A new type of symmetry,ren-symmetry,describing anyon physics and corresponding topological physics,is proposed.Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such...A new type of symmetry,ren-symmetry,describing anyon physics and corresponding topological physics,is proposed.Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as super-symmetric quantum mechanics,super-symmetric gravity,super-symmetric string theory,super-symmetric integrable systems and so on.Supersymmetry and Grassmann numbers are,in some sense,dual conceptions,and it turns out that these conceptions coincide for the ren situation,that is,a similar conception of ren-number(R-number)is devised for ren-symmetry.In particular,some basic results of the R-number and ren-symmetry are exposed which allow one to derive,in principle,some new types of integrable systems including ren-integrable models and ren-symmetric integrable systems.Training examples of ren-integrable KdV-type systems and ren-symmetric KdV equations are explicitly given.展开更多
This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application...This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.展开更多
To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4(CKP34)equation is found b...To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4(CKP34)equation is found by means of a direct method.The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group.The point symmetries of the CKP34 equation constitute an infinite dimensional KacMoody-Virasoro algebra.The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.展开更多
Multi-place nonlocal systems have attracted attention from many scientists.In this paper,we mainly review the recent progresses on two-place nonlocal systems(Alice-Bob systems)and four-place nonlocal models.Multi-plac...Multi-place nonlocal systems have attracted attention from many scientists.In this paper,we mainly review the recent progresses on two-place nonlocal systems(Alice-Bob systems)and four-place nonlocal models.Multi-place systems can firstly be derived from many physical problems by using a multiple scaling method with a discrete symmetry group including parity,time reversal,charge conjugates,rotations,field reversal and exchange transformations.Multiplace nonlocal systems can also be derived from the symmetry reductions of coupled nonlinear systems via discrete symmetry reductions.On the other hand,to solve multi-place nonlocal systems,one can use the symmetry-antisymmetry separation approach related to a suitable discrete symmetry group,such that the separated systems are coupled local ones.By using the separation method,all the known powerful methods used in local systems can be applied to nonlocal cases.In this review article,we take two-place and four-place nonlocal nonlinear Schr?dinger(NLS)systems and Kadomtsev-Petviashvili(KP)equations as simple examples to explain how to derive and solve them.Some types of novel physical and mathematical points related to the nonlocal systems are especially emphasized.展开更多
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical...The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.展开更多
A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimen...A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the(1+1)-dimensional Boussinesq equation.Because of the effects of the deformation,the(2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multivalued.展开更多
基金sponsored by the National Natural Science Foundation of China(Nos.12235007,11975131)。
文摘A new type of symmetry,ren-symmetry,describing anyon physics and corresponding topological physics,is proposed.Ren-symmetry is a generalization of super-symmetry which is widely applied in super-symmetric physics such as super-symmetric quantum mechanics,super-symmetric gravity,super-symmetric string theory,super-symmetric integrable systems and so on.Supersymmetry and Grassmann numbers are,in some sense,dual conceptions,and it turns out that these conceptions coincide for the ren situation,that is,a similar conception of ren-number(R-number)is devised for ren-symmetry.In particular,some basic results of the R-number and ren-symmetry are exposed which allow one to derive,in principle,some new types of integrable systems including ren-integrable models and ren-symmetric integrable systems.Training examples of ren-integrable KdV-type systems and ren-symmetric KdV equations are explicitly given.
基金sponsored by the National Natural Science Foundations of China(Nos.12301315,12235007,11975131)the Natural Science Foundation of Zhejiang Province(No.LQ20A010009).
文摘This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.
基金sponsored by the National Natural Science Foundations of China(Nos.11975131,11435005)
文摘To find symmetries,symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics.In this paper,the finite point symmetry group of the combined KP3 and KP4(CKP34)equation is found by means of a direct method.The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group.The point symmetries of the CKP34 equation constitute an infinite dimensional KacMoody-Virasoro algebra.The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.
基金sponsored by the National Natural Science Foundations of China(No.11975131,11435005)K C Wong Magna Fund in Ningbo University。
文摘Multi-place nonlocal systems have attracted attention from many scientists.In this paper,we mainly review the recent progresses on two-place nonlocal systems(Alice-Bob systems)and four-place nonlocal models.Multi-place systems can firstly be derived from many physical problems by using a multiple scaling method with a discrete symmetry group including parity,time reversal,charge conjugates,rotations,field reversal and exchange transformations.Multiplace nonlocal systems can also be derived from the symmetry reductions of coupled nonlinear systems via discrete symmetry reductions.On the other hand,to solve multi-place nonlocal systems,one can use the symmetry-antisymmetry separation approach related to a suitable discrete symmetry group,such that the separated systems are coupled local ones.By using the separation method,all the known powerful methods used in local systems can be applied to nonlocal cases.In this review article,we take two-place and four-place nonlocal nonlinear Schr?dinger(NLS)systems and Kadomtsev-Petviashvili(KP)equations as simple examples to explain how to derive and solve them.Some types of novel physical and mathematical points related to the nonlocal systems are especially emphasized.
基金sponsored by the National Natural Science Foundations of China(Nos.12235007,11975131,11435005,12275144,11975204)KC Wong Magna Fund in Ningbo UniversityNatural Science Foundation of Zhejiang Province No.LQ20A010009。
文摘The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.
基金support of the National Natural Science Foundation of China(Nos.12275144,12235007 and 11975131)the K C Wong Magna Fund at Ningbo University。
文摘A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the(1+1)-dimensional Boussinesq equation.Because of the effects of the deformation,the(2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multivalued.
