About the Structure of Meson Algebras

Superintegrable systems from block separation of variables and unified derivation of their quadratic algebras

Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras

Equations in Hirota's bilinear form appear naturally whenever one needs to write integrable differential equations in terms of "nice" functions. In the case of soliton equations nice typically means polynomials of exponentials with lin- ear exponents, whereas in the context of Painleve equations nice often means entire functions. The natural or original dependent variables of the equation are usually not the best in this respect and a change of the dependent vari- ables is necessary. This dependent variable transformation can be somewhat involved. Since the solutions of Painleve equations are meromorphic by definition (the movable singularities can be at worst poles) it is natural to express them as ratios of entire functions, and this leads to homogeneous equations and often to equations in Hirota's bilinear form. Indeed, bilinear forms were already derived by Painleve himself (1). It should be noted, however, that the converse of the above is not necessarily true: a bilinear form does not by itself imply that the independent functions are regular in any sense, in fact bilinear forms exist even for non integrable equations. In the field of integrable partial differential equations, the so-called τ - functions play a major role. These functions actually provide a huge pool of such "nice" functions, in both of the aforementioned interpretations, and they also possess an extremely rich algebraic and algebro-geometric structure. This is the main theme of Sato-theory, one of the great unifying theories for the description of integrable systems. The relevance of τ -functions for the study of the Painleve equations has been recognized ever since the seminal work of Jimbo and Miwa on isomonodromy deformations (2) and in particular since the results obtained by Okamoto, regarding the algebro-geometric properties of the Painleve equations (3).

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.

Quadratic Lie algebras with 2-plectic structures

An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, a bilinear form on $\mathbb{V}$ with values in another vector space $\mathbb{W}$, and a map $t \colon \mathbb{W} \to \mathbb{V}$, satisfying altogether four compatibility relations. Our structure theorem asserts that an enhanced Leibniz algebra is uniquely determined by the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, an appropriate abelian ideal ${\mathfrak i}$ inside it, as well as a cohomology 2-class $[\Delta]$ which only effects the $\mathbb{W}$-valued product. Positive quadratic enhanced Leibniz algebras, as needed for the definition of a Yang-Mills type action functional, turn out to be rather restrictive on the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \dot ])$: $\mathbb{V}$ has to be the hemisemidirect product of a positive quadratic Lie algebra ${\mathfrak g}$ with a ${\mathfrak g}$-module ${\mathfrak i}$, $\mathbb{V} \cong {\mathfrak g}\ltimes{\mathfrak i}$, with ${\mathfrak i}$ the above-mentioned ideal in this case. The second main result of this article is the construction of a functor from the category of such enhanced Leibniz algebras to the category of (semi-strict) Lie 2-algebras or, equivalentely, of two-term $L_\infty$-algebras.

The object of this note is to construct a set of real three-dimensional Lie groups such that every real three-dimensional Lie group is locally isomorphic with some group in the set. The construction is effected by first finding canonical forms for the constants of structure of real three-dimensional Lie algebras; these canonical forms give rise to certain bilinear forms, and the Lie groups are obtained as linear groups isomorphic with groups of automorphisms which leave these bilinear forms invariant.

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field Bij(x,x′,t) and isometries of the correlation space K3 equipped with a (pseudo-) Riemannian metrics ds2(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds2(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (Mt ,ds2(t)), Mt ⊂ K3. Here the metric ds2(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds2(t) in K3 and present symmetries (or isometric motions in K3) of the metric ds2(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds2(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λg is a second-order differential invariant and show how λg can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.

In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

Lax pair, Bäcklund transformation and conservation laws for the (2 + 1)-dimensional extended shallow water wave equation

Bilinear form and soliton solutions for a higher order wave equation

The present article is motivated by the theorem of Cartan-Dieudonn\'e which states that every orthogonal transformation is a product of reflections. Its purpose is to determine, for each orthogonal transformation, the minimal number of factors in a decomposition into a product of reflections, and to propose an effective algorithm giving such a decomposition. With the orthogonal transformations $g$ of a quadratic space $(V,q)$, it associates couples $(S,\phi)$ where $S$ is a subspace of $V$, and $\phi$ an non-degenerate bilinear form on $S$ such that $\phi(y,y)=q(y)$ for every $y$ in $S$. In general, the minimal decompositions of $g$ into a product of reflections correspond to the bases of $S$ in which the matrix of $\phi$ is lower triangular. Therefore, we need an algorithm of triangularization of bilinear forms. Affine isometries are also taken into consideration.

A brief survey of the bilinear formalism discovered by Hirota is given. First, the procedure to obtain solition solutions of nonlinear evolution equations is discussed. Then the algebraic structure of the equations in bilinear form is explained in a simple way. A few extensions of the formalism are also presented.

PurposeThis paper aims to study the breather, lump-kink and interaction solutions of a (3 + 1)-dimensional generalized shallow water waves (GSWW) equation, which describes water waves propagating in the ocean or is used for simulating weather.Design/methodology/approachHirota bilinear form and the direct method are used to construct breather and lump-kink solutions of the GSWW equation. The “rational-cosh-cos-type” test function is applied to obtain three kinds of interaction solutions.FindingsThe fusion and fission of the interaction solutions between a lump wave and a 1-kink soliton of the GSWW equation are studied. The dynamics of three kinds of interaction solutions between lump, kink and periodic waves are discussed graphically.Originality/valueThis paper studies the breather, lump-kink and interaction solutions of the GSWW equation by using various approaches and provides some phenomena that have not been studied.