Lie symmetry analysis and invariant solutions of (3+1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this paper, the (3+1)-dimensional constant coefficient of Date–Jimbo–Kashiwara–Miwa (CCDJKM) equation is studied. All of the vector fields, infinitesimal generators, Lie symmetry reductions and different similarity reduction solutions are constructed. Due to the arbitrary functions in the infinitesimal generators, the (3+1)-dimensional CCDJKM equation can further be reduced to many (2+1)-dimensional partial differential equations. The explicit solutions of the similarity reduction equations, which include the quasi-periodic wave solution, the interaction solution between the periodic wave and a kink soliton and the trigonometric function solutions, are constructed with proper arbitrary function selection, and these new exact solutions are given out analytically and graphically.

New interaction solutions of the similarity reduction for the integrable (2+1)-dimensional Boussinesq equation

The nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation is studied by the standard truncated Painlevé expansion. This nonlocal symmetry can be localized to the Lie point symmetry of the prolonged system by introducing two auxiliary dependent variables. The corresponding finite symmetry transformation and similarity reduction related to the nonlocal symmetry of the new integrable [Formula: see text]-dimensional Boussinesq equation are studied. The rational solution, the triangle solution, two solitoff-interaction solution and the soliton–cnoidal interaction solutions for the new [Formula: see text]-dimensional Boussinesq equation are presented analytically and graphically by selecting the proper arbitrary constants.

Lie Symmetry Analysis for Soliton Solutions of Generalised Kadomtsev-Petviashvili-Boussinesq Equation in (3+1)-dimensions

The Lie group of infinitesimal transformations technique and similarity reduction is performed for obtaining an exact invariant solution to generalized Kadomstev-Petviashvili-Boussinesq (gKPB) equation in (3+1)-dimensions. We obtain generators of infinitesimal transformations, which provide us a set of Lie algebras. In addition, we get geometric vector fields, a commutator table of Lie algebra, and a group of symmetries. It is observed that the analytic solution (closed-form solutions) to the nonlinear gKPB evolution equations can easily be treated employing the Lie symmetry technique. A detailed geometrical framework related to the nature of the solutions possessing traveling wave, bright and dark soliton, standing wave with multiple breathers, and one-dimensional kink, for the appropriate values of the parameters involved.

Nonlocal Symmetry, Painlevé Integrable and Interaction Solutions for CKdV Equations

In this paper, we provide a method to construct nonlocal symmetry of nonlinear partial differential equation (PDE), and apply it to the CKdV (CKdV) equations. In order to localize the nonlocal symmetry of the CKdV equations, we introduce two suitable auxiliary dependent variables. Then the nonlocal symmetries are localized to Lie point symmetries and the CKdV equations are extended to a closed enlarged system with auxiliary dependent variables. Via solving initial-value problems, a finite symmetry transformation for the closed system is derived. Furthermore, by applying similarity reduction method to the enlarged system, the Painlevé integral property of the CKdV equations are proved by the Painlevé analysis of the reduced ODE (Ordinary differential equation), and the new interaction solutions between kink, bright soliton and cnoidal waves are given. The corresponding dynamical evolution graphs are depicted to present the property of interaction solutions. Moreover, With the help of Maple, we obtain the numerical analysis of the CKdV equations. combining with the two and three-dimensional graphs, we further analyze the shapes and properties of solutions u and v.

Invariant Solutions and Bifurcation Analysis of the Nonlinear Transmission Line Model

In this paper, the nonlinear transmission line model with the power law nonlinearity and the constant capacitance and voltage relationship is studied using Lie symmetry analysis. Corre- sponding to the infinitesimals obtained, using commutation relations, abelian and non abelian Lie subalgebras are obtained. Also, using the adjoint table, the one dimensional optimal system of subalgebra is presented. Based on the optimal system, corresponding Lie symmetry reduc- tions are obtained. Moreover, variety of new similarity solutions in the form of trigonometric functions, hyperbolic functions are obtained. Corresponding to one similarity reduction, by bifurcation of dynamical system, the stable and unstable regions are determined, which show the existence of soliton solutions from the nonlinear dynamics view point. Some of the obtained solutions are represented graphically and observations are also discussed.

Development of a Multiphase Flamelet Generated Manifold for Spray Combustion Simulations

In this study, a model flame of quasi-one-dimensional (1D) counterflow spray flame has been developed. The two-dimensional (2D) multiphase convection-diffusion-reaction equations have been simplified to one dimension using similarity reduction under the Eulerian framework. This model flame is able to directly account for nonadiabatic heat loss, preferential evaporation, as well as multiple combustion regimes present in realistic spray combustion processes. A spray flamelet library was generated based on the model flame. To retrieve data from the spray flamelet library, the enthalpy was used as an additional controlling variable to represent the interphase heat transfer, while the mixing and chemical reaction processes were mapped to the mixture fraction and the progress variable. The spray flamelet generated manifolds (SFGM) approach was validated against the results from the direct integration of finite rate chemistry as a benchmark. The SFGM approach was found to give a better performance in terms of predictions of temperature and species mass fractions.

Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3 + 1)-dimensional extended quantum Zakharov–Kuznetsov equation in quantum physics

A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for obtaining the exact solution of some reduced transform equations. Finally, by invoking the new conservation theorem developed by Nail H. Ibragimov, the conservation laws of QZK equation have been derived.

A deep learning framework combined with word embedding to identify DNA replication origins

The DNA replication influences the inheritance of genetic information in the DNA life cycle. As the distribution of replication origins (ORIs) is the major determinant to precisely regulate the replication process, the correct identification of ORIs is significant in giving an insightful understanding of DNA replication mechanisms and the regulatory mechanisms of genetic expressions. For eukaryotes in particular, multiple ORIs exist in each of their gene sequences to complete the replication in a reasonable period of time. To simplify the identification process of eukaryote’s ORIs, most of existing methods are developed by traditional machine learning algorithms, and target to the gene sequences with a fixed length. Consequently, the identification results are not satisfying, i.e. there is still great room for improvement. To break through the limitations in previous studies, this paper develops sequence segmentation methods, and employs the word embedding technique, ‘Word2vec’, to convert gene sequences into word vectors, thereby grasping the inner correlations of gene sequences with different lengths. Then, a deep learning framework to perform the ORI identification task is constructed by a convolutional neural network with an embedding layer. On the basis of the analysis of similarity reduction dimensionality diagram, Word2vec can effectively transform the inner relationship among words into numerical feature. For four species in this study, the best models are obtained with the overall accuracy of 0.975, 0.765, 0.885, 0.967, the Matthew’s correlation coefficient of 0.940, 0.530, 0.771, 0.934, and the AUC of 0.975, 0.800, 0.888, 0.981, which indicate that the proposed predictor has a stable ability and provide a high confidence coefficient to classify both of ORIs and non-ORIs. Compared with state-of-the-art methods, the proposed predictor can achieve ORI identification with significant improvement. It is therefore reasonable to anticipate that the proposed method will make a useful high throughput tool for genome analysis.