Symbolic computation and observable effect for the (2 + 1)-dimensional symmetric regularized-long-wave equation from strongly magnetized cold-electron plasmas

In this paper, with the aid of computerized symbolic computation, we use the generalized hyperbolic-function method to obtain new families of exact analytic solutions for the (2 + 1)-dimensional symmetric regularized-long-wave equation. This equation describes weakly nonlinear ion-acoustic and space-charge waves in strongly magnetized cold-electron plasmas. The families we obtain consist of solitary waves and trigonometric functions. We outline an observable (2 + 1)-dimensional effect that could be of interest to future experiments on space and laboratory plasma systems. The usage of the Wu elimination method has also been addressed.     

Towered waves and anti-waves in the generalized Degasperis-Procesi equation

New traveling wave solutions of the generalized Degasperis-Procesi equation are investigated. The solutions are characterized by three parameters. Using an improved qualitative method, abundant traveling wave solutions, such as smooth waves, peaked waves, cusped waves, compacted waves, looped waves and fractal-like waves, are obtained. Especially, some strange composite wave solutions such as towered waves and their anti-waves are first given. We also study the limiting behavior of all periodic solutions as the parameters trend to some special values.     

New family of overturning soliton solutions for a typical breaking soliton equation

The breaking soliton equations are of current interest, while the application of computer algebra to sciences has a bright future. In this paper, a new family of overturning soliton solutions for a typical breaking soliton equation is obtained via a computer-algebra-based method. An example of explicit solutions from the family is given. Solitary waves are also shown to be merely a simple case belonging to the family.     

Energy-Preserving Algorithms for the Benjamin Equation

This paper presents several hybrid algorithms to preserve the global energy of the Benjamin equation. The Benjamin equation is a non-local partial differential equation involving the Hilbert transform. For this sake, quite few structure-preserving integrators have been proposed so far. Our schemes are derived based on an extended multi-symplectic Hamiltonian system of the Benjamin equation by using Fourier pseudospectral method, finite element method and wavelet collocation method spatially coupled with the AVF method temporally. The local and global properties of the proposed schemes are studied. Numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods and study the evolutions of the numerical solutions of solitary waves and wave breaking.     

The influence of the computational mesh on accuracy for initial value problems with discontinuous or nonunique solutions

Discontinuous, or weak, solutions of the wave equation, the inviscid form of Burgers equation, and the tine-dependent, two-dimensional Euler equations are studied. A numerical method of second-order accuracy in two forms, differential and integral, is used to calculate the weak solutions of these equations for several initial value problems, including supersonic flow past a wedge, a double symmetric wedge, and a sphere. The effect of the computational mesh on the accuracy of computed weak solutions including shock waves and expansion phenomena is studied. Modifications to the finite-difference method are presented which aid in obtaining desired solutions for initial value problems in which the solutions are nonunique.     

A finite difference method for korteweg-de vries like equation with nonlinear dispersion

The Korteweg-de Vries (KdV) equation has been generalized by Rosenau and Hyman [3] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons. Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local consevation laws. In this paper, an implicit numerical method has been developed to solve the K(2,3) equation. Accuracy and stability of the method have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested method. The numerical results have shown that this compacton exhibits true soliton behavior.

     

Simulation of spilling breaking waves using a two phase flow CFD model

A two phase flow CFD model has been developed for 2D spilling breaking wave simulations. A mass conservative level set method similar to Olsson and Kreiss [Olsson E, Kreiss G. A conservative level set method for two phase flow. J Comput Phys 2005;210(1):225–46] is implemented for capturing the air–water interface. The solver is discretised using a finite volume method based on a curvilinear coordinate system. A fully implicit fractional step method is used to advance simulations in time. The solver has been tested and validated by repeating benchmark results of dam breaking simulation and travelling solitary wave simulation. Finally, we employ this solver to simulate spilling breaking waves in the surf zone. Our results show that surface elevations, the location of the breaking point and undertow profiles can generally be well captured. We have also found that temporal and spatial schemes may have significant impacts on computational results.     

对称在破裂孤立子方程中的应用

首先对带有积分项的破裂孤立子方程（breaking soliton equation）进行变换,然后利用待定系数法求出它的对称,通过验证知道原方程的李群能构成李代数,再利用优化系统对原方程进行约化,求出了原方程的一些新解。     

(2+1)-dimensional ZK–Burgers equation with the generalized beta effect and its exact solitary solution

The (1 ＋1)-dimensional mathematical model had been extensively derived to describe Rossby solitary waves in a line in the past few decades. But as is well known, the (1 ＋1)-dimensional model cannot reflect the generation and evolution of Rossby solitary waves in a plane. In this paper, a (2 ＋1)-dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived to describe the evolution of Rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi-geostrophic potential vorticity equations with the generalized beta effect. The effects of the generalized beta and dissipation are presented by the Zakharov–Kuznetsov–Burgers equation. We also obtain the new solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent with the help of the Bernoulli equation, which is different from the common classical solitary solution. Based on the solution, the features of the variable coefficient are discussed by geometric figures Meanwhile, the approximate solitary solution of Zakharov–Kuznetsov–Burgers equation is given by using the homotopy perturbation method. And the amplitude of solitary waves changing with time is depicted by figures. Undoubtedly, these solitary solutions will extend previous results and better help to explain the feature of Rossby solitary waves.     

Interaction solutions of a (2+1)-dimensional dispersive long wave system

With the help of the consistent $tanh$ expansion, this paper obtains the interaction solutions between solitons and potential Burgers waves of a (2$+$1)-dimensional dispersive long wave system. Based on some known solutions of the potential Burgers equation, the multiple resonant soliton wave solutions, soliton–error function wave solutions, soliton–rational function wave solutions and soliton–periodic wave solutions are obtained directly.     

A numerical study of k(3,2) equation

The Korteweg-de Vries (Kdv) equation has been generalized by Rosenau and Hyman [7] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons

Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local conservation laws

In this paper, an implicit finite difference method and a finite element method have been developed to solve the K(3,2) equation. Accuracy and stability of the methods have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested methods. The umerical results have shown that this compacton exhibits true soliton behavior.     

On multi-soliton solutions for the (2+1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide

In this work, we construct multi-soliton solutions of the $(2+1)$-dimensional breaking soliton equation with variable coefficients by using the generalized unified method. We employ this method to obtain double- and triple-soliton solutions. Furthermore, we study the nonlinear interactions between these solutions in a graded-index waveguide. The physical insight and the movement role of the waves are discussed and analyzed graphically for different choices of the arbitrary functions in the obtained solutions. The interactions between the solitons are elastic whether the coefficients of the equation are constants or variables.     

Solitons to rogue waves transition,lump solutions and interaction solutions for the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics

ABSTRACT

In this work, we investigate the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation in fluid dynamics, which plays an important role in depicting weakly dispersive waves propagated in a quasi-media and fluid mechanics. By employing Hirota's bilinear method, we derive the one- and two-soliton solutions of the equation. Moreover, we reduce those soliton solutions to the periodic line waves and exact breather waves by considering different parameters. A long wave limit is used to derive the rogue wave solutions. Based on the resulting bilinear representation, we introduce two types of special polynomial functions, which are employed to find the lump solutions and interaction solutions between lump and stripe soliton. It is hoped that our results can be used to enrich dynamic behaviours of the (3+1)-dimensional BKP-type equations.     

Exact wave solutions for a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation

In this work, we investigate a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation. Based on the simplified Hirota bilinear method, we first construct its soliton solutions. Meanwhile, we correct the formula of N-soliton solution for this equation. On the basis of these solitons we further calculate its lump solutions, periodic waves. Meanwhile, rogue waves as well as interaction solutions of this equation are also obtained by a direct algebraic method. Some figures are given to display the behavior of these solutions.     

Estimated surface-wave contributions to radar Doppler velocity measurements of the ocean surface

For simulated ocean conditions, we estimate the magnitude of the Doppler velocity contributions produced by unresolved surface waves that typical spaceborne synthetic aperture radars (SAR) would measure. The mechanism for generating Doppler velocities is the correlation between wave phase and radar cross section. The contributions analyzed include those of linear gravity waves, second-order wave-wave interactions, Bragg-wave scatterers and breaking waves. For gravity waves, we consider both wave tilt and hydrodynamic modulation transfer functions (MTFs). We find that for nominal sea conditions, the Doppler velocity is significant, on the order of 1 m/s, and exhibits large variation as a function of incidence angle and look with respect to the sea direction. The most important contributors are gravity waves and the Bragg scatterers, followed by sea spikes. Effects produced by second-order wave solutions are argued to be inconsequential.     

具优势对称部分的非对称非线性问题的不精确Newton分裂算法

本文讨论了处理具优势对称部分的非对称非线性问题的不精确Newton方法.利用矩阵分裂技术,建立了求解此类问题的一类不精确Newton分裂极小参量法、不精确Newton分裂对称LQ法(简记:Newton-SMINRES,Newton-SSYMMLQ),并在合理的假设下,证明了算法的收敛性.数值计算表明:Newton-SMINRES,Newton-SSYMMLQ算法的收敛行为要好于一般求解非线性方程组的Newton-Krylov子空间方法:Newton-BiCGSTAB,Newton-GMRES和Newton-MINRES等算法.     

On closed form solutions of (2+1)-breaking soliton system by similarity transformations method

In the present research, similarity transformation method via Lie-group theory is proposed to seek some more exact closed form solutions of the (2+1)-dimensional breaking soliton system. The system describes the interactions of the Riemann wave along y-axis and long wave along x-axis. Some explicit solutions of breaking soliton system are attained with appropriate choices of the arbitrary functions and making use of arbitrary constants involved in the infinitesimals. In order to obtain physically meaningful solutions, numerical simulation is performed. On the basis of graphical representation, the physical analysis of solutions reveals into multi-solitons, periodic, quadratic, asymptotic and stationary profiles.     

Long range numerical simulation of short waves as nonlinear solitary waves

A new numerical method has been developed to propagate short wave equation pulses over indefinite distances and through regions of varying index of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain 2–3 grid cells in thickness indefinitely with no numerical spreading. With the method, only a simple discretized equation is solved each time step at each grid node. The method can be contrasted to Lagrangian Ray Tracing: it is an Eulerian based method that captures the waves directly on the computational grid, where the basic objects are codimension 1 surfaces (in the fine grid limit), defined on a regular grid, rather than collections of markers. In this way, the complex logic of current ray tracing methods, which involves allocation of markers to each surface and interpolation as the markers separate, is avoided.     

Numerical solution of the ‘classical’ Boussinesq system

We consider the ‘classical’ Boussinesq system of water wave theory, which belongs to the class of Boussinesq systems modelling two-way propagation of long waves of small amplitude on the surface of water in a horizontal channel. (We also consider its completely symmetric analog.) We discretize the initial-boundary-value problem for these systems, corresponding to homogeneous Dirichlet boundary conditions on the velocity variable at the endpoints of a finite interval, using fully discrete Galerkin-finite element methods of high accuracy. We use the numerical schemes as exploratory tools to study the propagation and interactions of solitary-wave solutions of these systems, as well as other properties of their solutions.     

Non-hydrostatic finite volume model for non-linear waves interacting with structures

An efficient non-hydrostatic finite volume model is developed and applied to simulate non-linear waves interacting with structures. The unsteady Navier–Stokes equations are solved in a 3D grid made of polyhedrons, which are built from a 2D horizontal unstructured grid by adding several horizontal layers. A new grid arrangement in the vertical direction is proposed, which renders the resulting model is relatively simple. Moreover, the discretized Poisson equation for pressure is symmetric and positive definite, and thus it can be solved effectively by the preconditioned conjugate gradient method. Several test cases including solitary wave interacting with a submerged structure, solitary wave scattering from a vertical circular cylinder and an array of four circular cylinders are used to demonstrate the capability of the model on simulating non-linear waves interacting with structures. In all cases, the model gives satisfactory results in comparison with analytical solutions, experimental data and other published numerical results.