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"exact solutions"
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Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation
A new method named bilinear neural network is introduced in this paper, and the corresponding tensor formula is proposed to obtain the exact analytical solutions of nonlinear partial differential equations (PDEs). This is the first time that the neural network model is used to find the exact analytical solution, and this method covers almost all methods of constructing a function after bilinearization to solve nonlinear PDEs. Furthermore, this method is most likely a universal method to obtain the exact analytical solutions of nonlinear PDEs. Abundant arbitrary functions solutions of the reduced p-gBKP equation are obtained by using this method. Various beautiful plots of the presented solutions, which show diversity of exact solutions to PDEs, are made. By choosing appropriate values and functions, the fractal solitons waves are obtained and the self-similar characteristics of these waves are observed by reducing the observation range and magnifying local images. Via various three-dimensional plots, the evolution characteristics of these waves are exhibited.
Journal Article
Dynamical analysis of diverse exact solutions, soliton surfaces and continuum limit theory for a semidiscrete Gardner equation
In this paper, our attention is given to the semidiscrete Gardner equation, which is a discrete analogue of the continuous Gardner equation describing the long wave propagation in a two-layer fluid. Firstly, a generalized (n,N-n)-fold Darboux transformation (DT) for this semi-discrete Gardner equation is constructed based on its recognized Lax pair. Secondly, the resulting DT is used to obtain exact solutions including ordinary soliton, rational soliton (RS) and their hybrid solutions within the non-zero seed background, and then analyze their asymptotic states as well as physical characteristics. Numerical simulations are also carried out to exhibit the dynamic characteristics of certain exact solutions. Thirdly, the soliton surface corresponding to this semi-discrete equation is investigated. Finally, employing continuum limit theory, we map the semi-discrete equation to the continuous equation, and obtain corresponding continuum limit for its Lax pair and DT. The findings given in this paper are conducive to a more profound understanding of the physical properties depicted by this equation.
Journal Article
Free-Fermion Models and Two-Dimensional Ising Models Under Zero Field and Imaginary Field i(π/2)kBT
The Ising model is famous in condensed matter and statistical physics. In this work we present a free-fermion formulation of the two-dimensional classical Ising models on honeycomb, triangular and Kagomé lattices. Each Ising model is studied in the cases of a zero field and of an imaginary field i(π/2)kBT. We employ the decorated lattice technique, star-triangle transformation, and weak-graph expansion method to exactly map each Ising model in both cases into an eight-vertex model on the square lattice. The resulting vertex weights are shown to satisfy the free-fermion condition. In the zero-field case, each Ising model is an even free-fermion model. In the case of the imaginary field, the Ising model on the honeycomb lattice is an even free-fermion model, while the models on the triangular and Kagomé lattices are odd free-fermion models. We obtain the exact solution of the Kagomé lattice Ising model under the imaginary field i(π/2)kBT, a result not previously reported in the literature. We also show that the frustrated Ising models on the triangular and Kagomé lattices in the imaginary field still exhibit a non-zero residual entropy.
Journal Article
The Exact Solutions of Stochastic Fractional-Space Kuramoto-Sivashinsky Equation by Using (G′G)-Expansion Method
In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation.
Journal Article
A high dimensional evolution model and its rogue wave solution, breather solution and mixed solutions
In this study, a bilinear neural network method is used to solve the exact solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili equation, which is a new geophysical fluid mechanics model. This equation is derived from Charney equation of geomagnetic field by integrating the multiscale expansion and the perturbation method. To obtain the exact solutions of the model, we built test functions by using the bilinear neural network method. Compared to conventional methods, this method is shown to have faster results and better accuracy. Based on the construction of single-layer models, the rational solution, rogue wave solution, breather type solution and mixed solutions are obtained, and exact solution is also produced based on double-layer models. By increasing the number of activation functions or the number of layers in a neural network, the method's effectiveness is tested from different angles, and more importantly, the method of solving nonlinear equations is expanded. Then, various three-dimensional graphs, contour plots, and density plots with time and selection of activation function are used to depict the typical evolution of these waves.
Journal Article
A regularized point cloud registration approach for orthogonal transformations
An important part of the well-known iterative closest point algorithm (ICP) is the variational problem. Several variants of the variational problem are known, such as point-to-point, point-to-plane, generalized ICP, and normal ICP (NICP). This paper proposes a closed-form exact solution for orthogonal registration of point clouds based on the generalized point-to-point ICP algorithm. We use points and normal vectors to align 3D point clouds, while the common point-to-point approach uses only the coordinates of points. The paper also presents a closed-form approximate solution to the variational problem of the NICP. In addition, the paper introduces a regularization approach and proposes reliable algorithms for solving variational problems using closed-form solutions. The performance of the algorithms is compared with that of common algorithms for solving variational problems of the ICP algorithm. The proposed paper is significantly extended version of Makovetskii et al. (CCIS 1090, 217–231, 2019).
Journal Article
MHD flow and heat transfer over a permeable stretching/shrinking sheet in a hybrid nanofluid with a convective boundary condition
Purpose
The purpose of this study is to present both effective analytic and numerical solutions to MHD flow and heat transfer past a permeable stretching/shrinking sheet in a hybrid nanofluid with suction/injection and convective boundary conditions. Water (base fluid) nanoparticles of alumina and copper were considered as a hybrid nanofluid.
Design/methodology/approach
Proper-similarity variables were applied to transform the system of partial differential equations into a system of ordinary (similarity) differential equations. Exact analytical solutions were then presented for the dimensionless stream and temperature functions. Further, the authors introduce a very nice analytic and numerical solutions for both small and large values of the magnetic parameter.
Findings
It was found that no/unique/two equal/dual physical solutions exist for the investigated boundary value problem. The physically realizable practice of these solutions depends on the range of the governing parameters. For a stretching/shrinking sheet, it was deduced that a hybrid nanofluid works as a cooler on increasing some of the investigated parameters. Moreover, in the case of a shrinking sheet, the first solutions of hybrid nanofluid are stable and physically realizable rather than the nanofluid, while those of the second solutions are not for both hybrid nanofluid and nanofluid.
Originality/value
The present results for the hybrid nanofluids are new and original, as they successfully extend (generalize) the problems previously considered by different authors for the case of nanofluids.
Journal Article
Bifurcations, stationary optical solitons and exact solutions for complex Ginzburg–Landau equation with nonlinear chromatic dispersion in non-Kerr law media
This paper obtains the stationary optical solitons and new exact solutions for complex Ginzburg–Landau equation (CGLE) with nonlinear chromatic dispersion and Kudryashov’s reflective index structure in a non-Kerr law media. The research work is carried out according to the following route: first, the CGLE is transformed into a second order nonlinear ordinary differential equation by an appropriate substitution. Then, the bifurcation, stationary soliton solution and exact solution of CGLE are obtained by using dynamic system theory and polynomial complete discriminant system method. Abundant solutions are obtained, including periodic solutions, doubly-periodic solutions, hyperbolic function solutions, rational function solutions and exponential function solutions. Finally, the 3D and 2D graphics for the solutions are drawn. Since the appearance of stationary soliton means the stop of signal transmission, the research results provide a way to avoid the disaster of superconducting propagation in nonlinear media.
Journal Article
One-component delocalized nonlinear vibrational modes of square lattices
All possible one-component delocalized nonlinear vibrational modes (DNVMs) in a square lattice are analyzed. DNVMs are obtained taking into account exclusively the symmetry of a square lattice, and therefore, they exist regardless of the type of interactions between particles. In this work, the interactions of the nearest and next nearest neighbors are described by the
β
-FPUT potential. For each DNVM, frequency, kinetic and potential energies are described as functions of amplitude. The mechanical stresses caused by DNVMs and the effect of DNVMs on the stiffness constants of the lattice are presented. DNVMs with higher vibration frequencies have a stronger effect on the mechanical properties of the lattice. Examples of analytical analysis of DNVMs are given. It is found that two of the sixteen one-component DNVMs can have frequencies above the phonon band in the entire range of their amplitudes. It is shown that these DNVMs can be used to construct discrete breathers by applying localizing functions. The modulational instability of these two DNVMs leads to the formation of chaotic DBs. The presented results contribute to a better understanding of the nonlinear dynamics of a square lattice by analyzing the properties of a class of delocalized exact solutions and demonstrating their connection with discrete breathers.
Journal Article