In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and...In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach.The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.展开更多
In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated...In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.展开更多
By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth...By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.展开更多
We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond num...We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.展开更多
In this paper, we propose two families of nonconforming finite elements on n-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the ...In this paper, we propose two families of nonconforming finite elements on n-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the H^(3) problems, we establish the quasi-optimal error estimate for the triharmonic equation in the broken H^(3) norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.展开更多
An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction...An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.The auxiliary parameter has an effect on the convergence of homotopy series solutions.Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.展开更多
The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system,which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects ...The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system,which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11861050,11261037)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2020LH01010)the Inner Mongolia Normal University Graduate Students Research and Innovation Fund(Grant No.CXJJS21119)。
文摘In this paper,we construct the rogue wave solutions of the sixth-order nonlinear Schrodinger equation on a background of Jacobian elliptic functions dn and cn by means of the nonlinearization of a spectral problem and Darboux transformation approach.The solutions we find present the dynamic phenomena of higher-order nonlinear wave equations.
基金the National Natural Science Foundation of China(91641107,91852116,12071470)Fundamental Research of Civil Aircraft(MJ-F-2012-04)of Ministry of Industrialization and Information of China.
文摘In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.
文摘By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.
文摘We consider the singularly perturbed sixth-order Boussinesq-type equation, which describes the bidirectional propagation of small amplitude and long capillary gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to 1/3. The sufficient conditions of blow-up of solution to the Cauchy problem for this equation are given.
基金supported in part by the National Natural Science Foundation of China(Grant No.12222101).
文摘In this paper, we propose two families of nonconforming finite elements on n-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the H^(3) problems, we establish the quasi-optimal error estimate for the triharmonic equation in the broken H^(3) norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 10735030, 10475055, 10675065, and 90503006)the National Basic Research Pro-gram of China (Grant No. 2007CB814800)+2 种基金the Program for Changjiang Scholars and Innovative Research Team (Grant No. IRT0734)the Research Fund of Postdoctoral of China (Grant No. 20070410727)the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20070248120) Recommended by LIAO ShiJun
文摘An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.The auxiliary parameter has an effect on the convergence of homotopy series solutions.Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.
基金Project supported by the National Natural Science Foundation of China (Nos. 10871112, 10771008)the Research Fund for the Doctoral Program of Higher Education of China (No. 20090005120009)+1 种基金 the Fundamental Research Funds for the Central Universities (No. BUPT2009RC0702)the Talents Scheme Funds of BUPT
文摘The authors investigate the global existence and semiclassical limit of weak solutions to a sixth-order parabolic system,which is a quantum-corrected macroscopic model derived recently to simulate the quantum effects in miniaturized semiconductor devices.
