Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d-1)n arbitrary functions of one variable. Given a PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions (if consistent).Comment: 16 page

Integrable equations of the dispersionless Hirota type and hypersurfaces in the Lagrangian Grassmannian

We investigate integrable second order equations of the form F(u_{xx}, u_{xy}, u_{yy}, u_{xt}, u_{yt}, u_{tt})=0. Familiar examples include the Boyer-Finley equation, the potential form of the dispersionless Kadomtsev-Petviashvili equation, the dispersionless Hirota equation, etc. The integrability is understood as the existence of infinitely many hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.Comment: 32 page

On Boussinesq-type models for long longitudinal waves in elastic rods

In this paper we revisit the derivations of model equations describing long nonlinear longitudinal bulk strain waves in elastic rods within the scope of the Murnaghan model in order to derive a Boussinesq-type model, and extend these derivations to include axially symmetric loading on the lateral boundary surface, and longitudinal pre-stretch. We systematically derive two forced Boussinesq-type models from the full equations of motion and non-zero surface boundary conditions, utilising the presence of two small parameters characterising the smallness of the wave amplitude and the long wavelength compared to the radius of the waveguide. We compare the basic dynamical properties of both models (linear dispersion curves and solitary wave solutions). We also briefly describe the laboratory experiments on generation of bulk strain solitary waves in the Ioffe Institute, and suggest that this generation process can be modelled using the derived equations.Comment: 19 pages, 5 figures, submitted to the Special Issue of Wave Motion, "Nonlinear Waves in Solids", in Memory of Professor Alexander M. Samsono

On a class of three-dimensional integrable Lagrangians

We characterize non-degenerate Lagrangians of the form $\int f(u_x, u_y, u_t) dx dy dt$ such that the corresponding Euler-Lagrange equations $(f_{u_x})_x+ (f_{u_y})_y+ (f_{u_t})_t=0$ are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an over-determined system of fourth order PDEs for the Lagrangian density $f$, which is in involution and possess interesting differential-geometric properties. The moduli space of integrable Lagrangians, factorized by the action of a natural equivalence group, is three-dimensional. Familiar examples include the dispersionless Kadomtsev-Petviashvili (dKP) and the Boyer-Finley Lagrangians, $f=u_x^3/3+u_y^2-u_xu_t$ and $f=u_x^2+u_y^2-2e^{u_t}$, respectively. A complete description of integrable cubic and quartic Lagrangians is obtained. Up to the equivalence transformations, the list of integrable cubic Lagrangians reduces to three examples, $f=u_xu_yu_t, f=u_x^2u_y+u_yu_t, and f=u_x^3/3+u_y^2-u_xu_t ({\rm dKP}).$ There exists a unique integrable quartic Lagrangian, $f=u_x^4+2u_x^2u_t-u_xu_y-u_t^2.$ We conjecture that these examples exhaust the list of integrable polynomial Lagrangians which are essentially three-dimensional (it was verified that there exist no polynomial integrable Lagrangians of degree five). We prove that the Euler-Lagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless `Lax pair'. MSC: 35Q58, 37K05, 37K10, 37K25. Keywords: Multi-dimensional Dispersionless Integrable Systems, Hydrodynamic Reductions, Pseudopotentials.Comment: 12 pages A4 format, standard Latex 2e. In the file progs.tar we include the programs needed for computations performed in the paper. Read 1-README first. The new version includes two new section

S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.Comment: 35 pages, uses AMS-Latex, Hyperref, Geometry, Array and Babel package

Involvement of transposable elements in neurogenesis

The article is about the role of transposons in the regulation of functioning of neuronal stem cells and mature neurons of the human brain. Starting from the first division of the zygote, embryonic development is governed by regular activations of transposable elements, which are necessary for the sequential regulation of the expression of genes specific for each cell type. These processes include differentiation of neuronal stem cells, which requires the finest tuning of expression of neuron genes in various regions of the brain. Therefore, in the hippocampus, the center of human neurogenesis, the highest transposon activity has been identified, which causes somatic mosai cism of cells during the formation of specific brain structures. Similar data were obtained in studies on experimental animals. Mobile genetic elements are the most important sources of long non-coding RNAs that are coexpressed with important brain protein-coding genes. Significant activity of long non-coding RNA was detected in the hippocampus, which confirms the role of transposons in the regulation of brain function. MicroRNAs, many of which arise from transposon transcripts, also play an important role in regulating the differentiation of neuronal stem cells. Therefore, transposons, through their own processed transcripts, take an active part in the epigenetic regulation of differentiation of neurons. The global regulatory role of transposons in the human brain is due to the emergence of protein-coding genes in evolution by their exonization, duplication and domestication. These genes are involved in an epigenetic regulatory network with the participation of transposons, since they contain nucleotide sequences complementary to miRNA and long non-coding RNA formed from transposons. In the memory formation, the role of the exchange of virus-like mRNA with the help of the Arc protein of endogenous retroviruses HERV between neurons has been revealed. A possible mechanism for the implementation of this mechanism may be reverse transcription of mRNA and site-specific insertion into the genome with a regulatory effect on the genes involved in the memory

Non-coding parts of genomes as the basis of epigenetic heredity

We hypothesized that the basis of epigenetic regulation of genomes in ontogenesis is the specificity of the distribution, number and composition of transposons. Transposons constitute the major part of the genomes of multicellular eukaryotes. The evolutionary preservation of transposons is associated with universal mechanisms for controlling cell differentiation: processing of non-coding RNAs and splicing regulation. These universal mechanisms were originally aimed at protecting against viruses and transposons. The cooperation of these protective systems with mechanisms for controlling the interrelation of cells and their differentiation became the basis for the emergence and evolution of multicellular eukaryotes. The evolutionary conservation of a complex enzymes Drosha, Dicer, Argonaut, RdRP and their homologues in all multicellular eukaryotes, and their absence in unicellular organisms supports this assumption. Introns originated from mobile genetic elements. Transposons played an important role in the propagation of introns in evolution and their regulation in ontogenesis. Transposons regulate the expression of genes in cis and in trans, and also indirectly by the production of small RNAs that affect their own activity, both by altering the DNA methylation and modifying histones, and at the posttranscriptional level. Tissue-specific and stage-specific changes in the activity of transposons in ontogenesis are associated with the expression of transposon-derived noncoding RNAs and altering the activity of genes, which leads to cell differentiation. We proposed that the species-specific features of activation of transposons for each subsequent cell division undergo evolutionary selection and are key regulators of the growth and development of the organism. We proposed that transposons in the genome affect their inherited activation in each subsequent cell division, which causes a change in cell differentiation

The role of transposable elements in the ecological morphogenesis under the influence of stress

In natural selection, insertional mutagenesis is an important source of genome variability. Transposons are sensors of environmental stress effects, which contribute to adaptation and speciation. These effects are due to changes in the mechanisms of morphogenesis, since transposons contain regulatory sequences that have cis and trans effects on specific protein-coding genes. In variability of genomes, the horizontal transfer of transposons plays an important role, because it contributes to changing the composition of transposons and the acquisition of new properties. Transposons are capable of site-specific transpositions, which lead to the activation of stress response genes. Transposons are sources of non-coding RNA, transcription factors binding sites and protein-coding genes due to domestication, exonization, and duplication. These genes contain nucleotide sequences that interact with non-coding RNAs processed from transposons transcripts, and therefore they are under the control of epigenetic regulatory networks involving transposons. Therefore, inherited features of the location and composition of transposons, along with a change in the phenotype, play an important role in the characteristics of responding to a variety of environmental stressors. This is the basis for the selection and survival of organisms with a specific composition and arrangement of transposons that contribute to adaptation under certain environmental conditions. In evolution, the capability to transpose into specific genome sites, regulate gene expression, and interact with transcription factors, along with the ability to respond to stressors, is the basis for rapid variability and speciation by altering the regulation of ontogenesis. The review presents evidence of tissue-specific and stage-specific features of transposon activation and their role in the regulation of cell differentiation to confirm their role in ecological morphogenesis