An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfull...An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfullyreduced based on the property of uniformity in rank.Furthermore,we discard the integral operation in the traditionalalgorithm.The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety solitonequations.展开更多
The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé...The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.展开更多
This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-co...This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.展开更多
A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedi...A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.展开更多
We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies (despite using ...We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies (despite using the same group theoretical methods) and the well known couplings existing between elasticity and electromagnetism (piezzo electricity, photo elasticity, streaming birefringence). The purpose of this paper is to avoid such contradictions by using new mathematical methods coming from the formal theory of systems of partial differential equations and Lie pseudo groups. These results finally allow unifying the previous independent tentatives done by the brothers E. and F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by using respectively the group of rigid motions of space or the conformal group of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra and algebraic analysis. We insist on the fact that these results could not have been obtained before 1975 as the corresponding tools were not known before.展开更多
基金Supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant No.20070979the National Natural Science Foundations of China under Grant Nos.10675065 and 10735030+1 种基金the Scientific Research Found of Ningbo University under Grant No.XKL09059the K.C.Wong Magana Fund in Ningbo University
文摘An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations withlogarithmic transformations is presented.In the algorithm,the general assumption of Hirota bilinear form is successfullyreduced based on the property of uniformity in rank.Furthermore,we discard the integral operation in the traditionalalgorithm.The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety solitonequations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975131 and 11435005)the K C Wong Magna Fund in Ningbo University。
文摘The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.
文摘This paper is concerned with the solvability and waveform relaxation methods of linear variable-coefficient differential-algebraic equations （DAEs）. Most of the previous works have been focused on linear variable-coefficient DAEs with smooth coefficients and data, yet no results related to the convergence rate of the corresponding waveform relaxation methods has been obtained. In this paper, we develope the solvability theory for the linear variable-coefficient DAEs on Legesgue square-integrable function space in both traditional and least squares senses, and determine the convergence rate of the waveform relaxation methods for solving linear variable-coefficient DAEs.
基金the Natural Science Foundation of Shandong Province under Grant No.Q2006A04
文摘A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.
文摘We start recalling with critical eyes the mathematical methods used in gauge theory and prove that they are not coherent with continuum mechanics, in particular the analytical mechanics of rigid bodies (despite using the same group theoretical methods) and the well known couplings existing between elasticity and electromagnetism (piezzo electricity, photo elasticity, streaming birefringence). The purpose of this paper is to avoid such contradictions by using new mathematical methods coming from the formal theory of systems of partial differential equations and Lie pseudo groups. These results finally allow unifying the previous independent tentatives done by the brothers E. and F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by using respectively the group of rigid motions of space or the conformal group of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra and algebraic analysis. We insist on the fact that these results could not have been obtained before 1975 as the corresponding tools were not known before.
