Quantum integrability provides a unique and powerful framework for accurately understanding quantum magnetism.In this review,we focus specifically on several quantum integrable low-dimensional quantum Ising models.We ...Quantum integrability provides a unique and powerful framework for accurately understanding quantum magnetism.In this review,we focus specifically on several quantum integrable low-dimensional quantum Ising models.We begin with the transverse field Ising chain(TFIC)at quantum critical point and examine how it evolves under perturbations,such as an applied longitudinal field or weak coupling to another quantum critical TFIC.展开更多
In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p...In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.展开更多
Some properties such as oscillation, stability, existence of periodic solutions and quadratic integrability of solutions based on a class of second order nonlinear delayed systems are analyzed by using the V-function,...Some properties such as oscillation, stability, existence of periodic solutions and quadratic integrability of solutions based on a class of second order nonlinear delayed systems are analyzed by using the V-function, the Lyapunov functional or the Beuman-Bihari inequality, and some sufficient conditions based on those properties are given. Finally, the conclusions are applied to over-voltage models based on three-phase nonsynchronous closing of switches appearing in the power systems, the results in accord with the background physical meaning are obtained. And all the conditions of the conclusions are easy to validate, so the conclusions have definite theoretical meaning and are easy to apply in practice.展开更多
A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discr...A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.展开更多
This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equartions as ■The existence result of weak so...This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equartions as ■The existence result of weak solutions can be get based on Galerkin approximation. With the linear operator B constructed by BOGOVSKII, we show that the density ■is square integrable up to the boundary.展开更多
Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fer...Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.展开更多
This paper is concerned with the generalized nonlinear second-order equation.By the direct construction method,all of the first-order multipliers of the equation are obtained,and the corresponding complete conservatio...This paper is concerned with the generalized nonlinear second-order equation.By the direct construction method,all of the first-order multipliers of the equation are obtained,and the corresponding complete conservation laws(CLs) of such equations are provided.Furthermore,the integrability of the equation is considered in terms of the conservation laws.In addition,the relationship of multipliers and symmetries of the equations is investigated.展开更多
The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equa...The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equations of motion are also found.展开更多
The integrability and the TST transformation of the bosonic string on AdS3× S^3 are studied. The Lax connection for gauge fixed theory is constructed and proved to be fiat.
Painleve integrability has been tested for （2＋1）D Boussinesq equation with disturbance term using the standard WTC approach after introducing the Kruskai＇s simplification. New breather solitary solutions depending...Painleve integrability has been tested for （2＋1）D Boussinesq equation with disturbance term using the standard WTC approach after introducing the Kruskai＇s simplification. New breather solitary solutions depending on constant equilibrium solution are obtained by using Extended Homoclinic Test Method. Moreover, the spatiotemporal feature of breather solitary wave is exhibited.展开更多
It is shown that the two-component Camassa-Holm and Hunter-Saxton systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws are ...It is shown that the two-component Camassa-Holm and Hunter-Saxton systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws are directly constructed. In addition, a class of nonlocal symmetries depending on the pseudo-potentials are obtained.展开更多
By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distin...By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.展开更多
Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct me...Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou. As a result, the nonlinear wave equation is not integrable.展开更多
The integrability of the （2＋l）-dimensional Broer-Kaup equation with variable coefficients （VCBK） is verified by finding a transformation mapping it to the usual （2＋l）-dimensional Broer-Kaup equation （BK）. Th...The integrability of the （2＋l）-dimensional Broer-Kaup equation with variable coefficients （VCBK） is verified by finding a transformation mapping it to the usual （2＋l）-dimensional Broer-Kaup equation （BK）. Thus the solutions of the （2＋1）-dimensional VCBK are obtained by making full use of the known solutions of the usual （2＋1）dimensional IRK. Two new integrable models are given by this transformation, their dromion-like solutions and rogue wave solutions are also obtained. Further, the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.展开更多
We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first resul...We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.展开更多
This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integra...This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.展开更多
This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied （TsT） transformations on S^3 and AdS3, respectively. It finds that the U（...This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied （TsT） transformations on S^3 and AdS3, respectively. It finds that the U（1） currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds generated by TsT transformations on both S3 and ADS3. Applying the TsT transformations, it derives the local Lax connections and the monodromy matrices in γ-deformed backgrounds with the spectral parameter which ensure the classical integrability of the string theories.展开更多
In this paper, we show that there exists a twisted duality symmetry between the Maurer-Cartan equations and the equations of motion in the hybrid formalism for the type liB superstring in an AdS2 ×S2 background w...In this paper, we show that there exists a twisted duality symmetry between the Maurer-Cartan equations and the equations of motion in the hybrid formalism for the type liB superstring in an AdS2 ×S2 background with Ramond-Ramond flux. As a result, from the twisted duality transformation, we construct the Lax connection with the spectral parameter, which ensures the integrability of the system.展开更多
This paper is concerned with the generalized variable-coefficient nonlinear evolution equation（vc-NLEE）.The complete integrability classification is presented,and the integrable conditions for the generalized variab...This paper is concerned with the generalized variable-coefficient nonlinear evolution equation（vc-NLEE）.The complete integrability classification is presented,and the integrable conditions for the generalized variable-coefficient equations are obtained by the Painlevé analysis.Then,the exact explicit solutions to these vc-NLEEs are investigated by the truncated expansion method,and the Lax pairs（LP） of the vc-NLEEs are constructed in terms of the integrable conditions.展开更多
A quantum integrability index was proposed in Komatsu et al(2019 SciPost Phys.7065).It systematizes the Goldschmidt and Witten's operator counting argument(Goldschmidt and Witten 1980 Phys.Lett.B 91392)by using th...A quantum integrability index was proposed in Komatsu et al(2019 SciPost Phys.7065).It systematizes the Goldschmidt and Witten's operator counting argument(Goldschmidt and Witten 1980 Phys.Lett.B 91392)by using the conformal symmetry.In this work we compute the quantum integrability indexes for the symmetric coset models SU(N)/SO(N)and SO(2N)/SO(N)×SO(N).The indexes of these theories are all non-positive except for the case of SO(4)/SO(2)×SO(2).Moreover we extend the analysis to the theories with fermions and consider a concrete theory:the CP~N model coupled with a massless Dirac fermion.We find that the indexes for this class of models are non-positive as well.展开更多
基金supported by the National Natural Science Foundation of China Grant Nos.12450004,12274288the Innovation Program for Quantum Science and Technology Grant No.2021ZD0301900。
文摘Quantum integrability provides a unique and powerful framework for accurately understanding quantum magnetism.In this review,we focus specifically on several quantum integrable low-dimensional quantum Ising models.We begin with the transverse field Ising chain(TFIC)at quantum critical point and examine how it evolves under perturbations,such as an applied longitudinal field or weak coupling to another quantum critical TFIC.
文摘In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.
文摘Some properties such as oscillation, stability, existence of periodic solutions and quadratic integrability of solutions based on a class of second order nonlinear delayed systems are analyzed by using the V-function, the Lyapunov functional or the Beuman-Bihari inequality, and some sufficient conditions based on those properties are given. Finally, the conclusions are applied to over-voltage models based on three-phase nonsynchronous closing of switches appearing in the power systems, the results in accord with the background physical meaning are obtained. And all the conditions of the conclusions are easy to validate, so the conclusions have definite theoretical meaning and are easy to apply in practice.
基金The project supported by the Scientific Research Award Foundation for Outstanding Young and Middle-Aged Scientists of Shandong Province of China
文摘A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.
基金supported by the National Natural Science Foundation of China(11271305,11531010)
文摘This work consider boundary integrability of the weak solutions of a non-Newtonian compressible fluids in a bounded domain in dimension three, which has the constitutive equartions as ■The existence result of weak solutions can be get based on Galerkin approximation. With the linear operator B constructed by BOGOVSKII, we show that the density ■is square integrable up to the boundary.
文摘Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11171041 and 10971018the Natural Science Foundation of Shandong Province under Grant No.ZR2010AM029+1 种基金the Promotive Research Fund for Young and Middle-Aged Scientists of Shandong Province under Grant No.BS2010SF001 the Doctoral Foundation of Binzhou University under Grant No.2009Y01
文摘This paper is concerned with the generalized nonlinear second-order equation.By the direct construction method,all of the first-order multipliers of the equation are obtained,and the corresponding complete conservation laws(CLs) of such equations are provided.Furthermore,the integrability of the equation is considered in terms of the conservation laws.In addition,the relationship of multipliers and symmetries of the equations is investigated.
文摘The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equations of motion are also found.
基金National Natural Science Foundation of China under Grant Nos.90403019 and 10575080Science and Technology Foundation of Xi'an Shiyou University under Grant No.2006-43
文摘The integrability and the TST transformation of the bosonic string on AdS3× S^3 are studied. The Lax connection for gauge fixed theory is constructed and proved to be fiat.
基金Supported by National Natural Science Foundation of China under Grant Nos. 11061028, 11261049, Yunnan Natural Science Foundation under Grant Nos. 2010CD086 and 2011Y012 and Qujing Normal University Natural Science Foundation under Grant Nos. 2009ZD002 and 2012QN016
文摘Painleve integrability has been tested for （2＋1）D Boussinesq equation with disturbance term using the standard WTC approach after introducing the Kruskai＇s simplification. New breather solitary solutions depending on constant equilibrium solution are obtained by using Extended Homoclinic Test Method. Moreover, the spatiotemporal feature of breather solitary wave is exhibited.
基金Supported by the China NSF for Distinguished Young Scholars under Grant No.10925104
文摘It is shown that the two-component Camassa-Holm and Hunter-Saxton systems are geometrically integrable, namely they describe pseudo-spherical surfaces. As a consequence, their infinite number of conservation laws are directly constructed. In addition, a class of nonlocal symmetries depending on the pseudo-potentials are obtained.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11201290 and 71103118)
文摘By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.
文摘Four types of similarity reductions are obtained for the nonlinear wave equation arising in the elasto-plasticmicrostructure model by using both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou. As a result, the nonlinear wave equation is not integrable.
基金Supported by the National Natural Science Foundation of China under Grant No.10971109K.C. Wong Magna Fund in Ningbo Universitythe Natural Science Foundation of Ningbo under Grant No.2011A610179
文摘The integrability of the （2＋l）-dimensional Broer-Kaup equation with variable coefficients （VCBK） is verified by finding a transformation mapping it to the usual （2＋l）-dimensional Broer-Kaup equation （BK）. Thus the solutions of the （2＋1）-dimensional VCBK are obtained by making full use of the known solutions of the usual （2＋1）dimensional IRK. Two new integrable models are given by this transformation, their dromion-like solutions and rogue wave solutions are also obtained. Further, the velocity of the dromion-like solutions can be designed and the center of the rogue wave solutions can be controlled artificially because of the appearance of the four arbitrary functions in the transformation.
文摘We generalize several classical results on the integrability of trigonometric series and relations among the best approximation and the coefficients of trigonometric series. Theorem 3 and Theorem 4 are the first results on the relations among the weighted best approximation and the coefficients of trigonometric series.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11171041 and 11505090Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009the doctorial foundation of Liaocheng University under Grant No.31805
文摘This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 90403019 and 10575080)
文摘This paper considers classical strings propagating in γ-deformed AdS3 γ S^3 backgrounds generated by certain shift T-dualities accompanied （TsT） transformations on S^3 and AdS3, respectively. It finds that the U（1） currents of strings with the twisted boundary conditions are equal to those in γ-deformed backgrounds generated by TsT transformations on both S3 and ADS3. Applying the TsT transformations, it derives the local Lax connections and the monodromy matrices in γ-deformed backgrounds with the spectral parameter which ensure the classical integrability of the string theories.
基金The project supported by National Natural Science Foundation of China under Grant No. 90403019 Acknowledgments We would like to thank Prof. Kang-Jie Shi for many stimulating discussions. Xie is also grateful to Zhan-Yun Wang, Xiao-Lin Cai, Pei Song, and Jun Feng for their helpful discussions, and especially A-Ping Yang＇s encouragements.
文摘In this paper, we show that there exists a twisted duality symmetry between the Maurer-Cartan equations and the equations of motion in the hybrid formalism for the type liB superstring in an AdS2 ×S2 background with Ramond-Ramond flux. As a result, from the twisted duality transformation, we construct the Lax connection with the spectral parameter, which ensures the integrability of the system.
基金Project supported by the National Natural Science Foundation of China(Grant No.11171041)the High-Level Personnel Foundation of Liaocheng University(Grant No.31805)
文摘This paper is concerned with the generalized variable-coefficient nonlinear evolution equation（vc-NLEE）.The complete integrability classification is presented,and the integrable conditions for the generalized variable-coefficient equations are obtained by the Painlevé analysis.Then,the exact explicit solutions to these vc-NLEEs are investigated by the truncated expansion method,and the Lax pairs（LP） of the vc-NLEEs are constructed in terms of the integrable conditions.
基金JT would like to thank Shota Komatsu for his inspiring lectures on integrability at the 13th Kavli Asian Winter School.The work was in part supported by NSFC Grant No.11335012,No.11325522 and No.11735001。
文摘A quantum integrability index was proposed in Komatsu et al(2019 SciPost Phys.7065).It systematizes the Goldschmidt and Witten's operator counting argument(Goldschmidt and Witten 1980 Phys.Lett.B 91392)by using the conformal symmetry.In this work we compute the quantum integrability indexes for the symmetric coset models SU(N)/SO(N)and SO(2N)/SO(N)×SO(N).The indexes of these theories are all non-positive except for the case of SO(4)/SO(2)×SO(2).Moreover we extend the analysis to the theories with fermions and consider a concrete theory:the CP~N model coupled with a massless Dirac fermion.We find that the indexes for this class of models are non-positive as well.
