The interference wave and the bright and dark soliton for two integro-differential equation by using BNNM

Interference wave is an important research target in the field of navigation, electromagnetic and earth science. In this work, the nonlinear property of neural network is used to study the interference wave and the bright and dark soliton solutions. The generalized broken soliton-like equation is derived through the generalized bilinear method. Three neural network models are presented to fit explicit solutions of generalized broken soliton-like equations and Boiti–Leon–Manna–Pempinelli-like equation with 100% accuracy. Interference wave solutions of the generalized broken soliton-like equation and the bright and dark soliton solutions of the Boiti–Leon–Manna–Pempinelli-like equation are obtained with the help of the bilinear neural network method. Interference waves and the bright and dark soliton solutions are shown via three-dimensional plots and density plots.

     

Nonlocal symmetry and similarity reductions for a $$\varvec{}$$-dimensional Korteweg–de Vries equation

Based on the Lax pair, the nonlocal symmetries to \((2+1)\)-dimensional Korteweg–de Vries equation are investigated, which are also constructed by the truncated Painlevé expansion method. Through introducing some internal spectrum parameters, infinitely many nonlocal symmetries are given. By choosing four suitable auxiliary variables, nonlocal symmetries are localized to a closed prolonged system. Via solving the initial-value problems, the finite symmetry transformations are obtained to generate new solutions. Moreover, rich explicit interaction solutions are presented by similarity reductions. In particular, bright soliton, dark soliton, bell-typed soliton and soliton interacting with elliptic solutions are found. Through computer numerical simulation, the dynamical phenomena of these interaction solutions are displayed in graphical way, which show meaningful structures.     

A general nonlocal nonlinear Schrödinger equation with shifted parity,charge-conjugate and delayed time reversal

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.     

A new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function

This paper proposes a new approach to investigate the nonlinear dynamics in a (3 + 1)-dimensional nonlinear evolution equation via Wronskian condition with a free function. Firstly, a Wronskian condition involving a free function is introduced for the equation. Secondly, by solving the Wronskian condition, some exact solutions are presented. Thirdly, the dynamical behaviors are analyzed by choosing specific functions in the Wronskian condition. In addition, some exact solutions are graphically illustrated by using Mathematica symbolic computations. The dynamical behaviors include stationary y-breather, line-soliton resonance, line-soliton-like phenomenon, parabola–soliton interaction, cubic–parabola–soliton resonance, kink behavior, and singular waves. These results not only illustrate the merits of the proposed method in deriving new exact solutions but also novel dynamical behaviors in the (3 + 1)-dimensional nonlinear evolution equation.

     

Solving Huxley equation using an improved PINN method

Although many effective methods for solving partial differential equations (PDEs) have been proposed, there is no universal method that can solve all PDEs. Therefore, solving partial differential equations has always been a difficult problem in mathematics, such as deep neural network (DNN). In recent years, a method of embedding some basic physical laws into traditional neural networks has been proposed to reveal the dynamic behavior of equations directly from space-time data [i.e., physics-informed neural network (PINN)]. Based on the above, an improved deep learning method to recover the new soliton solution of Huxley equation has been proposed in this paper. As far as we know, this is the first time that we have used an improved method to study the numerical solution of the Huxley equation. In order to illustrate the advantages of the improved method, we use the same network depth, the same hidden layer and neurons contained in the hidden layer, and the same training sample points. We analyze the dynamic behavior and error of Huxley’s exact solution and the new soliton solution and give vivid graphs and detailed analysis. Numerical results show that the improved algorithm can use fewer sample points to reconstruct the exact solution of the Huxley equation with faster convergence speed and better simulation effect.

     

Improvement of positive position feedback controller for suppressing compressor blade oscillations

By using various techniques, we investigate the (2\(+\)1)-dimensional modified KdV-Calogero–Bogoyavlenskii–Schiff equation. This equation is integrable under the mean of the consistent Riccati expansion method. The truncated Painlevé expansion, the simplified Hirota’s method and other methods are used as powerful vehicles to conduct the analysis. We formally derive, in explicit forms, abundant solutions of distinct physical structures, including multiple soliton solutions, multiple complex soliton solutions, kink solutions and singular solutions.     

Reduction approach and three types of multi-soliton solutions of the shifted nonlocal mKdV equation

Nonlinear Dynamics - In this paper, a reduction approach is proposed for a shifted nonlocal mKdV equation, from which we obtain its three types of multi-soliton solutions. Specifically, we proceed...     

Lattice hydrodynamic traffic flow model with explicit drivers’ physical delay

An extended lattice hydrodynamic model is presented by considering the effect of drivers’ delay in sensing relative flux. The linear stability criterion of the new model is obtained by employing the linear stability theory. By means of nonlinear analysis method, the modified Korteweg–deVries (mKdV) equation near the critical point is constructed and solved. The propagation behavior of traffic jam can thus be described by the kink–antikink soliton solution for the mKdV equation. The good agreement between the simulation results and the analytical results show that the drivers’ delay in sensing relative flux effect plays an important role in traffic jamming transition.     

Soliton solutions and Bäcklund transformations of a (2 + 1)-dimensional nonlinear evolution equation via the Jaulent–Miodek hierarchy

In this paper, a (2 + 1)-dimensional nonlinear evolution equation generated via the Jaulent–Miodek hierarchy is investigated. Based on the Bell polynomials and Hirota method, bilinear forms and Bäcklund transformations are derived. One- and two-soliton solutions are constructed via symbolic computation. Soliton solutions are obtained through the Bäcklund transformations. We can get three types by choosing different parameters: the kink, bell-shape, and anti-bell-shape solitons. Propagation of the one soliton and elastic interactions between the two solitons are discussed graphically. After the interaction of the two bell-shape or anti-bell-shape solitons, solitonic shapes and amplitudes keep invariant except for some phase shifts, while after the interaction of the kink soliton and anti-bell-shape soliton, the anti-bell-shape soliton turns into a bell-shape one, and the kink soliton keeps its shape, with their amplitudes unchanged.     

Analyses of the driver’s anticipation effect in a new lattice hydrodynamic traffic flow model with passing

In this paper, we studied the effect of driver’s anticipation with passing in a new lattice model. The effect of driver’s anticipation is examined through linear stability analysis and shown that the anticipation term can significantly enlarge the stability region on the phase diagram. Using nonlinear stability analysis, we obtained the range of passing constant for which kink soliton solution of mKdV equation exist. For smaller values of passing constant, uniform flow and kink jam phase are present on the phase diagram and jamming transition occurs between them. When passing constant is greater than the critical value depending on the anticipation coefficient, jamming transitions occur from uniform traffic flow to kink-bando traffic wave through chaotic phase with decreasing sensitivity. The theoretical findings are verified using numerical simulation which confirm that traffic jam can be suppressed efficiently by considering the anticipation effect in the new lattice model.     

机器学习在求解一维双曲守恒律方程中的应用

双曲守恒律方程对空气动力学、物理学和海洋学等众多领域问题的计算有着重大意义,本文应用机器学习框架下的BP神经网络对双曲守恒律方程近似求解.首先,采用熵稳定格式及基于自适应移动网格的熵稳定格式所得多个时间层的数值解构造网络输入,采用高分辨率熵稳定格式所得对应的多个时间层的数值解构造网络输出,并对数据集作归一化处理.随后,...     

Novel solitons and higher-order solitons for the nonlocal generalized Sasa–Satsuma equation of reverse-space-time type

Nonlinear Dynamics - The general soliton solutions and higher-order soliton solutions for the nonlocal generalized Sasa–Satsuma (SS) equation of reverse-space-time type are explored. Firstly,...     

Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

Dark and gray solitons of ($$$$)-dimensional nonlocal nonlinear media with periodic response function

By using the standard symmetry reduction method, some exact analytical solutions including gray solitons and gray soliton lattice solutions are derived for the (\(2+1\))-dimensional nonlinear optical media with periodic nonlocal response. Furthermore, dark/gray soliton solutions and dark soliton lattice solutions are found by means of hyperbolic function expansion method and elliptic function expansion method for the nonlocal nonlinear system, respectively. It is found that two critical points exist for soliton solutions, and the switching dynamics of solitons may be described by the critical points.     

Families of nonsingular soliton solutions of a nonlocal Schrödinger–Boussinesq equation

Nonlocal nonlinear evolution equations with self-induced parity–time symmetric potential have received intensive attention, due to their good applications in nonlinear optics. A nonlocal Schrödinger–Boussinesq equation is proposed in this paper. By using the Hirota bilinear method and the Kadomtsev–Petviashvili hierarchy reduction method, explicit soliton solution with the nonzero boundary condition is succinctly constructed in terms of determinant. Typical dynamics and asymptotic behaviours of three types of two-soliton solutions are discussed in detail.     

Nonlocal symmetry,CRE solvability and soliton–cnoidal solutions of the ($$$$)-dimensional modified KdV-Calogero–Bogoyavlenkskii–Schiff equation

In this paper, a modified KdV-CBS equation is investigated by using the truncated Painlevé expansion and consistent Riccati expansion method, respectively. It is shown that the modified KdV-CBS equation has a nonlocal symmetry related to the residue of its truncated Painlevé expansion. It is also proved that the modified KdV-CBS equation is consistent Riccati expansion solvable. Furthermore, with the help of the consistent Riccati expansion method, the soliton–cnoidal wave interaction solutions are explicitly given.     

Dissipative acoustic solitons under a weakly-nonlinear, Lagrangian-averaged Euler-α model of single-phase lossless fluids

A one-dimensional weakly-nonlinear model equation based on a Lagrangian-averaged Euler-α model of compressible flow in lossless fluids is presented. Traveling wave solutions (TWS)s, in the form of a topological soliton (or kink), admitted by this fourth-order partial differential equation are derived and analyzed. An implicit finite-difference scheme with internal iterations is constructed in order to study soliton collisions. It is shown that, for certain parameters, the TWSs interact as solitons, i.e., they retain their “identity” after a collision. Kink-like solutions with an oscillatory tail are found to emerge in a signaling-type initial-boundary-value problem for the linearized equation of motion. Additionally, connections are drawn to related weakly-nonlinear acoustic models and the Korteweg-de Vries equation from shallow-water wave theory.     

Spatiotemporal soliton clusters in strongly nonlocal media with variable potential coefficients

We study analytically and numerically spatiotemporal solitons in three-dimensional strongly nonlocal nonlinear media. A broad class of exact self-similar solutions to the strongly nonlocal Schrödinger equation with variable potential coefficients has been obtained. We find robust soliton cluster solutions of the accessible type, constructed with the help of Kummer and Hermite functions. They are characterized by the set of three quantum numbers. Dynamical features of these spatiotemporal accessible solitons are discussed. The validity of the analytical solutions and their stability is verified by means of direct numerical simulations.