Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potential...Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems （FDHS） in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.展开更多
In order to simulate the airflow in anhydrous case and the water-air flow in groundwater case, a numerical model of airflow in soil was developed. For the nonlinearity of the governing partial differential equation, t...In order to simulate the airflow in anhydrous case and the water-air flow in groundwater case, a numerical model of airflow in soil was developed. For the nonlinearity of the governing partial differential equation, the corresponding discretization and linearization methods were given. Due to the mass transfer between air-phase and water-phase, phase states of the model elements were constantly changing. Thus, parameters of the model were divided into primary ones and secondary ones, and the primary variables changing with phase states and the secondary variables can be obtained by their functional relationship with the primary variables. Additionally, the special definite condition of this numerical model was illustrated. Two examples were given to simulate the airflow in soil whether there was groundwater or not, and the effectiveness of the numerical model is verified by comparing the results of simulation with that of exoeriment.展开更多
基金The project supported by National Natural Science Foundation of China under Grant No. 10371023 and Shanghai Shuguang Project of China under Grant No. 02SG02
文摘Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems （FDHS） in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.
基金Project(Y5080022) supported by the Natural Science Foundation of Zhejiang Province,ChinaProject(RC1202) supported by Scientific and Technological Program of Water Resources Department of Zhejiang Province in 2012,ChinaProject(Y201224384) supported by Scientific Research Program of Education Department of Zhejiang Province in 2012,China
文摘In order to simulate the airflow in anhydrous case and the water-air flow in groundwater case, a numerical model of airflow in soil was developed. For the nonlinearity of the governing partial differential equation, the corresponding discretization and linearization methods were given. Due to the mass transfer between air-phase and water-phase, phase states of the model elements were constantly changing. Thus, parameters of the model were divided into primary ones and secondary ones, and the primary variables changing with phase states and the secondary variables can be obtained by their functional relationship with the primary variables. Additionally, the special definite condition of this numerical model was illustrated. Two examples were given to simulate the airflow in soil whether there was groundwater or not, and the effectiveness of the numerical model is verified by comparing the results of simulation with that of exoeriment.
