In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more genera...In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more general exact travelling wave solutions for (2+ 1)-dimensional KK equation by the symmetry method and the (G1/G)-expansion method. Consequently, we find some new solutions of (2+1)-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.展开更多
A new nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained from the mKP equation by means of an asymptotically exact reduction method based on Fourier expansion and spatio-temporal resealing....A new nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained from the mKP equation by means of an asymptotically exact reduction method based on Fourier expansion and spatio-temporal resealing. In order to demonstrate integrability property of the new equation, the corresponding Lax pair is obtained by applying the reduction technique to the Lax pair of the mKP equation.展开更多
In this paper, we investigate symmetries of the new (4+1)-dimensional Fokas equation, including point symmetries and the potential symmetries. We firstly employ the algorithmic procedure of computing the point symm...In this paper, we investigate symmetries of the new (4+1)-dimensional Fokas equation, including point symmetries and the potential symmetries. We firstly employ the algorithmic procedure of computing the point symmetries. And then we transform the Fokas equation into a potential system and gain the potential symmetries of Fokas equation. Finally, we use the obtained point symmetries wave solutions and other solutions of the Fokas equation. and some constructive methods to get some doubly periodic In particular, some solitary wave solutions are also given.展开更多
Abstract By applying the Lie group method, the (2+l)-dimensional soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional brea...Abstract By applying the Lie group method, the (2+l)-dimensional soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.展开更多
By means of the generalized direct method,a relationship is constructed between the new solutions andthe old ones of the (3+1)-dimensional breaking soliton equation.Based on the relationship,a new solution is obtained...By means of the generalized direct method,a relationship is constructed between the new solutions andthe old ones of the (3+1)-dimensional breaking soliton equation.Based on the relationship,a new solution is obtainedby using a given solution of the equation.The symmetry is also obtained for the (3+1)-dimensional breaking solitonequation.By using the equivalent vector of the symmetry,we construct a seven-dimensional symmetry algebra and getthe optimal system of group-invariant solutions.To every case of the optimal system,the (3+1)-dimensional breakingsoliton equation is reduced and some solutions to the reduced equations are obtained.Furthermore,some new explicitsolutions are found for the (3+1)-dimensional breaking soliton equation.展开更多
On the basis of dialectics basic laws and mathematical theorems of variable sets,this paper proposes a variable sets method for urban flood vulnerability assessment.In this method,the comprehensive relative membership...On the basis of dialectics basic laws and mathematical theorems of variable sets,this paper proposes a variable sets method for urban flood vulnerability assessment.In this method,the comprehensive relative membership degree of multi-indices is represented by an index relative difference degree,which follows the characteristics of dialectical philosophy and mathematics.According to the quality-quantity exchange theorem,the relative difference degree of two adjacent levels(h and h+1),whose index standard interval values cross the boundaries,equals 0 in the urban flood vulnerability assessment.On the basis of the opposite unity theorem,the sum of relative membership degrees should be equal to 1 when indices lie in the adjacent degrees h and h+1.The variable sets method is proved to be theoretically rigorous and computationally simple.This paper takes 29 cities of Hunan province as an example to assess the urban flood vulnerability,and then compares the results from this newly developed method with the assessment results obtained from the fuzzy comprehensive evaluation and fuzzy set pair analysis methods.展开更多
The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedur...The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev~ equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.展开更多
基金Supported by the Natural Science Foundation of Shandong Province in China under Grant No.Q2005A01
文摘In this paper, by using the symmetry method, the relationships between new explicit solutions and old ones of the (2+1)-dimensional Kaup-Kupershmidt (KK) equation are presented. We successfully obtain more general exact travelling wave solutions for (2+ 1)-dimensional KK equation by the symmetry method and the (G1/G)-expansion method. Consequently, we find some new solutions of (2+1)-dimensional KK equation, including similarity solutions, solitary wave solutions, and periodic solutions.
基金supported by National Natural Science Foundation of China under Grant No. 10575087the Natural Science Foundation of Zhejiang Province under Grant No. 102053
文摘A new nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained from the mKP equation by means of an asymptotically exact reduction method based on Fourier expansion and spatio-temporal resealing. In order to demonstrate integrability property of the new equation, the corresponding Lax pair is obtained by applying the reduction technique to the Lax pair of the mKP equation.
基金supported by the National Natural Science Foundation of China under Grant No.60821002the National Key Basic Research Program of China under Grant No.2004CB318000
文摘In this paper, we investigate symmetries of the new (4+1)-dimensional Fokas equation, including point symmetries and the potential symmetries. We firstly employ the algorithmic procedure of computing the point symmetries. And then we transform the Fokas equation into a potential system and gain the potential symmetries of Fokas equation. Finally, we use the obtained point symmetries wave solutions and other solutions of the Fokas equation. and some constructive methods to get some doubly periodic In particular, some solitary wave solutions are also given.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004zx16
文摘Abstract By applying the Lie group method, the (2+l)-dimensional soliton equation is reduced to some (1+1)-dimensional nonlinear equations. Based upon some new explicit solutions of the (2+1)-dimensional breaking soliton equation are obtained.
基金National Natural Science Foundation of China under Grant No.10735030Shanghai Leading Academic Discipline Project under Grant No.B412+1 种基金Natural Science Foundations of Zhejiang Province of China under Grant No.Y604056the Doctoral Foundation of Ningbo City under Grant No.2005A61030
文摘By means of the generalized direct method,a relationship is constructed between the new solutions andthe old ones of the (3+1)-dimensional breaking soliton equation.Based on the relationship,a new solution is obtainedby using a given solution of the equation.The symmetry is also obtained for the (3+1)-dimensional breaking solitonequation.By using the equivalent vector of the symmetry,we construct a seven-dimensional symmetry algebra and getthe optimal system of group-invariant solutions.To every case of the optimal system,the (3+1)-dimensional breakingsoliton equation is reduced and some solutions to the reduced equations are obtained.Furthermore,some new explicitsolutions are found for the (3+1)-dimensional breaking soliton equation.
基金supported by the National Natural Science Foundation of China(Grant Nos.51209032,51379027,51109025)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20100041120004)the Fundamental Research Funds for the Central Universities(Grnat No.DUT13JS06)
文摘On the basis of dialectics basic laws and mathematical theorems of variable sets,this paper proposes a variable sets method for urban flood vulnerability assessment.In this method,the comprehensive relative membership degree of multi-indices is represented by an index relative difference degree,which follows the characteristics of dialectical philosophy and mathematics.According to the quality-quantity exchange theorem,the relative difference degree of two adjacent levels(h and h+1),whose index standard interval values cross the boundaries,equals 0 in the urban flood vulnerability assessment.On the basis of the opposite unity theorem,the sum of relative membership degrees should be equal to 1 when indices lie in the adjacent degrees h and h+1.The variable sets method is proved to be theoretically rigorous and computationally simple.This paper takes 29 cities of Hunan province as an example to assess the urban flood vulnerability,and then compares the results from this newly developed method with the assessment results obtained from the fuzzy comprehensive evaluation and fuzzy set pair analysis methods.
基金supported by the National Natural Science Foundation of China(Nos.11275072,11435005)the Research Fund for the Doctoral Program of Higher Education of China(No.20120076110024)+3 种基金the Innovative Research Team Program of the National Natural Science Foundation of China(No.61321064)the Shanghai Knowledge Service Platform for Trustworthy Internet of Things(No.ZF1213)the Shanghai Minhang District Talents of High Level Scientific Research Project and the Talent FundK.C.Wong Magna Fund in Ningbo University
文摘The nonlocal symmetry of the Boussinesq equation is obtained from the known Lax pair. The explicit analytic interaction solutions between solitary waves and cnoidal waves are obtained through the localization procedure of nonlocal symmetry. Some other types of solutions, such as rational solutions and error function solutions, are given by using the fourth Painlev~ equation with special values of the parameters. For some interesting solutions, the figures are given out to show their properties.