In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the p...In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.展开更多
From Lax representations,recursion operators for the supersymmetric KdV and the supersymmetric Kaup-Kupershimdt (SKK) equations are proposed explicitly.Under some special conditions,the recursion operator of the super...From Lax representations,recursion operators for the supersymmetric KdV and the supersymmetric Kaup-Kupershimdt (SKK) equations are proposed explicitly.Under some special conditions,the recursion operator of the supersymmetric Sawada-Kotera equation can be recovered by the one of the SKK equation.展开更多
A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of disc...A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.展开更多
Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n...Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n,n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.展开更多
The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH...The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are 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The project supported by National Natural Science Foundation of China under Grant No.10401021
文摘In this paper, the pseudo-differential operators and the generalized Lax equations in integrable systems are implemented in symbolic software Mathematica. A great deal of differential polynomials which appear in the procedure are dealt with by differential characteristic chain method. Using the program, several classical examples are given.
基金Supported by Zhejiang Provincial Natural Science Foundations of China under Grant No.Y6090592National Natural Science Foundation of China under Grant Nos.10735030 and 11041003+1 种基金Ningbo Natural Science Foundation under Grant Nos.2009B21003,2010A610103 and 2009B21003K.C.Wong Magna Fund in Ningbo University
文摘From Lax representations,recursion operators for the supersymmetric KdV and the supersymmetric Kaup-Kupershimdt (SKK) equations are proposed explicitly.Under some special conditions,the recursion operator of the supersymmetric Sawada-Kotera equation can be recovered by the one of the SKK equation.
基金Supported by the Science and Technology Plan project of the Educational Department of Shandong Province of China under Grant No. J09LA54the research project of "SUST Spring Bud" of Shandong university of science and technology of China under Grant No. 2009AZZ071
文摘A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.
基金supported by the National Natural Science Foundation of China (Nos.10971109,10971209,10825101)
文摘Based on the Lax operator L and Orlov-Shulman's M operator,the string equations of the q-KP hierarchy are established from the special additional symmetry flows,and the negative Virasoro constraint generators {L-n,n ≥ 1} of the 2-reduced q-KP hierarchy are also obtained.
基金supported by the National Natural Science Foundation of China(Nos.11201251,10971109)the Natural Science Foundation of Zhejiang Province(No.LY12A01007)the K.C.Wong Magna Fundin Ningbo University
文摘The authors give finite dimensional exponential solutions of the bigraded Toda hierarchy （BTH）. As a specific example of exponential solutions of the BTH, the authors consider a regular solution for the （1, 2）-BTH with a 3 × 3-sized Lax matrix, and discuss some geometric structures of the solution from which the difference between the （1, 2）- BTH and the original Toda hierarchy is shown. After this, the authors construct another kind of Lax representation of （N, 1）-BTH which does not use the fractional operator of Lax operator. Then the authors introduce the lattice Miura transformation of （N, 1）-BTH which leads to equations depending on one field, and meanwhile some specific examples which contain the Volterra lattice equation （a useful ecological competition model） are 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.
