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Author(s):  
Guanqi Tao ◽  
Jalil Manafian ◽  
Onur Alp İlhan ◽  
Syed Maqsood Zia ◽  
Latifa Agamalieva

In this paper, we check and scan the (3+1)-dimensional variable-coefficient nonlinear wave equation which is considered in soliton theory and generated by considering the Hirota bilinear operators. We retrieve some novel exact analytical solutions, including cross-kink soliton solutions, breather wave solutions, interaction between stripe and periodic, multi-wave solutions, periodic wave solutions and solitary wave solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota bilinear forms and their generalized equivalences. Lastly, the graphical simulations of the exact solutions are depicted.


2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Xuejun Zhou ◽  
Onur Alp Ilhan ◽  
Fangyuan Zhou ◽  
Sutarto Sutarto ◽  
Jalil Manafian ◽  
...  

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.


Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin ◽  
Hanze Liu

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yuan Zhou ◽  
Botao Fa ◽  
Ting Wei ◽  
Jianle Sun ◽  
Zhangsheng Yu ◽  
...  

AbstractInvestigation of the genetic basis of traits or clinical outcomes heavily relies on identifying relevant variables in molecular data. However, characteristics such as high dimensionality and complex correlation structures of these data hinder the development of related methods, resulting in the inclusion of false positives and negatives. We developed a variable importance measure method, termed the ECAR scores, that evaluates the importance of variables in the dataset. Based on this score, ranking and selection of variables can be achieved simultaneously. Unlike most current approaches, the ECAR scores aim to rank the influential variables as high as possible while maintaining the grouping property, instead of selecting the ones that are merely predictive. The ECAR scores’ performance is tested and compared to other methods on simulated, semi-synthetic, and real datasets. Results showed that the ECAR scores improve the CAR scores in terms of accuracy of variable selection and high-rank variables’ predictive power. It also outperforms other classic methods such as lasso and stability selection when there is a high degree of correlation among influential variables. As an application, we used the ECAR scores to analyze genes associated with forced expiratory volume in the first second in patients with lung cancer and reported six associated genes.


2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Feng Zhang ◽  
Yuru Hu ◽  
Xiangpeng Xin

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.


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