Separation of variables for soliton equations via their binary constrained flows

Binary constrained flows of soliton equations admitting $2\times 2$ Lax matrices have 2N degrees of freedom, which is twice as many as degrees of freedom in the case of mono-constrained flows. For their separation of variables only N pairs of canonical separated variables can be introduced via their Lax matrices by using the normal method. A new method to introduce the other N pairs of canonical separated variables and additional separated equations is proposed. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.Comment: 39 pages, Amste

A Study on the Improvement of English Writing Competence for College Students

English writing has long been viewed as a challenge to most Chinese college students because of their weak language foundation, limited expanse of ideas and inadequate targeted practice for the production of satisfactory written work. Attention has constantly been directed to the low efficiency in the writing teaching. By adopting some widely used approaches, this paper tries to explore efficient ways to integrate writing into the whole process of English learning and teaching. The focus of this paper is put on the suggested solutions to writing problems and the improvement in the efficiency of English writing learning and teaching

Two binary Darboux transformations for the KdV hierarchy with self-consistent sources

Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non auto-B\"{a}cklund transformation between two n-th KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.Comment: 19 pages, LaTeX, no figures, to be published in Journal of Mathematical Physic

Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems

A systematic construction of St\"{a}ckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is derived. One Casimir bi-Hamiltonian case is studed in details and in this case, a systematic construction of related hydrodynamic systems in arbitrary coordinates is presented, using a cofactor method and soliton symmetry constraints.Comment: to appear in Journal of Geometry and Physic

3-Amino-4,6-dimethyl­thieno[2,3-b]pyridine-2-carbonitrile

The mol­ecule of the title compound, C10H9N3S, is almost planar, with a dihedral angle of 1.38 (4)° between the thio­phene and pyridine rings. In the crystal packing, mol­ecules are linked into layers parallel to the ab plane by inter­molecular N—H⋯N hydrogen bonds and by π⋯π stacking inter­actions involving adjacent pyridine and thio­phene rings with a centroid–centroid distance of 3.537 (3) Å

Identification of a novel submergence response gene regulated by the Sub1A gene

Submergence is one of the major constraints to rice production in many rice growing areas in the world. The Sub1A gene has been demonstrated to dramatically improve submergence tolerance in rice. Here, we report the identification of a novel submergence response (RS1) gene that is specifically induced in the Sub1A-mediated submergence tolerance response. Under submergence, RS1 was upregulated in M202 (Sub1A) but downregulated in M202 in RNA-seq and microarray assays. Expression analyses of various tissues and developmental stages show that RS1 mRNA levels are high in leaves and sheaths, but low in roots, stems, and panicles. Our results also show that RS1 is highly expressed under submergence, drought, and NaCl stresses, but not under cold or dehydration stress. Hormone ABA treatment induces, whereas GA treatment decreases, RS1 expression. The RS1 and Sub1A genes are co-regulated under submergence. Overexpression of RS1 in transgenic Kitaake (without Sub1A) and M202(Sub1A)×Kitaake do not result in enhanced submergence tolerance. Conversely, down-regulation of RS1 in M202(Sub1A)×Kitaake lead to weaken submergence tolerance. We hypothesize that RS1 may play a role in the Sub1A-mediated submergence tolerance pathway.Key word: Rice (Oryza sativa L.), submergence, RNA-seq, Sub1A, abiotic stress

Higher Order Potential Expansion for the Continuous Limits of the Toda Hierarchy

A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the lower ones and can be continued to be performed indefinitly. By introducing the higher order terms, the fewer equations in the Toda hierarchy are needed in the so-called recombination method to recover the KdV hierarchy. It is shown that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend towards the corresponding ones of the KdV hierarchy in continuous limit.Comment: 20 pages, Latex, to be published in Journal of Physics