In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not unifor...In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.展开更多
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 ...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.展开更多
Convergence of modified truncated Euler-Maruyama(MTEM)method for stochastic differential equations(SDEs)with(1/2+α)-Holder continuous diffusion coefficients are investigated in this paper.We prove that the MTEM metho...Convergence of modified truncated Euler-Maruyama(MTEM)method for stochastic differential equations(SDEs)with(1/2+α)-Holder continuous diffusion coefficients are investigated in this paper.We prove that the MTEM method for SDE converges to the exact solution in L9 sense under given conditions.Two examples are provided to support our conclusions.展开更多
This paper extracts some analytical solutions of simplified modified Camassa-Holm(SMCH)equations with various derivative operators,namely conformable and M-truncated derivatives that have been recently introduced.The ...This paper extracts some analytical solutions of simplified modified Camassa-Holm(SMCH)equations with various derivative operators,namely conformable and M-truncated derivatives that have been recently introduced.The SMCH equation is used to model the unidirectional propagation of shallowwater waves.The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived.We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system.Furthermore,some 2D and 3D graphical illustrations of the obtained solutions have been presented.The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted.The derived solutions may contribute to comprehending wave propagation in shallow water.So,the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management,which are critical topics in the ocean and coastal engineering.展开更多
The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a ...The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested;2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.展开更多
This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann ...This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.展开更多
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 equati...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.展开更多
基金supported by the National Natural Science Foundation of China(11226159)
文摘In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt ＋ umx ＋ 2ux m = 0, m =（1- δx^2）^2u,u（x, 0） = u0（x） ∈ H^s（R）, x ∈ R, t 〉 0,and show that the solution map is not uniformly continuous in Sobolev spaces H^s（R） for s 〉 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.
文摘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.
基金supported by the Natural Science Foundation of Beijing Municipality(Grant No.1192013).
文摘Convergence of modified truncated Euler-Maruyama(MTEM)method for stochastic differential equations(SDEs)with(1/2+α)-Holder continuous diffusion coefficients are investigated in this paper.We prove that the MTEM method for SDE converges to the exact solution in L9 sense under given conditions.Two examples are provided to support our conclusions.
基金Scientific and Technological Research Council of Turkey(TUBITAK)for the finan-cial support of the 2211-A Fellowship Program.
文摘This paper extracts some analytical solutions of simplified modified Camassa-Holm(SMCH)equations with various derivative operators,namely conformable and M-truncated derivatives that have been recently introduced.The SMCH equation is used to model the unidirectional propagation of shallowwater waves.The extended rational sine−cosine and sinh−cosh techniques have been successfully implemented to the considered equations and some kinds of the solitons such as kink and singular have been derived.We have checked that all obtained solutions satisfy the main equations by using a computer algebraic system.Furthermore,some 2D and 3D graphical illustrations of the obtained solutions have been presented.The effect of the parameters in the solutions on the wave propagation has been examined and all figures have been interpreted.The derived solutions may contribute to comprehending wave propagation in shallow water.So,the solutions might help further studies in the development of autonomous ships/underwater vehicles and coastal zone management,which are critical topics in the ocean and coastal engineering.
文摘The solution of an n-dimensional stochastic differential equation driven by Gaussian white noises is a Markov vector. In this way, the transition joint probability density function (JPDF) of this vector is given by a deterministic parabolic partial differential equation, the so-called Fokker-Planck-Kolmogorov (FPK) equation. There exist few exact solutions of this equation so that the analyst must resort to approximate or numerical procedures. The finite element method (FE) is among the latter, and is reviewed in this paper. Suitable computer codes are written for the two fundamental versions of the FE method, the Bubnov-Galerkin and the Petrov-Galerkin method. In order to reduce the computational effort, which is to reduce the number of nodal points, the following refinements to the method are proposed: 1) exponential (Gaussian) weighting functions different from the shape functions are tested;2) quadratic and cubic splines are used to interpolate the nodal values that are known in a limited number of points. In the applications, the transient state is studied for first order systems only, while for second order systems, the steady-state JPDF is determined, and it is compared with exact solutions or with simulative solutions: a very good agreement is found.
基金supported by the National Natural Science Foundation of China(Nos.11901302,11401180)the Natural Science Foundation from Jiangsu province BK20171029the Academic Discipline Project of Shanghai Dianji University(No.16JCXK02)。
文摘This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.
基金supported by the National Natural Science Foundation of China(Nos.11631107,11471174)。
文摘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.
基金Supported by the National Natural Science Foundation of China(11526101)the National Natural Science Foundation of China(11461028)+1 种基金the Natural Science Foundation of Jiangxi Province(20151BAB211016)the National Social Science Foundation of China(14CJY053)
