parameters of the device were limited to a small range: the spring stiffness coefficient ⅛=0.96-4.19 N/mm and the angle of the furrow compaction wheel ф=30°-60°. In the soil-bin experiment, the rotary combination design was adopted to study the effects of the parameters of the furrow compaction device with opener on the seeds location variation. The regression model of two factors with respect to each indicator was established in the Design-Expert software, revealing the effects of two factors on the indicators. Finally, the optimal structural parameters obtained were: the spring stiffness coefficient ⅛=4.0 N/mm, and the angle of furrow compaction wheel ф=42.4°. The field test was carried out to verify the effect of the furrow compaction device with opener on the performance of precision seeder. The results showed that the average values of the sowing depth variable coefficient, the lateral deviation and the seed spacing variable coefficient respectively were 5.77%, 5.1 mm and 9.54% in the treatment of the furrow compaction device with opener. All indicators were superior to the traditional double-disc opener. This research can provide references for the design of furrow opening device and maize precision seeder.