The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equ...The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equation by the Hirota bilinear method and test function method.With the help of Maple,we draw three-dimensional plots of the lump solution and lump-type solutions,and by observing the plots,we analyze the dynamic behavior of the(2+1)-dimensional dissipative AKNS equation.We find that the interaction solutions come in a variety of interesting forms.展开更多
The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal r...The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.展开更多
Infinitely many conservation laws for some （1＋1）-dimension soliton hierarchy with self-consistent sources are constructed from their corresponding Lax pairs directly. Three examples are given. Besides, infinitely m...Infinitely many conservation laws for some （1＋1）-dimension soliton hierarchy with self-consistent sources are constructed from their corresponding Lax pairs directly. Three examples are given. Besides, infinitely many conservation laws for Kadomtsev-Petviashvili （KP） hierarchy with self-consistent sources are obtained from the pseudo-differential operator and the Lax pair.展开更多
The n-fold Darboux transform (DT) is a 2×2 matrix for the Ablowitz-Kaup-Newell-Segur (AKNS) system.In this paper,each element of this matrix is expressed by 2n + 1 ranks'determinants.Using these formulae,the ...The n-fold Darboux transform (DT) is a 2×2 matrix for the Ablowitz-Kaup-Newell-Segur (AKNS) system.In this paper,each element of this matrix is expressed by 2n + 1 ranks'determinants.Using these formulae,the determinant expressions of eigenfunctions generated by the n-fold DT are obtained.Furthermore,we give out the explicit forms of the n-soliton surface of the Nonlinear Schrodinger Equation (NLS) by the determinant of eigenfunctions.展开更多
文摘The lump solution is one of the exact solutions of the nonlinear evolution equation.In this paper,we study the lump solution and lump-type solutions of(2+1)-dimensional dissipative Ablowitz-Kaup-Newell-Segure(AKNS)equation by the Hirota bilinear method and test function method.With the help of Maple,we draw three-dimensional plots of the lump solution and lump-type solutions,and by observing the plots,we analyze the dynamic behavior of the(2+1)-dimensional dissipative AKNS equation.We find that the interaction solutions come in a variety of interesting forms.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11305031,11365017,and 11305106)the Natural Science Foundation of Guangdong Province,China(Grant No.S2013010011546)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Grant No.LQ13A050001)the Science and Technology Project Foundation of Zhongshan,China(Grant Nos.2013A3FC0264 and 2013A3FC0334)the Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province,China(Grant No.Yq2013205)
文摘The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.
基金supported by the National Natural Science Foundation of China (Grant Nos.10371070, 10671121)the Shanghai Leading Academic Discipline Project (Grant No.J50101)the President Foundation of East China Institute of Technology (Grant No.DHXK0810)
文摘Infinitely many conservation laws for some （1＋1）-dimension soliton hierarchy with self-consistent sources are constructed from their corresponding Lax pairs directly. Three examples are given. Besides, infinitely many conservation laws for Kadomtsev-Petviashvili （KP） hierarchy with self-consistent sources are obtained from the pseudo-differential operator and the Lax pair.
基金This work was supported partly by the project 973"Nonlinear Science"the National Natural Science Foundation of China(Grant No.10301030)SRFDP of China.
文摘The n-fold Darboux transform (DT) is a 2×2 matrix for the Ablowitz-Kaup-Newell-Segur (AKNS) system.In this paper,each element of this matrix is expressed by 2n + 1 ranks'determinants.Using these formulae,the determinant expressions of eigenfunctions generated by the n-fold DT are obtained.Furthermore,we give out the explicit forms of the n-soliton surface of the Nonlinear Schrodinger Equation (NLS) by the determinant of eigenfunctions.
