A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations.The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation,the coupled Korte...A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations.The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation,the coupled Korteweg-de Vries system,the Kadomtsev–Petviashvili equation,the Ablowitz–Kaup–Newell–Segur equation and the potential Korteweg-de Vries equation.The facts show that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations.展开更多
We investigate the Lax equation that can be employed to describe motions of long waves in shallow water under gravity.A nonlocal symmetry of this equation is given and used to find exact solutions and derive lower int...We investigate the Lax equation that can be employed to describe motions of long waves in shallow water under gravity.A nonlocal symmetry of this equation is given and used to find exact solutions and derive lower integrable models from higher ones.It is interesting that this nonlocal symmetry links with its corresponding Riccati-type pseudopotential.By introducing suitable and simple auxiliary dependent variables,the nonlocal symmetry is localized and used to generate new solutions from trivial solutions.Meanwhile,this equation is reduced to an ordinary differential equation by means of this nonlocal symmetry and some local symmetries.展开更多
Employing the modified Clarkson-Kruskal direct method,we realize the symmetries of the nonlinear(2+1)-dimensional modified Kadomtzev-Patvishvili-Ⅱequation.Applying the given Lie symmetry,we obtain the similarity redu...Employing the modified Clarkson-Kruskal direct method,we realize the symmetries of the nonlinear(2+1)-dimensional modified Kadomtzev-Patvishvili-Ⅱequation.Applying the given Lie symmetry,we obtain the similarity reduction and new exact solutions.We also obtain conservation laws of the equations with the corresponding Lie symmetry.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos 11075055 and 11275072the Innovative Research Team Program of the National Natural Science Foundation of China(No 61021004)+1 种基金the National High-Technology Research and Development Program of China(No 2011AA010101)Shanghai Knowledge Service Platform for Trustworthy Internet of Things(No ZF1213)。
文摘A method is proposed to seek the nonlocal symmetries of nonlinear evolution equations.The validity and advantages of the proposed method are illustrated by the applications to the Boussinesq equation,the coupled Korteweg-de Vries system,the Kadomtsev–Petviashvili equation,the Ablowitz–Kaup–Newell–Segur equation and the potential Korteweg-de Vries equation.The facts show that this method can obtain not only the nonlocal symmetries but also the general Lie point symmetries of the given equations.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11275072,11075055the Innovative Research Team Program of the National Natural Science Foundation of China(No 61021004)+1 种基金Shanghai Leading Academic Discipline Project(No B412)the National High-Technology Research and Development Program(No 2011AA010101).
文摘We investigate the Lax equation that can be employed to describe motions of long waves in shallow water under gravity.A nonlocal symmetry of this equation is given and used to find exact solutions and derive lower integrable models from higher ones.It is interesting that this nonlocal symmetry links with its corresponding Riccati-type pseudopotential.By introducing suitable and simple auxiliary dependent variables,the nonlocal symmetry is localized and used to generate new solutions from trivial solutions.Meanwhile,this equation is reduced to an ordinary differential equation by means of this nonlocal symmetry and some local symmetries.
基金Supported by the Natural Science Foundation of Shandong Province under Grant Nos Y2008A35 and Y2007G64.
文摘Employing the modified Clarkson-Kruskal direct method,we realize the symmetries of the nonlinear(2+1)-dimensional modified Kadomtzev-Patvishvili-Ⅱequation.Applying the given Lie symmetry,we obtain the similarity reduction and new exact solutions.We also obtain conservation laws of the equations with the corresponding Lie symmetry.
