More recently,sixteen families of Jacobian elliptic function solutions of mKdV equation have been found by using our extended jacobian elliptic function expansion method.In this paper,we continue to improve our method...More recently,sixteen families of Jacobian elliptic function solutions of mKdV equation have been found by using our extended jacobian elliptic function expansion method.In this paper,we continue to improve our method by using another eight pairs of the closed Jacobian elliptic functions.The mKdV equation is chosen to illustrate the improved method such that another eight families of new Jacobian elliptic functioin solutions are obtained again.The new method can be more powerful to be applied to other nonlinear differential equations.展开更多
We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as ...We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction, in the and even models, and dromion solutions (exponentially decaying solutions in all direction) in many and models. In this paper, symmetry reductions in are considered for the break soliton-type equation with fully nonlinear dispersion (called equation) , which is a generalized model of break soliton equation , by using the extended direct reduction method. As a result, six types of symmetry reductions are obtained. Starting from the reduction equations and some simple transformations, we obtain the solitary wave solutions of equations, compacton solutions of equations and the compacton-like solution of the potential form (called ) . In addition, we show that the variable admits dromion solutions rather than the field itself in equation.展开更多
Searching for special solitary wave solutions with compact support is of important significance in soliton theory. In this paper, to understand the role of nonlinear dispersion in pattern formation, a family of the re...Searching for special solitary wave solutions with compact support is of important significance in soliton theory. In this paper, to understand the role of nonlinear dispersion in pattern formation, a family of the regularized long-wave Boussinesq equations with fully nonlinear dispersion (simply called equations), ( const.), is studied. New solitary wave solutions with compact support of equations are found. In addition we find another compacton solutions of the two special cases, equation and equation. It is found that the nonlinear dispersion term in a nonlinear evolution equation is not a necessary condition of that it possesses compacton solutions.展开更多
Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct me...Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou. As a result, the nonlinear wave equation is not integrable.展开更多
The two-parameter family of Estevez-Mansfield-Clarkson equations with fully nonlinear dispersion (called E(m, n) equations), (uzm)zzr + γ(unzur)z + urr = 0 which is a generalized model of the integrable Estevez-Mansf...The two-parameter family of Estevez-Mansfield-Clarkson equations with fully nonlinear dispersion (called E(m, n) equations), (uzm)zzr + γ(unzur)z + urr = 0 which is a generalized model of the integrable Estevez-MansfieldClarkson equation u + γ(uzuzr +uzzur) +urr = 0, is presented. Five types of symmetries of the E(m, n) equation are obtained by making use of the direct reduction method. Using these obtained reductions and some simple tranaformations,we obtain the solitary-like wave solutions of E(1, n) equation. In addition, we also find the compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they reemerge with the same coherent shape) orE(3, 2) equation and E(m, m- 1) for its potentials, say, uz, and compacton-like solutions of E(m, m- 1)equations, respectively. Whether there exist compacton-like solutions of the other E(m, n) equation with m ≠ n + 1 is still an open problem.展开更多
We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2+1)-dimensional space,ut+1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance metho...We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2+1)-dimensional space,ut+1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance method.As is well known,the introduction of the concept of dromions (the exponentially localized solutions in (2+1)-dimensional space)has triggered renewed interest in (2+1)-dimensional soliton systems.The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself.In addition,it is shown that the equation passes Painleve test.展开更多
文摘More recently,sixteen families of Jacobian elliptic function solutions of mKdV equation have been found by using our extended jacobian elliptic function expansion method.In this paper,we continue to improve our method by using another eight pairs of the closed Jacobian elliptic functions.The mKdV equation is chosen to illustrate the improved method such that another eight families of new Jacobian elliptic functioin solutions are obtained again.The new method can be more powerful to be applied to other nonlinear differential equations.
文摘We have found two types of important exact solutions, compacton solutions, which are solitary waves with the property that after colliding with their own kind, they re-emerge with the same coherent shape very much as the solitons do during a completely elastic interaction, in the and even models, and dromion solutions (exponentially decaying solutions in all direction) in many and models. In this paper, symmetry reductions in are considered for the break soliton-type equation with fully nonlinear dispersion (called equation) , which is a generalized model of break soliton equation , by using the extended direct reduction method. As a result, six types of symmetry reductions are obtained. Starting from the reduction equations and some simple transformations, we obtain the solitary wave solutions of equations, compacton solutions of equations and the compacton-like solution of the potential form (called ) . In addition, we show that the variable admits dromion solutions rather than the field itself in equation.
基金National Key Basic Research Development Project Program of China under Grant，Doctoral Foundation of China under Grant，国家自然科学基金
文摘Searching for special solitary wave solutions with compact support is of important significance in soliton theory. In this paper, to understand the role of nonlinear dispersion in pattern formation, a family of the regularized long-wave Boussinesq equations with fully nonlinear dispersion (simply called equations), ( const.), is studied. New solitary wave solutions with compact support of equations are found. In addition we find another compacton solutions of the two special cases, equation and equation. It is found that the nonlinear dispersion term in a nonlinear evolution equation is not a necessary condition of that it possesses compacton solutions.
文摘Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou. As a result, the nonlinear wave equation is not integrable.
文摘The two-parameter family of Estevez-Mansfield-Clarkson equations with fully nonlinear dispersion (called E(m, n) equations), (uzm)zzr + γ(unzur)z + urr = 0 which is a generalized model of the integrable Estevez-MansfieldClarkson equation u + γ(uzuzr +uzzur) +urr = 0, is presented. Five types of symmetries of the E(m, n) equation are obtained by making use of the direct reduction method. Using these obtained reductions and some simple tranaformations,we obtain the solitary-like wave solutions of E(1, n) equation. In addition, we also find the compacton solutions (which are solitary waves with the property that after colliding with other compacton solutions, they reemerge with the same coherent shape) orE(3, 2) equation and E(m, m- 1) for its potentials, say, uz, and compacton-like solutions of E(m, m- 1)equations, respectively. Whether there exist compacton-like solutions of the other E(m, n) equation with m ≠ n + 1 is still an open problem.
文摘We obtain Backlund transformation and some new kink-like solitary wave solutions for the generalized Burgers equation in (2+1)-dimensional space,ut+1/2(uδy^-1ux)x-uxx=0,by using the extended homogeneous balance method.As is well known,the introduction of the concept of dromions (the exponentially localized solutions in (2+1)-dimensional space)has triggered renewed interest in (2+1)-dimensional soliton systems.The solutions obtained are used to show that the variable ux admits exponentially localized solutions rather than the physical field u(x,y,t) itself.In addition,it is shown that the equation passes Painleve test.
