For a given truncated Painleve′ expansion of an arbitrary nonlinear Painleve′ integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, wh...For a given truncated Painleve′ expansion of an arbitrary nonlinear Painleve′ integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, which is proved to be localized to Lie point symmetries for suitable prolonged systems. Taking the Korteweg–de Vries equation as an example, the n-th binary Darboux–Ba¨cklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.展开更多
An invariant function(IF) is defined as a multiplier of a symmetry that means a symmetry multiplied by an IF is still a symmetry. Primary branch solutions of arbitrary first order scalar systems can be obtained by mea...An invariant function(IF) is defined as a multiplier of a symmetry that means a symmetry multiplied by an IF is still a symmetry. Primary branch solutions of arbitrary first order scalar systems can be obtained by means of the IF and its related symmetry approach. Especially, one recursion operator and some sets of infinitely many high order symmetries are also explicitly given for arbitrary(1+1)-dimensional first order autonomous systems. Because of the intrusion of the arbitrary function, various implicit special exact solutions can be found by fixing the arbitrary functions and selecting different seed solutions.展开更多
The Chinese ancient sage Laozi said that everything comes from 'nothing'. In the work [Chin. Phys. Lett. 30?(2013)?080202], infinitely many discrete integrable systems have been obtained from nothing via simple ...The Chinese ancient sage Laozi said that everything comes from 'nothing'. In the work [Chin. Phys. Lett. 30?(2013)?080202], infinitely many discrete integrable systems have been obtained from nothing via simple principles (Dao). In this study, a new idea, the consistent correlated bang, is introduced to obtain nonlinear dynamic systems including some integrable ones such as the continuous nonlinear Schr?dinger equation, the (potential) Korteweg de Vries equation, the (potential) Kadomtsev–Petviashvili equation and the sine-Gordon equation. These nonlinear systems are derived from nothing via suitable 'Dao', the shifted parity, the charge conjugate, the delayed time reversal, the shifted exchange, the shifted-parity-rotation and so on.展开更多
The dark Korteweg-de Vries(KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar cl...The dark Korteweg-de Vries(KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV(MKdV) systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Furthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The recursion operators of two classes of dark MKdV systems are also given.展开更多
By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.Fo...By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11675055,11175092,and 11205092)the Program from Shanghai Knowledge Service Platform for Trustworthy Internet of Things(Grant No.ZF1213)K C Wong Magna Fund in Ningbo University
文摘For a given truncated Painleve′ expansion of an arbitrary nonlinear Painleve′ integrable system, the residue with respect to the singularity manifold is known as a nonlocal symmetry, called the residual symmetry, which is proved to be localized to Lie point symmetries for suitable prolonged systems. Taking the Korteweg–de Vries equation as an example, the n-th binary Darboux–Ba¨cklund transformation is re-obtained by the Lie point symmetry approach accompanied by the localization of the n-fold residual symmetries.
基金Supported by the National Natural Science Foundations of China under Grant Nos.11435005,11471004,11175092,and 11205092Shanghai Knowledge Service Platform for Trustworthy Internet of Things No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘An invariant function(IF) is defined as a multiplier of a symmetry that means a symmetry multiplied by an IF is still a symmetry. Primary branch solutions of arbitrary first order scalar systems can be obtained by means of the IF and its related symmetry approach. Especially, one recursion operator and some sets of infinitely many high order symmetries are also explicitly given for arbitrary(1+1)-dimensional first order autonomous systems. Because of the intrusion of the arbitrary function, various implicit special exact solutions can be found by fixing the arbitrary functions and selecting different seed solutions.
基金Supported by the Global Change Research Program of China under Grant No 2015CB953904the National Natural Science Foundation of China under Grant No 11435005+1 种基金the Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No ZF1213the K.C.Wong Magna Fund in Ningbo University
文摘The Chinese ancient sage Laozi said that everything comes from 'nothing'. In the work [Chin. Phys. Lett. 30?(2013)?080202], infinitely many discrete integrable systems have been obtained from nothing via simple principles (Dao). In this study, a new idea, the consistent correlated bang, is introduced to obtain nonlinear dynamic systems including some integrable ones such as the continuous nonlinear Schr?dinger equation, the (potential) Korteweg de Vries equation, the (potential) Kadomtsev–Petviashvili equation and the sine-Gordon equation. These nonlinear systems are derived from nothing via suitable 'Dao', the shifted parity, the charge conjugate, the delayed time reversal, the shifted exchange, the shifted-parity-rotation and so on.
基金Supported by the Global Change Research Program of China under Grant No.2015Cb953904National Natural Science Foundation of China under Grant Nos.11675054,11435005,11175092,and 11205092+1 种基金Shanghai Knowledge Service Platform for Trustworthy Internet of Things(No.ZF1213)K.C.Wong Magna Fund in Ningbo University
文摘The dark Korteweg-de Vries(KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV(MKdV) systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Furthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The recursion operators of two classes of dark MKdV systems are also given.
基金Supported by the National Natural Science Foundation of China Grant under Nos.11435005,11175092,and 11205092Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘By using a general symmetry theory related to invariant functions,strong symmetry operators and hereditary operators,we find a general integrable hopf heirarchy with infinitely many general symmetries and Lax pairs.For the first order Hopf equation,there exist infinitely many symmetries which can be expressed by means of an arbitrary function in arbitrary dimensions.The general solution of the first order Hopf equation is obtained via hodograph transformation.For the second order Hopf equation,the Hopf-diffusion equation,there are five sets of infinitely many symmetries.Especially,there exist a set of primary branch symmetry with which contains an arbitrary solution of the usual linear diffusion equation.Some special implicit exact group invariant solutions of the Hopf-diffusion equation are also given.