A simple and direct method is applied to solving the (2+1)-dimensional perturbed Ablowitz–Kaup–Newell–Segur system (PAKNS). Starting from a special B?cklund transformation and the variable separation approach, we c...A simple and direct method is applied to solving the (2+1)-dimensional perturbed Ablowitz–Kaup–Newell–Segur system (PAKNS). Starting from a special B?cklund transformation and the variable separation approach, we convert the PAKNS system into the simple forms, which are four variable separation equations, then obtain a quite general solution. Some special localized coherent structures like fractal dromions and fractal lumps of this model are constructed by selecting some types of lower-dimensional fractal patterns.展开更多
The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger...The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.展开更多
An explicit N-flod Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed.By using the Darboux transformation,the solutions of the evolution equations are reduced to s...An explicit N-flod Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed.By using the Darboux transformation,the solutions of the evolution equations are reduced to solving a linear algebraic system,from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given.Furthermore,a reduction, technique for MKdV equation is presented.and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique.A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.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
1. IntroductionThere has been nowadays much progress in investigating the algebraic structures related to integrable nonlinear evolution equations (NEEs), most of which can be brought together as: giving approaches to...1. IntroductionThere has been nowadays much progress in investigating the algebraic structures related to integrable nonlinear evolution equations (NEEs), most of which can be brought together as: giving approaches to construct hierarchies of NEEs starting from the representation theo-展开更多
The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system an...The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.展开更多
文摘A simple and direct method is applied to solving the (2+1)-dimensional perturbed Ablowitz–Kaup–Newell–Segur system (PAKNS). Starting from a special B?cklund transformation and the variable separation approach, we convert the PAKNS system into the simple forms, which are four variable separation equations, then obtain a quite general solution. Some special localized coherent structures like fractal dromions and fractal lumps of this model are constructed by selecting some types of lower-dimensional fractal patterns.
基金supported by China Postdoctoral Science Foundation and National Natural Science Foundation of China under Grant No.10771207
文摘The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.
文摘An explicit N-flod Darboux transformation for evolution equations determined by general 2×2 AKNS system is constructed.By using the Darboux transformation,the solutions of the evolution equations are reduced to solving a linear algebraic system,from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given.Furthermore,a reduction, technique for MKdV equation is presented.and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique.A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.
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.
基金Project supported by the Fok Ying-Tung Education Foundation and Science Fund of State Education Committee of China.
文摘1. IntroductionThere has been nowadays much progress in investigating the algebraic structures related to integrable nonlinear evolution equations (NEEs), most of which can be brought together as: giving approaches to construct hierarchies of NEEs starting from the representation theo-
文摘The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.
