Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2+1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.Comment: Latex, 34 page

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

We obtain the necessary and sufficient conditions for a two-component (2+1)-dimensional system of hydrodynamic type to possess infinitely many hydrodynamic reductions. These conditions are in involution, implying that the systems in question are locally parametrized by 15 arbitrary constants. It is proved that all such systems possess three conservation laws of hydrodynamic type and, therefore, are symmetrizable in Godunov's sense. Moreover, all such systems are proved to possess a scalar pseudopotential which plays the role of the `dispersionless Lax pair'. We demonstrate that the class of two-component systems possessing a scalar pseudopotential is in fact identical with the class of systems possessing infinitely many hydrodynamic reductions, thus establishing the equivalence of the two possible definitions of the integrability. Explicit linearly degenerate examples are constructed.Comment: 15 page

The Peptidoglycan-Binding Protein SjcF1 Influences Septal Junction Function and Channel Formation in the Filamentous Cyanobacterium Anabaena

Filamentous, heterocyst-forming cyanobacteria exchange nutrients and regulators between cells for diazotrophic growth. Two alternative modes of exchange have been discussed involving transport either through the periplasm or through septal junctions linking adjacent cells. Septal junctions and channels in the septal peptidoglycan are likely filled with septal junction complexes. While possible proteinaceous factors involved in septal junction formation, SepJ (FraG), FraC, and FraD, have been identified, little is known about peptidoglycan channel formation and septal junction complex anchoring to the peptidoglycan. We describe a factor, SjcF1, involved in regulation of septal junction channel formation in the heterocyst-forming cyanobacterium Anabaena sp. strain PCC 7120. SjcF1 interacts with the peptidoglycan layer through two peptidoglycan-binding domains and is localized throughout the cell periphery but at higher levels in the intercellular septa. A strain with an insertion in sjcF1 was not affected in peptidoglycan synthesis but showed an altered morphology of the septal peptidoglycan channels, which were significantly wider in the mutant than in the wild type. The mutant was impaired in intercellular exchange of a fluorescent probe to a similar extent as a sepJ deletion mutant. SjcF1 additionally bears an SH3 domain for protein-protein interactions. SH3 binding domains were identified in SepJ and FraC, and evidence for interaction of SjcF1 with both SepJ and FraC was obtained. SjcF1 represents a novel protein involved in structuring the peptidoglycan layer, which links peptidoglycan channel formation to septal junction complex function in multicellular cyanobacteria. Nonetheless, based on its subcellular distribution, this might not be the only function of SjcF1.Peer reviewe

Riemann Invariants and Rank-k Solutions of Hyperbolic Systems

In this paper we employ a "direct method" in order to obtain rank-k solutions of any hyperbolic system of first order quasilinear differential equations in many dimensions. We discuss in detail the necessary and sufficient conditions for existence of these type of solutions written in terms of Riemann invariants. The most important characteristic of this approach is the introduction of specific first order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method of solving multi-dimensional systems of PDEs. We have demonstrated the usefulness of our approach through several examples of hydrodynamic type systems; new classes of solutions have been obtained in a closed form.Comment: 30 page

Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant Poisson brackets of hydrodynamic type. A complete list of two- and three-component integrable Hamiltonians is obtained. All our examples possess dispersionless Lax pairs and an infinity of hydrodynamic reductions.Comment: 34 page

Multimode solutions of first-order elliptic quasilinear systems obtained from Riemann invariants

Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links between two techniques, that of the symmetry reduction method and of the generalized method of characteristics. A variant of the conditional symmetry method for constructing this type of solution is proposed. A specific feature of that approach is an algebraic-geometric point of view, which allows the introduction of specific first-order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method for solving elliptic homogeneous systems of PDEs. A further generalization of the Riemann invariants method to the case of inhomogeneous systems, based on the introduction of specific rotation matrices, enables us to weaken the integrability condition. It allows us to establish a connection between the structure of the set of integral elements and the possibility of constructing specific classes of simple mode solutions. These theoretical considerations are illustrated by the examples of an ideal plastic flow in its elliptic region and a system describing a nonlinear interaction of waves and particles. Several new classes of solutions are obtained in explicit form, including the general integral for the latter system of equations

Cell Envelope Components Influencing Filament Length in the Heterocyst-Forming Cyanobacterium Anabaena sp. Strain PCC 7120

Heterocyst-forming cyanobacteria grow as chains of cells (known as trichomes or filaments) that can be hundreds of cells long. The filament consists of individual cells surrounded by a cytoplasmic membrane and peptidoglycan layers. The cells, however, share a continuous outer membrane, and septal proteins, such as SepJ, are important for cell-cell contact and filament formation. Here, we addressed a possible role of cell envelope components in filamentation, the process of producing and maintaining filaments, in the model cyanobacterium Anabaena sp. strain PCC 7120. We studied filament length and the response of the filaments to mechanical fragmentation in a number of strains with mutations in genes encoding cell envelope components. Previously published peptidoglycan- and outer membrane-related gene mutants and strains with mutations in two genes (all5045 and alr0718) encoding class B penicillin-binding proteins isolated in this work were used. Our results show that filament length is affected in most cell envelope mutants, but the filaments of alr5045 and alr2270 gene mutants were particularly fragmented. All5045 is a DD-transpeptidase involved in peptidoglycan elongation during cell growth, and Alr2270 is an enzyme involved in the biosynthesis of lipid A, a key component of lipopolysaccharide. These results indicate that both components of the cell envelope, the murein sacculus and the outer membrane, influence filamentation. As deduced from the filament fragmentation phenotypes of their mutants, however, none of these elements is as important for filamentation as the septal protein SepJ.Peer reviewe