In this paper, the symmetry method has been carried over to the generalized variable coefficients Zakharov- Kuznetsov equation. The infinitesimal symmetries and the optimal system are deduced and from this optimal sys...In this paper, the symmetry method has been carried over to the generalized variable coefficients Zakharov- Kuznetsov equation. The infinitesimal symmetries and the optimal system are deduced and from this optimal system seven basic fields are determined, and for every vector field in the optimal system the admissible forms of the coefficients are found and this also leads us to transform the given equation into partial differential equations in two variables. After using some referenced transformations the mentioned partial differential equations eventually reduce to ordinary differential equations. The search for solutions to those equations has yielded many exact solutions in most cases.展开更多
A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedi...A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.展开更多
A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a...A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.展开更多
A new loop algebra G is established to obtain integrable coupling of GJ hierarchy. In particular, iiitegrable coupling of the well-known AKNS hierarchy is presented. This method can be used generally.
This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear SchrSdinger eq...This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear SchrSdinger equations and systems developed by Martel et al. to the present non-integrable generalized Davey- Stewartson system and overcoming some new difficulties caused by the nonlocal operator B, we prove the existence of multi-solitons for this system. Furthermore, we also give a generalization of this result to a more general class of equations with nonlocal nonlinearities.展开更多
It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons i...It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiserete integrable system （the time variable is continuous and the space one is discrete） that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized KdV system （the both of time and space variables are discrete） introduced by Kanki et al. In this paper, for the generalized discrete KdV （gdKdV） equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gdKdV equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis.展开更多
基金Supported by the Natural Science Foundation of Inner Mongolia(Grant No 2009 MS0108)the High Education Science Research of Inner Mongolia Autonomous Region(Grant No.NJ10045)~~
文摘In this paper, the symmetry method has been carried over to the generalized variable coefficients Zakharov- Kuznetsov equation. The infinitesimal symmetries and the optimal system are deduced and from this optimal system seven basic fields are determined, and for every vector field in the optimal system the admissible forms of the coefficients are found and this also leads us to transform the given equation into partial differential equations in two variables. After using some referenced transformations the mentioned partial differential equations eventually reduce to ordinary differential equations. The search for solutions to those equations has yielded many exact solutions in most cases.
基金the Natural Science Foundation of Shandong Province under Grant No.Q2006A04
文摘A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.
文摘A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.
文摘A new loop algebra G is established to obtain integrable coupling of GJ hierarchy. In particular, iiitegrable coupling of the well-known AKNS hierarchy is presented. This method can be used generally.
基金supported by National Natural Science Foundation of China (Grant No. 11571381)
文摘This paper studies multi-solitons of non-integrable generalized Davey-Stewartson system in the elliptic-elliptic case. By extending the method for constructing multi-solitons of non-integrable nonlinear SchrSdinger equations and systems developed by Martel et al. to the present non-integrable generalized Davey- Stewartson system and overcoming some new difficulties caused by the nonlocal operator B, we prove the existence of multi-solitons for this system. Furthermore, we also give a generalization of this result to a more general class of equations with nonlocal nonlinearities.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11501353,11271254,11428102,and 11671255supported by the Ministry of Economy and Competitiveness of Spain under contracts MTM2012-37070 and MTM2016-80276-P(AEI/FEDER,EU)
文摘It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiserete integrable system （the time variable is continuous and the space one is discrete） that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized KdV system （the both of time and space variables are discrete） introduced by Kanki et al. In this paper, for the generalized discrete KdV （gdKdV） equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gdKdV equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis.
