By using the constraint relating potential and eigenfunctions, the decomposition of each equation in the Boussinesq hierarchy into two commuting finite-dimensional integrable Hamiltonian system (FDIHS) is presented. A...By using the constraint relating potential and eigenfunctions, the decomposition of each equation in the Boussinesq hierarchy into two commuting finite-dimensional integrable Hamiltonian system (FDIHS) is presented. A method to construct the Lax representations for both x- and t(n)- constrained flows via reduction of the adjoint representations of the auxiliary linear problems is developed.展开更多
Within framework of zero-curvature representation theory, the Lax representations for x- andtn-constrained flows of soliton hierarchy are obtained from reductions of adjoint representationsof the auxiliary linear prob...Within framework of zero-curvature representation theory, the Lax representations for x- andtn-constrained flows of soliton hierarchy are obtained from reductions of adjoint representationsof the auxiliary linear problems. This method is applied to the third order spectral problem bytaking modified Boussinesq hierarchy as an illustrative example.展开更多
The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations...The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are n,n(n−1)/2,1,nwith a total of N=(n+1)(n+2)/2, that is 15 when n=4for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations, n(n−1)/2rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary n≥1because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when n=4, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through Nn(n−1)/2potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.展开更多
By using a general scheme for decomposing a zero-curvature equation into two commut- ing x-and t_n-finite-dimensional integrable Hamiltonian systems (FDIHS),a systematic deduction of the Lax representation for all con...By using a general scheme for decomposing a zero-curvature equation into two commut- ing x-and t_n-finite-dimensional integrable Hamiltonian systems (FDIHS),a systematic deduction of the Lax representation for all constrained flows of the AKNS hierarchy from the adjoint repre- sentation of the two auxiliary linear problems is presented.The Darboux transformation for these FDIHSs is derived.展开更多
文摘By using the constraint relating potential and eigenfunctions, the decomposition of each equation in the Boussinesq hierarchy into two commuting finite-dimensional integrable Hamiltonian system (FDIHS) is presented. A method to construct the Lax representations for both x- and t(n)- constrained flows via reduction of the adjoint representations of the auxiliary linear problems is developed.
基金Project supported by the National Basic Reseach Project "Nonlinear Scijence
文摘Within framework of zero-curvature representation theory, the Lax representations for x- andtn-constrained flows of soliton hierarchy are obtained from reductions of adjoint representationsof the auxiliary linear problems. This method is applied to the third order spectral problem bytaking modified Boussinesq hierarchy as an illustrative example.
文摘The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870) and the Maxwell/Weyl equations (1873, 1918) are among the most famous partial differential equations that can be found today in any textbook dealing with elasticity theory, continuum mechanics, thermodynamics or electromagnetism. Over a manifold of dimension n, their respective numbers are n,n(n−1)/2,1,nwith a total of N=(n+1)(n+2)/2, that is 15 when n=4for space-time. This is also just the number of parameters of the Lie group of conformal transformations with n translations, n(n−1)/2rotations, 1 dilatation and n highly non-linear elations introduced by E. Cartan in 1922. The purpose of this paper is to prove that the form of these equations only depends on the structure of the conformal group for an arbitrary n≥1because they are described as a whole by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained by applying totally new differential geometric methods in field theory. In particular, when n=4, the main idea is not to shrink the group from 10 down to 4 or 2 parameters by using the Schwarzschild or Kerr metrics instead of the Minkowski metric, but to enlarge the group from 10 up to 11 or 15 parameters by using the Weyl or conformal group instead of the Poincaré group of space-time. Contrary to the Einstein equations, these equations can be all parametrized by the adjoint of the second Spencer operator through Nn(n−1)/2potentials. These results bring the need to revisit the mathematical foundations of both General Relativity and Gauge Theory according to a clever but rarely quoted paper of H. Poincaré (1901). They strengthen the recent comments we already made about the dual confusions made by Einstein (1915) while following Beltrami (1892), both using the same Einstein operator but ignoring it is self-adjoint in the framework of differential double duality.
基金Supported by the Chinese National Basic Research Project"Nonlinear Science"
文摘By using a general scheme for decomposing a zero-curvature equation into two commut- ing x-and t_n-finite-dimensional integrable Hamiltonian systems (FDIHS),a systematic deduction of the Lax representation for all constrained flows of the AKNS hierarchy from the adjoint repre- sentation of the two auxiliary linear problems is presented.The Darboux transformation for these FDIHSs is derived.
