With the aid of the classical Lie group method and nonclassical Lie group method,we derive the classicalLie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS)equation.Usi...With the aid of the classical Lie group method and nonclassical Lie group method,we derive the classicalLie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS)equation.Usingthe symmetries,we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation.Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.展开更多
Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the (2+1)- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the gr...Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the (2+1)- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou's direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada-Kotera and Konopelchenko Dubrovsky equations, respectively.展开更多
The symmetries of the (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe the atmospheric gravity waves (GWs), are researched in this paper. The Lie symmetries...The symmetries of the (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe the atmospheric gravity waves (GWs), are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. Calculation shows the INHB equations are invariant under some Galilean transformations, scaling transformations, and space-time translations. The symmetry reduction equations and similar solutions of the INHB equations are proposed.展开更多
A complete approximate symmetry classification of a class of perturbed nonlinear wave equations isperformed using the method originated from Fushchich and Shtelen.Moreover,large classes of approximate invariantsolutio...A complete approximate symmetry classification of a class of perturbed nonlinear wave equations isperformed using the method originated from Fushchich and Shtelen.Moreover,large classes of approximate invariantsolutions of the equations based on the Lie group method are constructed.展开更多
Applying the Lie group method to the differential-difference equation, the Lie point symmetry of Blaszak- Marciniak four-field Lattice equation is obtained. Using the obtained symmetry, the similarity reduction equati...Applying the Lie group method to the differential-difference equation, the Lie point symmetry of Blaszak- Marciniak four-field Lattice equation is obtained. Using the obtained symmetry, the similarity reduction equations of Blaszak-Marciniak four-field Lattice equation are derived. Solving the reduction, we get the solution of Blaszak-Marciniak four-field Lattice equation which not only recovers one of the solutions obtained by Ma and Hu [J. Math. Phys. 40 (1999) 6071] but also has the singularity when we choose the arbitrary constants accurately.展开更多
Using the (2+1)-dimensional Broer-Kaup equation as an simple example, a new direct method is developed to find symmetry groups and symmetry algebras and then exact solutions of nonlinear mathematical physical equations.
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is ...Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.展开更多
We study the symmetries of a (2+1)-dimensional generalized Broer-Kaup system by means of the classical Lie group theory. The corresponding group algebra is constructed. Based on the symmetries, severaJ types of sim...We study the symmetries of a (2+1)-dimensional generalized Broer-Kaup system by means of the classical Lie group theory. The corresponding group algebra is constructed. Based on the symmetries, severaJ types of similarity solutions are obtained.展开更多
By use of a direct method, we discuss symmetries and reductions of the two-dimensional Burgers equation with variable coefficient (VCBurgers). Five types of symmetry-reducing VCBurgers to (1+1)-dimensional partial dif...By use of a direct method, we discuss symmetries and reductions of the two-dimensional Burgers equation with variable coefficient (VCBurgers). Five types of symmetry-reducing VCBurgers to (1+1)-dimensional partial differential equation and three types of symmetry reducing VCBurgers to ordinary differential equation are obtained.展开更多
The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and ...The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of f are estimated. Owren and Welfert (2000) stated that if the initial point is close sufficiently to a zero of f, then Newton’s method on Lie group converges to the zero; while this paper provides a Kantorovich’s criterion for the convergence of Newton’s method, not requiring the existence of a zero as a priori.展开更多
A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2+1)-dimensional asymmetric Nizhnik-Novikov-Ves...A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.展开更多
Using the modified CK's direct method, we derive a symmetry group theorem of (2+1)-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of (2+1)-...Using the modified CK's direct method, we derive a symmetry group theorem of (2+1)-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of (2+1)- dimensional dispersive long-wave equations are obtained.展开更多
Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few exp...Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.展开更多
基金Supported by National Natural Science Foundation of China and China Academy of Engineering Physics (NSAF 11076015)
文摘With the aid of the classical Lie group method and nonclassical Lie group method,we derive the classicalLie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS)equation.Usingthe symmetries,we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation.Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.
基金the State Key Basic Research Program of China under Grant No.2004CB318000
文摘Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the (2+1)- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou's direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada-Kotera and Konopelchenko Dubrovsky equations, respectively.
基金Supported by the Scientific Research Foundation for the Doctors of University of Electronic Science and Technology of China Zhongshan Institute under Grant No. 408YKQ09the National Natural Science Foundation of China under Grant No. 10735030
文摘The symmetries of the (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe the atmospheric gravity waves (GWs), are researched in this paper. The Lie symmetries and the corresponding reductions are obtained by means of classical Lie group approach. Calculation shows the INHB equations are invariant under some Galilean transformations, scaling transformations, and space-time translations. The symmetry reduction equations and similar solutions of the INHB equations are proposed.
文摘A complete approximate symmetry classification of a class of perturbed nonlinear wave equations isperformed using the method originated from Fushchich and Shtelen.Moreover,large classes of approximate invariantsolutions of the equations based on the Lie group method are constructed.
基金Supported by the National Natural Science Foundation of China under Grant No.10735030the National Natural Science Foundation of China under Grant No.90718041+1 种基金Shanghai Leading Academic Discipline Project under Grant No.B412Program for Changjiang Scholars and Innovative Research Team in University under Grant No.IRT0734
文摘Applying the Lie group method to the differential-difference equation, the Lie point symmetry of Blaszak- Marciniak four-field Lattice equation is obtained. Using the obtained symmetry, the similarity reduction equations of Blaszak-Marciniak four-field Lattice equation are derived. Solving the reduction, we get the solution of Blaszak-Marciniak four-field Lattice equation which not only recovers one of the solutions obtained by Ma and Hu [J. Math. Phys. 40 (1999) 6071] but also has the singularity when we choose the arbitrary constants accurately.
文摘Using the (2+1)-dimensional Broer-Kaup equation as an simple example, a new direct method is developed to find symmetry groups and symmetry algebras and then exact solutions of nonlinear mathematical physical equations.
文摘Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10475055 and 10547124 and partly by Shanghai Leading Academic Discipline Project under Grant No. T0401.Acknowledgments The authors would like to thank Prof. S.Y. Lou for his helpful discussions.
文摘We study the symmetries of a (2+1)-dimensional generalized Broer-Kaup system by means of the classical Lie group theory. The corresponding group algebra is constructed. Based on the symmetries, severaJ types of similarity solutions are obtained.
文摘By use of a direct method, we discuss symmetries and reductions of the two-dimensional Burgers equation with variable coefficient (VCBurgers). Five types of symmetry-reducing VCBurgers to (1+1)-dimensional partial differential equation and three types of symmetry reducing VCBurgers to ordinary differential equation are obtained.
基金Project supported by the National Natural Science Foundation of China (No. 10271025)the Program for New Century Excellent Talents in University of China
文摘The convergence criterion of Newton’s method to find the zeros of a map f from a Lie group to its corresponding Lie algebra is established under the assumption that f satisfies the classical Lipschitz condition, and that the radius of convergence ball is also obtained. Furthermore, the radii of the uniqueness balls of the zeros of f are estimated. Owren and Welfert (2000) stated that if the initial point is close sufficiently to a zero of f, then Newton’s method on Lie group converges to the zero; while this paper provides a Kantorovich’s criterion for the convergence of Newton’s method, not requiring the existence of a zero as a priori.
基金The project supported by the National 0utstanding Youth Foundation of China under Grant No. 19925522 and the National Natural Science Foundation of China under Grant Nos. 90203001, 10475055. The authors are in debt to thank helpful discussions with Drs. X.Y. Tang, C.L. Chen, Y. Chen, H.C. Hu, X.M. Qian, B. Tong, and W.R. Cai.
文摘A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation and Nizhnik Novikov-Vesselov equation, both the Lie point symmetry groups and the non-Lie symmetry groups are obtained. The Lie symmetry groups obtained via traditional Lie approaches are only speciai cases. Furthermore, the expressions of the exact finite transformations of the Lie groups are much simpler than those obtained via the standard approaches.
基金supported by the Natural Science Foundation of Shandong Province of China under Grant Nos.Q2005A01
文摘Using the modified CK's direct method, we derive a symmetry group theorem of (2+1)-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of (2+1)- dimensional dispersive long-wave equations are obtained.
基金The project supported by National Natural Science Foundation of China under Grant No. 10471139 and Hong Kong Research Grant Council under Grant No. HKBU RGC 2016/05p
文摘Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.