UNIQUE CONTINUATION AND PERSISTENCE PROPERTIES OF SOLUTIONS OF THE 2-COMPONENT DEGASPERIS-PROCESI EQUATIONS

In this article, the unique continuation and persistence properties of solutions of the 2-component Degasperis-Procesi equations are discussed. It is shown that strong solutions of the 2-component Degasperis-Procesi equations, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if they also decay exponentially at a later time.

An Approach for Solving Short-Wave Models for Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa-Holm equation and the Degasperis-Procesi equation respectively. One-cusp soliton solution of the Camassa-Flolm equation is obtained. One-loop soliton solution of the Degasperis- Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.

Analytic Approximations for Soliton Solutions of Short-Wave Models for Camassa-Holm and Degasperis-Procesi Equations

In this paper, the short-wave model equations are investigated, which are associated with the Camassa- Holm （CH） and Degasperis Procesi （DP） shallow-water wave equations. Firstly, by means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformatioas back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems.

Loop Soliton Solutions of a Short Wave Model for a Degasperis-Procesi Equation

An analytic method, i.e. the homotopy analysis method, was applied for constructing the solutions of the short waves model equations associated with the Degasperis-Procesi (DP) shallow water waves equation. The explicit analytic solutions of loop soliton governing the propagation of short waves were obtained. By means of the transformation of independent variables, an analysis one-loop soliton solution expressed by a series of exponential functions was obtained, which agreed well with the exact solution. The results reveal the validity and great potential of the homotopy analysis method in solving complicated solitary water wave problems.

A New Analytical Study of Modified Camassa-Holm and Degasperis-Procesi Equations

In this letter, variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of modified Camassa-Holm and Degasperis-Procesi equations. The results show that the VHPM is suitable for solving nonlinear differential equations with fully nonlinear dispersion term. The travelling wave solution for above equation compared with VIM, HPM, and exact solution. Also, it was shown that the present method is effective, suitable, and reliable for these types of equations.

Stability analysis solutions of the nonlinear modified Degasperis-Procesi water wave equation

In the present study,the solitary wave solutions of modified Degasperis-Procesi equation are developed.Unlike the standard Degasperis-Procesi equation,where multi-peakon solutions arise,the modification caused a change in the characteristic of these peakon solutions and changed it to bell-shaped solitons.By using the extended auxiliary equation method,we deduced some new soliton solutions of the fourthorder nonlinear modified Degasperis-Procesi equation with constant coefficient.These solutions include symmetrical,non-symmetrical kink solutions,solitary pattern solutions,weiestrass elliptic function solutions and triangular function solutions.We discuss the stability analysis for these solutions.

High Order Finite Difference WENO Methods with Unequal-Sized Sub-Stencils for the Degasperis-Procesi Type Equations

In this paper,we develop twofinite difference weighted essentially non-oscillatory(WENO)schemes with unequal-sized sub-stencils for solving the Degasperis-Procesi(DP)andµ-Degasperis-Procesi(µDP)equations,which contain nonlinear high order derivatives,and possibly peakon solutions or shock waves.By introducing auxiliary variable(s),we rewrite the DP equation as a hyperbolic-elliptic system,and theµDP equation as afirst order system.Then we choose a linearfinite difference scheme with suitable order of accuracy for the auxiliary variable(s),and twofinite difference WENO schemes with unequal-sized sub-stencils for the primal variable.One WENO scheme uses one large stencil and several smaller stencils,and the other WENO scheme is based on the multi-resolution framework which uses a se-ries of unequal-sized hierarchical central stencils.Comparing with the classical WENO scheme which uses several small stencils of the same size to make up a big stencil,both WENO schemes with unequal-sized sub-stencils are simple in the choice of the stencil and enjoy the freedom of arbitrary positive linear weights.Another advantage is that thefinal reconstructed polynomial on the target cell is a polynomial of the same de-gree as the polynomial over the big stencil,while the classicalfinite difference WENO reconstruction can only be obtained for specific points inside the target interval.Nu-merical tests are provided to demonstrate the high order accuracy and non-oscillatory properties of the proposed schemes.

THE STRUCTURE-PRESERVING METHODS FOR THE DEGASPERIS-PROCESI EQUATION

This paper gives several structure-preserving schemes for the Degasperis-Procesi equation which has bi-Hamiltonian structures consisted of both complex and non-local Hamiltonian differential operators. For this sake, few structure-preserving schemes have been proposed so far. In our work, based on one of the bi-Hamiltonian structures, a symplectic scheme and two new energy-preserving schemes are constructed. The symplecticity comes straightly from the application of the implicit midpoint method on the semi-discrete system which is proved to remain Hamiltonian, while the energy conservation is derived by the combination of the averaged vector field method of second and fourth order, respectively. Some numerical results are presented to show that the three schemes do have the advantages in numerical stability, accuracy in long time computing and ability to preserve the invariants of the DP equation.

On peaked solitary waves of the Degasperis-Procesi equation

The Degasperis-Procesi （DP） equation describing the propagation of shallow water waves contains a physical parameter co, and it is well-known that the DP equation admits solitary waves with a peaked crest when ω = 0. In this article, we illustrate, for the first time, that the DP equation admits peaked solitary waves even when ω≠ 0. This is helpful to enrich our knowledge and deepen our understandings about peaked solitary waves of the DP equation.

A Splitting Method for the Degasperis-Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

The Degasperis-Procesi(DP)equation is split into a system of a hyperbolic equation and an elliptic equation.For the hyperbolic equation,we use an optimized finite difference weighted essentially non-oscillatory(OWENO)scheme.New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation,which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations.For the elliptic equation,the Fourier pseudospectral method(FPM)is employed to discretize the high order derivative.Due to the combination of the WENO reconstruction and FPM,the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions.The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

An efficient computational technique for time-fractional modified Degasperis-Procesi equation arising in propagation of nonlinear dispersive waves

In this paper,an efficient hybrid numerical scheme which is based on a joint venture of the q-homotopy analysis method and Sumudu transform is applied to investigate the time-fractional modified Degasperis-Procesi(DP)equation.The present study considers the Caputo fractional derivative.The fractional order modified DP model is very important and plays a great role in study of ocean engineering and science.The proposed scheme provides a beautiful opportunity for proper selection of the auxiliary parameter h and the asymptotic parameterρ(≥1)to handle mainly the differential equations of nonlinear nature.The offered scheme produces the solution in the shape of a convergent series in a large admissible domain which is helpful to regulate the region of convergence of a series solution.The proposed work computes the approximate analytical solution of the fractional modified DP equation systematically and also presents graphically the variation of the obtained solution for diverse values of the fractional parameterβ.

Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

In this paper,we develop,analyze and test local discontinuous Galerkin(LDG)methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives,and possibly discontinuous or sharp transition solutions.The LDG method has the flexibility for arbitrary h and p adaptivity.We prove the L2 stability for general solutions.The proof of the total variation stability of the schemes for the piecewise constant P0 case is also given.The numerical simulation results for different types of solutions of the nonlinear Degasperis-Procesi equation are provided to illustrate the accuracy and capability of the LDG method.

一类非线性波方程的尖波解

利用动力系统分岔理论来分析一类非线性Degasperis-Procesi方程的全局动力学性质,给出了不同行波相互转换的分岔值,揭示了行波类型间的转换与参数变化的关系,合理的解释了该方程产生尖波的原因,并给出了相应的行波解的表达式。

一类非线性波方程尖波解及其动力学性质的分析

应用动力系统分岔理论和定性理论研究了一类非线性DegasperisProcesi方程的行波解及其动力学性质,并结合可积系统的特点,利用哈密尔顿系统的能量特征,通过Maple软件绘出其相轨图,再根据行波与相轨道间的对应关系,揭示了不同类型的行波解间的转变与参数变化的关系,并且给出了不同行波间相互转换的参数分岔值,从根本上解释了Peakon产生的原因.数值模拟验证了该方法的正确性.最后给出了相应行波解的表达式.

Peaked Traveling Wave Solutions to a Generalized Novikov Equation with Cubic and Quadratic Nonlinearities

The Camassa-Holm equation, Degasperis–Procesi equation and Novikov equation are the three typical integrable evolution equations admitting peaked solitons. In this paper, a generalized Novikov equation with cubic and quadratic nonlinearities is studied, which is regarded as a generalization of these three well-known studied equations. It is shown that this equation admits single peaked traveling wave solutions, periodic peaked traveling wave solutions, and multi-peaked traveling wave solutions.

COMPARISONS OF RAY-TRACING AND PARABOLIC EQUATION METHODS FOR THE LARGE-SCALE COMPLEX ELECTROMAGNETIC ENVIRONMENT SIMULATIONS

It is important for the wireless communication field to conduct research on large-scale complex electromagnetic environment(CEME)simulation.There exist many models for computing CEME simulation,including empirical models,half-empirical or halfdeterministic models and deterministic models.Most of these models cannot obtain satisfactory results due to the limitation of the capacity of computers.The ray tracing(RT)and parabolic equation(PE)methods are very suitable for large-scale CEME simulation.Based on the introduction of RT and PE,qualitative comparisons of the two methods are analyzed in view of algorithm theory,the category of the model,solution to the model and the application field,and then four specific indices are focused on to analyze the computational complexity,accuracy,speed and parallelism in details.The numerical experiments are presented by the three-dimensional(3D)RT method employing the software of Wireless InSite(WI)and a quasi-3DPE method using the sliced method.Although both RT and PE methods can achieve high speedup using coarse-grained parallel computing,the experimental results indicate that the PE method can obtain a higher speed than the RT method,and the two methods can acquire an approximate precision.A hybrid procedure using both RT and PE methods can obtain a better result for solving CEME problems.

Nonlocal Symmetries of the Camassa-Holm Type Equations

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies.