In the present paper, a super-extension of the Yang hierarchy is proposed by super-matrix Lie algebras, and the super-Yang hierarchy with self-consistent sources is established. Furthermore, we establish infinitely many conservation laws of the super-integrable hierarchy. The methods presented by us can be generalized to other nonlinear equation hierarchies with self-consistent sources. 相似文献
Although many effective methods for solving partial differential equations (PDEs) have been proposed, there is no universal method that can solve all PDEs. Therefore, solving partial differential equations has always been a difficult problem in mathematics, such as deep neural network (DNN). In recent years, a method of embedding some basic physical laws into traditional neural networks has been proposed to reveal the dynamic behavior of equations directly from space-time data [i.e., physics-informed neural network (PINN)]. Based on the above, an improved deep learning method to recover the new soliton solution of Huxley equation has been proposed in this paper. As far as we know, this is the first time that we have used an improved method to study the numerical solution of the Huxley equation. In order to illustrate the advantages of the improved method, we use the same network depth, the same hidden layer and neurons contained in the hidden layer, and the same training sample points. We analyze the dynamic behavior and error of Huxley’s exact solution and the new soliton solution and give vivid graphs and detailed analysis. Numerical results show that the improved algorithm can use fewer sample points to reconstruct the exact solution of the Huxley equation with faster convergence speed and better simulation effect.
Let n be a positive integer, and \(\mathfrak {A}(n)=\mathbb {F}[x]/(x^{p^{n}})\), the divided power algebra over an algebraically closed field \(\mathbb {F}\) of prime characteristic p >?2. Let π(n) be the tensor product of \(\mathfrak {A}(n)\) and the Grassmann superalgebra \(\bigwedge (1)\) in one variable. The Zassenhaus superalgebra \(\mathcal {Z}(n)\) is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra \(\mathcal {Z}(n)\) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained. 相似文献
In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada-Kotera, Sawada-Kotera-Parker-Dye, Caudrey-Dodd-Gibbon, Kaup-Kupershmidt, Kaup-Kupershmidt-Parker-Dye, and the Ito forms. 相似文献
Benzenehexapyrrole‐α,ω‐dialdehyde, composed of a pair of formyltripyrrole units with a 1,3‐phenylene linker, was metallated to give dinuclear single‐stranded helicates. X‐ray studies of the bis‐nickel(II) complex showed a helical C2 form with a pair of helical–metal coordination planes of a 3N+O donor set. The terminal aldehyde was readily converted into the imine by optically active amines, whereby helix‐sense bias was induced. Bis‐nickel(II) and bis‐palladium(II) complexes of the benzenehexapyrrole‐α,ω‐diimines were studied to show that an enantiomer pair of the helical C2 form are interchanged by slow flipping of each coordination plane and fast rotation around the C(benzene)?C(pyrrole) bond. The helical screw in the bis‐nickel(II) complexes was biased to one side in more than 95 % diastereoselectivity, which was achieved by using a variety of optically active amines, such as (R)‐1‐cyclohexylethylamine, (S)‐1‐ phenylethylamine, L ‐Phe(OEt) (Phe=phenylalanine), and (R)‐valinol. The nickel complexes showed much better diastereoselectivity than the corresponding palladium complexes. 相似文献
The hexapyrrole-α,ω-dicarbaldehydes 1 a and 1 b were metallated with CuII, NiII, and PdII to give bimetallic complexes where a pair of 3 N+O four-coordinate metal planes are helically distorted and the central 2,2′-bipyrrole subunit adopts a cis or trans conformation. X-ray crystallographic analysis of the bisCu complex revealed a closed form with a cis-2,2′-bipyrrole subunit and an open form with a trans-2,2′-bipyrrole subunit. The bisPd complexes took a closed form both in the solid state and in solution. They are regarded as single helicates of two turns and the energy barrier for the interchange between an M helix and a P helix was remarkably influenced by the bulky 3,3′-substituent of the central 2,2′-bipyrrole subunit. Although the bisNi complexes adopt a closed form in the solid state, they exist as a homohelical open C2-symmetric form or a heterohelical open Ci-symmetric form in solution. A theoretical study suggested that the closed form of 1 a Pd was stabilized by the Pd–Pd interaction. Compound 1 a Pd was reversibly oxidized by one electron at 0.14 V versus ferrocene/ferrocenium (Fc/Fc+) and this oxidized species showed Vis/NIR absorption bands at λ=767 and 1408 nm. 相似文献
The integrability and multi-shock wave solutions of the DJKM equation are studied by means of Bell polynomials scheme, Hirota bilinear method, and symbolic computation. A more generalized bilinear system of the DJKM equation is constructed via Bell polynomials scheme. Moreover, Lax pair and infinite conservation laws of this equation are first obtained via its corresponding Bell-polynomials-type Bäcklund transformation. Furthermore, the multi-shock wave solutions are also obtained by applying standard Hirota bilinear method, and the propagation and collision of shock waves are graphically demonstrated by graphs. 相似文献