在本文中,我们研究了扰动的(2 + 1)维Gardner方程孤立波解的存在性。首先,我们利用平面动力系统的相关知识对未扰系统的平衡点进行研究,得到了孤立波解存在的参数条件并画出了相图。然后,利用几何奇异摄动理论、不变流形理论和弗雷德霍...在本文中,我们研究了扰动的(2 + 1)维Gardner方程孤立波解的存在性。首先,我们利用平面动力系统的相关知识对未扰系统的平衡点进行研究,得到了孤立波解存在的参数条件并画出了相图。然后,利用几何奇异摄动理论、不变流形理论和弗雷德霍姆理论,通过构造相应微分方程的不变流形,我们得到了相应的同宿轨道。进一步,根据同宿轨道和孤立波之间的对应关系,我们得到了扰动系统孤立波解的存在性。In this paper, we investigate the existence of solitary wave solutions for perturbed (2 + 1) dimensional Gardner equation. Firstly, we utilize the relevant knowledge of planar dynamical systems to study the equilibrium points of undisturbed system, obtain the parameter conditions for the existence of solitary wave solutions, and drew a phase diagram. Then, using geometric singular perturbation theory, invariant manifold theory, and Fredholm theory, we obtain a homoclinic orbit by constructing the invariant manifold of the corresponding differential equation. Further, through the correspondence between homoclinic orbits and solitary waves, we prove the existence of solitary wave solutions for the perturbed system.展开更多
In this paper,two different methods for calculating the conservation laws are used,these are the direct construction of conservation laws and the conservation theorem proposed by Ibragimov.Using these two methods,we o...In this paper,two different methods for calculating the conservation laws are used,these are the direct construction of conservation laws and the conservation theorem proposed by Ibragimov.Using these two methods,we obtain the conservation laws of the Gardner equation,Landau-Ginzburg-Higgs equation and Hirota-Satsuma equation,respectively.展开更多
Recent studies have underscored the significance of the capillary fringe in hydrological and biochemical processes.Moreover,its role in shallow waters is expected to be considerable.Traditionally,the study of groundwa...Recent studies have underscored the significance of the capillary fringe in hydrological and biochemical processes.Moreover,its role in shallow waters is expected to be considerable.Traditionally,the study of groundwater flow has centered on unsaturated-saturated zones,often overlooking the impact of the capillary fringe.In this study,we introduce a steady-state two-dimensional model that integrates the capillary fringe into a 2-D numerical solution.Our novel approach employs the potential form of the Richards equation,facilitating the determination of boundaries,pressures,and velocities across different ground surface zones.We utilized a two-dimensional Freefem++finite element model to compute the stationary solution.The validation of the model was conducted using experimental data.We employed the OFAT(One_Factor-At-Time)method to identify the most sensitive soil parameters and understand how changes in these parameters may affect the behavior and water dynamics of the capillary fringe.The results emphasize the role of hydraulic conductivity as a key parameter influencing capillary fringe shape and dynamics.Velocity values within the capillary fringe suggest the prevalence of horizontal flow.By variation of the water table level and the incoming flow q0,we have shown the correlation between water table elevation and the upper limit of the capillary fringe.展开更多
文摘在本文中,我们研究了扰动的(2 + 1)维Gardner方程孤立波解的存在性。首先,我们利用平面动力系统的相关知识对未扰系统的平衡点进行研究,得到了孤立波解存在的参数条件并画出了相图。然后,利用几何奇异摄动理论、不变流形理论和弗雷德霍姆理论,通过构造相应微分方程的不变流形,我们得到了相应的同宿轨道。进一步,根据同宿轨道和孤立波之间的对应关系,我们得到了扰动系统孤立波解的存在性。In this paper, we investigate the existence of solitary wave solutions for perturbed (2 + 1) dimensional Gardner equation. Firstly, we utilize the relevant knowledge of planar dynamical systems to study the equilibrium points of undisturbed system, obtain the parameter conditions for the existence of solitary wave solutions, and drew a phase diagram. Then, using geometric singular perturbation theory, invariant manifold theory, and Fredholm theory, we obtain a homoclinic orbit by constructing the invariant manifold of the corresponding differential equation. Further, through the correspondence between homoclinic orbits and solitary waves, we prove the existence of solitary wave solutions for the perturbed system.
基金funded by the National Natural Science Foundation of China(Grant Nos.12371256&11971475)。
文摘In this paper,two different methods for calculating the conservation laws are used,these are the direct construction of conservation laws and the conservation theorem proposed by Ibragimov.Using these two methods,we obtain the conservation laws of the Gardner equation,Landau-Ginzburg-Higgs equation and Hirota-Satsuma equation,respectively.
文摘Recent studies have underscored the significance of the capillary fringe in hydrological and biochemical processes.Moreover,its role in shallow waters is expected to be considerable.Traditionally,the study of groundwater flow has centered on unsaturated-saturated zones,often overlooking the impact of the capillary fringe.In this study,we introduce a steady-state two-dimensional model that integrates the capillary fringe into a 2-D numerical solution.Our novel approach employs the potential form of the Richards equation,facilitating the determination of boundaries,pressures,and velocities across different ground surface zones.We utilized a two-dimensional Freefem++finite element model to compute the stationary solution.The validation of the model was conducted using experimental data.We employed the OFAT(One_Factor-At-Time)method to identify the most sensitive soil parameters and understand how changes in these parameters may affect the behavior and water dynamics of the capillary fringe.The results emphasize the role of hydraulic conductivity as a key parameter influencing capillary fringe shape and dynamics.Velocity values within the capillary fringe suggest the prevalence of horizontal flow.By variation of the water table level and the incoming flow q0,we have shown the correlation between water table elevation and the upper limit of the capillary fringe.
