We study solutions of the nonlinear Schrdinger equation (NLSE) and higher-order nonlinear Schrdingerequation (HONLSE) with variable coefficients.By considering all the higher-order effect of HONLSE as a new dependentv...We study solutions of the nonlinear Schrdinger equation (NLSE) and higher-order nonlinear Schrdingerequation (HONLSE) with variable coefficients.By considering all the higher-order effect of HONLSE as a new dependentvariable,the NLSE and HONLSE can be changed into one equation.Using the generalized Lie group reduction method(GLGRM),the abundant solutions of NLSE and HONLSE are obtained.展开更多
It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which ...It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.展开更多
基金National Natural Science Foundation of China under Grant No.10675065
文摘We study solutions of the nonlinear Schrdinger equation (NLSE) and higher-order nonlinear Schrdingerequation (HONLSE) with variable coefficients.By considering all the higher-order effect of HONLSE as a new dependentvariable,the NLSE and HONLSE can be changed into one equation.Using the generalized Lie group reduction method(GLGRM),the abundant solutions of NLSE and HONLSE are obtained.
基金Supported by the Natural Science Foundation of Henan Province of China(0111050200) Natural Science Foundation of Education Committee of Henan Province of China(2003110003)the Science Foundation of Henan University of Science and Technology(2003ZY03)
文摘It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes？ To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.
