Solitary wave solution of nonlinear Bogoyavlenskii system by variational analysis method

In this work, the Bogoyavlenskii system (BS) and fractal BS are investigated by variational method for the first time. An efficient and simple scheme is proposed to seek their exact solitary wave solutions, which is called variational analysis method. The novel scheme requires only two steps, making it much attractive in practical applications, and a good result is obtained. This paper cleans up the road to the exact solitions, and it sheds a new light on the soliton theory. Finally, the physical properties of solitary wave solutions obtained are analyzed by some simulation figures.

Abundant soliton wave solutions for the (3+1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles by bilinear analysis

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.

Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

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.

The Duffing Oscillator Equation and Its Applications in Physics

In this paper, we solve the Duffing equation for given initial conditions. We introduce the concept of the discriminant for the Duffing equation and we solve it in three cases depending on sign of the discriminant. We also show the way the Duffing equation is applied in soliton theory.

Improved spatial soliton theory on the example of photorefractive gallium arsenide

This article presents a critical look at the standard theory of bright and dark photorefractive screening solitons. We pay attention to the commonly overlooked fact of the inconsistency of the theory in the context of the accordance of soliton solution with the microscopic band transport models. Taking into account the material equations for the semi-insulating semiconductor (SI-GaAs) and including the nonlinear transport of hot electrons, a simple differential equation has been developed to determine the distribution of refractive index changes in the material for a localized optical beam. An amendment to the standard solution of (1 + 1)D solitons has been proposed, which particularly should be used for dark solitons to obtain the plausible self-consistent solutions

Lump and rational solutions for weakly coupled generalized Kadomtsev–Petviashvili equation

Through the [Formula: see text]-KP hierarchy, we present a new (3+1)-dimensional equation called weakly coupled generalized Kadomtsev–Petviashvili (wc-gKP) equation. Based on Hirota bilinear differential equations, we get rational solutions to wc-gKP equation, and further we obtain lump solutions by searching for a symmetric positive semi-definite matrix. We do some numerical analysis on the trajectory of rational solutions and fit the trajectory equation of wave crest. Some graphics are illustrated to describe the properties of rational solutions and lump solutions. The method used in this paper to get lump solutions by constructing a symmetric positive semi-definite matrix can be applied to other integrable equations as well. The results expand the understanding of lump and rational solutions in soliton theory.

A Hausdorff fractal Nizhnik-Novikov-Veselov model arising in the incompressible fluid

Purpose Nizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an incompressible fluid. The purpose of this paper is to present a fractal model of the NNVS based on the Hausdorff fractal derivative fundamental concept. Design/methodology/approach A two-scale transformation is used to convert the proposed fractal model into regular NNVS. The variational strategy of well-known Chinese scientist Prof. Ji Huan He is used to generate bright and exponential soliton solutions for the proposed fractal system. Findings The NNV fractal model and its variational principle are introduced in this paper. Solitons are created with a variety of restriction interactions that must all be applied equally. Finally, the three-dimensional diagrams are displayed using an appropriate range of physical parameters. The results of the solitary solutions demonstrated that the suggested method is very accurate and effective. The proposed methodology is extremely useful and nearly preferable for use in such problems. Practical implications The research study of the soliton theory has already played a pioneering role in modern nonlinear science. It is widely used in many natural sciences, including communication, biology, chemistry and mathematics, as well as almost all branches of physics, including nonlinear optics, plasma physics, fluid dynamics, condensed matter physics and field theory, among others. As a result, while constructing possible soliton solutions to a nonlinear NNV model arising from the field of an incompressible fluid is a popular topic, solving nonlinear fluid mechanics problems is significantly more difficult than solving linear ones. Originality/value To the best of the authors’ knowledge, for the first time in the literature, this study presents Prof. Ji Huan He's variational algorithm for finding and studying solitary solutions of the fractal NNV model. The reported solutions are novel and present a valuable addition to the literature in soliton theory.

Classical Soliton Theory for Studying the Dynamics and Evolution of in Network

This paper presents the dynamic model ofthe soliton. Based on this model, it is supposed to study the state of the network. The term neural networks refersto the networks of neurons in the mammalian brain. Neurons are its main units of computation. In the brain, they are connected together in a network to process data. This can be a very complex task, and so the dynamics of neural networks in the mammalian brain in response to external stimuli can be quite complex. The inputs and outputs of each neuron change as a function of time, in the form of so-called spike chains, but the network itself also changes. We learn and improve our data processing capabilities by establishing reconnections between neurons.

Reciprocal Transformations in Relativistic Gasdynamics. Lie Group Connections

Reciprocal transformations associated with admitted conservation laws were originally used to derive invariance properties in non-relativistic gasdynamics and applied to obtain reduction to tractable canonical forms. They have subsequently been shown to have diverse physical applications to nonlinear systems, notably in the analytic treatment of Stefan-type moving boundary problem and in linking inverse scattering systems and integrable hierarchies in soliton theory. Here,invariance under classes of reciprocal transformations in relativistic gasdynamics is shown to be linked to a Lie group procedure.