An H^1-Galerkin Expanded Mixed Element Method for Semi-linear Hyperbolic Wave Equation

An H1-Galerkin expanded mixed finite element method is discussed for a class of second order semi-linear hyperbolic wave equations. By using the mixed formulation, we can get the optimal approximation for three variables: the scalar unknown, its gradient and its flux(coefficient times the gradient), simultaneously. We also prove the existence and uniqueness of semi-discrete solution. Finally, we obtain some numerical results to illustrate the efficiency of the method.

Interaction of Conormal Waves With Strong and Weak Singularities For Semi－Linear Equations

We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place anti their strength will be no more than the original weak one.

mKdV Equation for Solitary Rossby Waves with Linear Topography Effect in Barotropic Fluids

The modify Korteweg-de Vries(mKdV) equations,governing the evolution of the amplitude of solitary Rossby waves,are derived from quasi-geostrophic vorticity equation by using the perturbation method.The result manifests that the linear topography effect with the change of latitude can induce solitary Rossby wave.

Classical and nonclassical symmetry classifications of nonlinear wave equation with dissipation

A complete classical symmetry classification and a nonclassical symmetry classification of a class of nonlinear wave equations are given with three arbitrary parameter functions. The obtained results show that such nonlinear wave equations admit richer classical and nonclassical symmetries, leading to the conservation laws and the reduction of the wave equations. Some exact solutions of the considered wave equations for particular cases are derived.

MODELING THE INTERACTION OF SOLITARY WAVES AND SEMI-CIRCULAR BREAKWATERS BY USING UNSTEADY REYNOLDS EQUATIONS

A vertical 2-D numerical wave model was developed based on unsteady Reynolds equations. In this model, the k-epsilon models were used to close the Reynolds equations, and volume of fluid(VOF) method was used to reconstruct the free surface. The model was verified by experimental data. Then the model was used to simulate solitary wave interaction with submerged, alternative submerged and emerged semi-circular breakwaters. The process of velocity field, pressure field and the wave surface near the breakwaters was obtained. It is found that when the semi-circular breakwater is submerged, a large vortex will be generated at the bottom of the lee side wall of the breakwater; when the still water depth is equal to the radius of the semi-circular breakwater, a pair of large vortices will be generated near the shoreward wall of the semi-circular breakwater due to wave impacting, but the velocity near the bottom of the lee side wall of the breakwater is always relatively small. When the semi-circular breakwater is emerged, and solitary wave cannot overtop it, the solitary wave surface will run up and down secondarily during reflecting from the breakwater. It can be further used to estate the diffusing and transportation of the contamination and transportation of suspended sediment.

Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation

A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K（2,2,1）, the Oliver water wave equation, and the k（n, n） equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

Linear superposition method for （2＋1）-dimensional nonlinear wave equations

New forms of different-periodic travelling wave solutions for the （2＋1）-dimensional Zakharov-Kuznetsov （ZK） equation and the Davey-Stewartson （DS） equation are obtained by the linear superposition approach of Jacobi elliptic function. A sequence of cyclic identities plays an important role in these procedures.

Different-Periodic Travelling Wave Solutions for Nonlinear Equations

Using Jacobi elliptic function linear superposition approach for the (1+1)-dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation and the (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equation, many new periodic travelling wave solutions with different periods and velocities are obtained based on the known periodic solutions. This procedure is crucially dependent on a sequence of cyclic identities involving Jacobi elliptic functions sn(), cn(), and dn().

A nonlinear Schrodinger equation for gravity waves slowly modulated by linear shear flow

Assume that a fluid is inviscid, incompressible, and irrotational. A nonlinear Schr?dinger equation(NLSE) describing the evolution of gravity waves in finite water depth is derived using the multiple-scale analysis method. The gravity waves are influenced by a linear shear flow, which is composed of a uniform flow and a shear flow with constant vorticity. The modulational instability(MI) of the NLSE is analyzed, and the region of the MI for gravity waves(the necessary condition for existence of freak waves) is identified. In this work, the uniform background flows along or against wave propagation are referred to as down-flow and up-flow, respectively. Uniform up-flow enhances the MI, whereas uniform down-flow reduces it. Positive vorticity enhances the MI, while negative vorticity reduces it. Hence, the influence of positive(negative)vorticity on MI can be balanced out by that of uniform down(up) flow. Furthermore, the Peregrine breather solution of the NLSE is applied to freak waves. Uniform up-flow increases the steepness of the free surface elevation, while uniform down-flow decreases it. Positive vorticity increases the steepness of the free surface elevation, whereas negative vorticity decreases it.

INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

We discuss the incomplete semi-iterative method (ISIM) for an approximate solution of a linear fixed point equations x=Tx+c with a bounded linear operator T acting on a complex Banach space X such that its resolvent has a pole of order k at the point 1. Sufficient conditions for the convergence of ISIM to a solution of x=Tx+c, where c belongs to the range space of R(I-T) k, are established. We show that the ISIM has an attractive feature that it is usually convergent even when the spectral radius of the operator T is greater than 1 and Ind 1T≥1. Applications in finite Markov chain is considered and illustrative examples are reported, showing the convergence rate of the ISIM is very high.

Effect of Weight Function in Nonlinear Part on Global Solvability of Cauchy Problem for Semi-Linear Hyperbolic Equations

In this paper, we investigate the effect of weight function in the nonlinear part on global solvability of the Cauchy problem for a class of semi-linear hyperbolic equations with damping.

Travelling Wave Solution of the Fisher-Kolmogorov Equation with Non-Linear Diffusion

In this paper we study one-dimensional Fisher-Kolmogorov equation with density dependent non-linear diffusion. We choose the diffusion as a function of cell density such that it is high in highly cell populated areas and it is small in the regions of fewer cells. The Fisher equation with non-linear diffusion is known as modified Fisher equation. We study the travelling wave solution of modified Fisher equation and find the approximation of minimum wave speed analytically, by using the eigenvalues of the stationary states, and numerically by using COMSOL (a commercial finite element solver). The results reveal that the minimum wave speed depends on the parameter values involved in the model. We observe that when diffusion is moderately non-linear, the eigenvalue method correctly predicts the minimum wave speed in our numerical calculations, but when diffusion is strongly non-linear the eigenvalues method gives the wrong answer.

THE SEMI-DISCREIXTE METHOD FOR SOLVING HIGH-DIMENSION WAVE EQUATION

The article gives a semi-discrete method for solving high-dimension wave equationBy the method, high-dimension wave equation is converted by, means of diseretizationinto I-D wave equation system which is well-posed. The convergence of the semidijcrete method is given. The numerical calculating resulis show that the speed of convergence is high.

ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R^2

In this article, we investigate the initial value problem（IVP） associated with the defocusing nonlinear wave equation on R^2 as follows：{δttu-△u=-u^3 u（0,x）=u0（x）,δtu＊（0,x）=u1（x,）where the initial data （uo,ul）∈H^s-1（R^2）It is shown that the IVP is global well-posedness in H^s（R^2）×H^s-1×H^s-1（R^2）for any 1 〉 s 〉2/5.The proof relies upon the almost conserved quantity in using multilinear correction term. The main difficulty is to control the growth of the variation of the almost conserved quantity. Finally, we utilize linear-nonlinear decomposition benefited from the ideas of Roy [1].

Domain Decomposition of an Optimal Control Problem for Semi-Linear Elliptic Equations on Metric Graphs with Application to Gas Networks

We consider optimal control problems for the flow of gas in a pipe network. The equations of motions are taken to be represented by a semi-linear model derived from the fully nonlinear isothermal Euler gas equations. We formulate an optimal control problem on a given network and introduce a time discretization thereof. We then study the well-posedness of the corresponding time-discrete optimal control problem. In order to further reduce the complexity, we consider an instantaneous control strategy. The main part of the paper is concerned with a non-overlapping domain decomposition of the semi-linear elliptic optimal control problem on the graph into local problems on a small part of the network, ultimately on a single edge.

Resonant multiple wave solutions to some integrable soliton equations

To transform the exponential traveling wave solutions to bilinear differential equations, a sufficient and necessary condition is proposed. Motivated by the condition, we extend the results to the(2+1)-dimensional Kadomtsev–Petviashvili(KP) equation, the(3+1)-dimensional generalized Kadomtsev–Petviashvili(g-KP) equation, and the B-type Kadomtsev–Petviashvili(BKP) equation. Aa a result, we obtain some new resonant multiple wave solutions through the parameterization for wave numbers and frequencies via some linear combinations of exponential traveling waves. Finally, these new resonant type solutions can be displayed in graphs to illustrate the resonant behaviors of multiple wave solutions.

The Evolution Equation for Second-order Internal Solitary Waves in Stratified Fluids of Great Depth

By using perturbation methods, the evolution equation is derived for the second-order internal solitarywaves in stratified fluids of great depth, which is a kind of inhomogeneous linearized Belljamin-Ono equation.

Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach

Starting from the symbolic computation system Maple and Riccati equation mapping approach and a linear variable separation approach, a new family of non-traveling wave solutions of the (1 + 1)-dimensional Burgers system is derived.