A direct method for obtaining the expanding integrable models of the hierarchies of evolution equations was proposed. By using the equivalent transformation between the matrices, a new isospectral problem was directly...A direct method for obtaining the expanding integrable models of the hierarchies of evolution equations was proposed. By using the equivalent transformation between the matrices, a new isospectral problem was directly established according to the known isospectral problem, which can be used to obtain the expanding integrable model of the known hierarchy.展开更多
In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy...In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy possessing bi-Hamiltonian structure is obtained by choosing V with derivatives in x and spectral potentials. Then integrable coupling, i.e. expanding Lax integrable model of the hierarchy obtained is presented by constructing a subalgebra of loop algebra A2.展开更多
In the article 'Evolution Model of the Earth’s Limited Expanding' published in Volume 45 Number (4) of Chinese Science Bulletin[1], the author suggests that the earth expands according to a law R(t) = R0+A(1 ...In the article 'Evolution Model of the Earth’s Limited Expanding' published in Volume 45 Number (4) of Chinese Science Bulletin[1], the author suggests that the earth expands according to a law R(t) = R0+A(1 -exp(β(t-ts))) (remark: this formula was mistakenly printed as R(t) = R0 + Aexp(β(t-ts)) in the and formula (12) of the text of ref. [1]). According to ref. [1], the earth was formed 4.6 billion years ago. After 0.3 billion years from its birth (ts), it started expansion from an initial radius R0 of 4651 km, and may reach a final maximum radius of R0+A = 6511 km. In the 4.6 billion years history, the radius of the earch has increased by 1720 km, or the density decreased from 14200 km/m3 (2.57 times the present density) to 5520 kg/m3 within the latest 4.3 billion years.展开更多
Under the framework of the complex column-vector loop algebra C^(p),we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu sche...Under the framework of the complex column-vector loop algebra C^(p),we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme.As applications of the scheme,we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model.The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems,including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell.Specially,we obtain the famous Fokker-Plank equation and its generalized form,which has extensive applications in the stochastic dynamic systems.Finally,we investigate the Lie group symmetries,fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H_(3)and the matrix linear group SL(2,R)for the generalized Fokker-Plank equation(GFPE).展开更多
基金Project supported by the National Natural Science Foundation of China （Grant Nos.10371070, 10547123）
文摘A direct method for obtaining the expanding integrable models of the hierarchies of evolution equations was proposed. By using the equivalent transformation between the matrices, a new isospectral problem was directly established according to the known isospectral problem, which can be used to obtain the expanding integrable model of the known hierarchy.
文摘In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy possessing bi-Hamiltonian structure is obtained by choosing V with derivatives in x and spectral potentials. Then integrable coupling, i.e. expanding Lax integrable model of the hierarchy obtained is presented by constructing a subalgebra of loop algebra A2.
基金the National Natural Science Foundation of China (Grant No. 49774236),
文摘In the article 'Evolution Model of the Earth’s Limited Expanding' published in Volume 45 Number (4) of Chinese Science Bulletin[1], the author suggests that the earth expands according to a law R(t) = R0+A(1 -exp(β(t-ts))) (remark: this formula was mistakenly printed as R(t) = R0 + Aexp(β(t-ts)) in the and formula (12) of the text of ref. [1]). According to ref. [1], the earth was formed 4.6 billion years ago. After 0.3 billion years from its birth (ts), it started expansion from an initial radius R0 of 4651 km, and may reach a final maximum radius of R0+A = 6511 km. In the 4.6 billion years history, the radius of the earch has increased by 1720 km, or the density decreased from 14200 km/m3 (2.57 times the present density) to 5520 kg/m3 within the latest 4.3 billion years.
基金Supported by the National Natural Science Foundation of China(Grant No.11971475)。
文摘Under the framework of the complex column-vector loop algebra C^(p),we propose a scheme for generating nonisospectral integrable hierarchies of evolution equations which generalizes the applicable scope of the Tu scheme.As applications of the scheme,we work out a nonisospectral integrable Schrodinger hierarchy and its expanding integrable model.The latter can be reduced to some nonisospectral generalized integrable Schrodinger systems,including the derivative nonlinear Schrodinger equation once obtained by Kaup and Newell.Specially,we obtain the famous Fokker-Plank equation and its generalized form,which has extensive applications in the stochastic dynamic systems.Finally,we investigate the Lie group symmetries,fundamental solutions and group-invariant solutions as well as the representation of the tensor of the Heisenberg group H_(3)and the matrix linear group SL(2,R)for the generalized Fokker-Plank equation(GFPE).
