Search Results (42)

A formalism for the systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is shown that this IVP has a formal solution, and within the framework of two particular subalgebras of the hereditary Lie algebra, the explicit forms of this formal solution are derived. Finally, this formalism is applied to the Korteveg-de Vries, dispersive water waves and Ablowitz–Kaup–Newell–Segur soliton hierarchies. The zero-curvature representations and Hamiltonian structures of the considered non-autonomous soliton hierarchies are also provided. Full article

This study sought to deepen our understanding of the dynamical properties of the newly extended $(3+1)$-dimensional integrable Kadomtsev–Petviashvili (KP) equation, which models the behavior of ion acoustic waves in plasmas and nonlinear optics. This paper aimed to perform Lie symmetry analysis and derive lump, breather, and soliton solutions using the extended hyperbolic function method and the generalized logistic equation method. It also analyzed the dynamical system using chaos detection techniques such as the Lyapunov exponent, return maps, and the fractal dimension. Initially, we focused on constructing lump and breather soliton solutions by employing Hirota’s bilinear method. Secondly, employing Lie symmetry analysis, symmetry generators were utilized to satisfy the Lie invariance conditions. This approach revealed a seven-dimensional Lie algebra for the extended $(3+1)$-dimensional integrable KP equation, incorporating translational symmetry (including dilation or scaling) as well as translations in space and time, which were linked to the conservation of energy. The analysis demonstrated that this formed an optimal sub-algebraic system via similarity reductions. Subsequently, a wave transformation method was applied to reduce the governing system to ordinary differential equations, yielding a wide array of exact solitary wave solutions. The extended hyperbolic function method and the generalized logistic equation method were employed to solve the ordinary differential equations and explore closed-form analytical solitary wave solutions for the diffusive system under consideration. Among the results obtained were various soliton solutions. When plotting the results of all the solutions, we obtained bright, dark, kink, anti-kink, peak, and periodic wave structures. The outcomes are illustrated using 2D, 3D, and contour plots. Finally, upon introducing the perturbation term, the system’s behavior was analyzed using chaos detection techniques such as the Lyapunov exponent, return maps, and the fractal dimension. The results contribute to a deeper understanding of the dynamic properties of the extended KP equation in fluid mechanics. Full article

In this study, we investigate the symmetry analysis and explicit solutions for the Estevez–Mansfield–Clarkson (EMC) equation. Our main objectives are to identify the Lie point symmetries of the EMC equation, construct an optimal system of one-dimensional subalgebras, and reduce the EMC equation to a set of ordinary differential equations (ODEs). We employ the Riccati–Bernoulli sub-ODE method (RBSODE) to solve these reduced ODEs and obtain explicit solutions for the EMC model. The obtained solutions are validated using numerical analyses, and corresponding figures are presented to illustrate the physical implications of the derived solutions. Full article

The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map $\alpha$ and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general $(n+1)$-dimensional n-Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. Full article

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let $\mathcal{A}$ be a Banach space nest algebra and $\mathcal{L}$ be a weakly closed Lie $\mathcal{A}$-module. We show that there exist a weakly closed $\mathcal{A}$-bimodule $\mathcal{K}$, a weakly closed subalgebra ${\mathcal{D}}_{\mathcal{K}}$ of $\mathcal{A}$, and a largest weakly closed $\mathcal{A}$-bimodule $\mathcal{J} \mathrm{contained} \mathrm{in} \mathcal{L}$$,\mathrm{such} \mathrm{that} \mathcal{J}\subseteq \mathcal{L}\subseteq \mathcal{K}+{\mathcal{D}}_{\mathcal{K}}$, $\mathrm{with} [\mathcal{K},\mathcal{A}]\subseteq \mathcal{L}$. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras. Full article

In this paper, the $(2+1)$-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of $(1+1)$-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation. Full article

In the present paper, we review the progress of the project of the classification and construction of invariant differential operators for non-compact, semisimple Lie groups. Our starting point is the class of algebras which we called earlier ‘conformal Lie algebras’ (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this purpose, we introduced recently the new notion of a parabolic relation between two non-compact, semi-simple Lie algebras $\mathcal{G}$ and ${\mathcal{G}}^{\prime }$ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Full article

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows. Full article

Over a field of characteristic $p>3$, let $KO(n,n+1;\underline{t})$ denote the odd contact Lie superalgebra. In this paper, the super-biderivations of odd Contact Lie superalgebra $KO(n,n+1;\underline{t})$ are studied. Let $T_{KO}$ be a torus of $KO(n,n+1;\underline{t})$, which is an abelian subalgebra of $KO(n,n+1;\underline{t})$. By applying the weight space decomposition approach of $KO(n,n+1;\underline{t})$ with respect to $T_{KO}$, we show that all skew-symmetric super-biderivations of $KO(n,n+1;\underline{t})$ are inner super-biderivations. Full article

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity $u_{tt}+\alpha \left(u\right)u_t={\left(u^{\beta }{u}_x\right)}_x+f\left(u\right)$. We carried out a complete Lie symmetry classification by finding forms for $\alpha \left(u\right)$ and $f\left(u\right)$. This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach. Full article

In the given study, we intended to gain familiarity with the idea of fuzzy Hom–Lie subalgebras (ideals) of Hom–Lie algebras. It primarily seeks to study a few of their properties. This research investigates the relationship between fuzzy Hom–Lie subalgebras (ideals) and Hom–Lie subalgebras (ideals). Additionally, this study constructs new fuzzy Hom–Lie subalgebras based on the direct sum of a finite number of existing ones. Finally, the properties of fuzzy Hom–Lie subalgebras and fuzzy Hom–Lie ideals are examined in the context of the morphisms of Hom–Lie algebras. Full article

Seaweeds or seaweed Lie algebras are subalgebras of the full-matrix algebra $\mathrm{Mat}\left(n\right)$ introduced by Dergachev and Kirillov to give an example of algebras for which it is possible to compute the Dixmier index via combinatorial methods. It is worth noting that finding such an index for general Lie algebras is a cumbersome problem. On the other hand, Brauer configuration algebras are multiserial and symmetric algebras whose representation theory can be described using combinatorial data. It is worth pointing out that the set of integer partitions and compositions of a fixed positive integer give rise to Brauer configuration algebras. However, giving a closed formula for the dimension of these kinds of algebras or their centers for all positive integer is also a tricky problem. This paper gives formulas for the dimension of Brauer configuration algebras (and their centers) induced by some restricted compositions. It is also proven that some of these algebras allow defining seaweeds of Dixmier index one. Full article

In a recent paper, a new conformally flat metric was introduced, describing an expanding scalar field in a spherically symmetric geometry. The spacetime can be interpreted as a Schwarzschild-like model with an apparent horizon surrounding the curvature singularity. For the above metric, we present the complete conformal Lie algebra consisting of a six-dimensional subalgebra of isometries (Killing Vector Fields or KVFs) and nine proper conformal vector fields (CVFs). An interesting aspect of our findings is that there exists a gradient (proper) conformal symmetry (i.e., its bivector F_ab vanishes) which verifies the importance of gradient symmetries in constructing viable cosmological models. In addition, the 9-dimensional conformal algebra implies the existence of constants of motion along null geodesics that allow us to determine the complete solution of null geodesic equations. Full article

The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors $\mathfrak{T}\mathcal{Q}$. From this procedure two complementary Lie subalgebras of $\mathfrak{T}\mathcal{Q}$ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order $n⩾2$. Full article

This article deals with Lie algebra $\mathfrak{G}$ of all infinitesimal affine transformations of the manifold $\mathcal{M}$ with an affine connection, its stationary subalgebra $ℌ\subset \mathfrak{G}$, the Lie group $\mathcal{G}$ corresponding to the algebra $\mathfrak{G}$, and its subgroup $\mathcal{H}\subset \mathcal{G}$ corresponding to the subalgebra $ℌ\subset \mathfrak{G}$. We consider the center $ℨ\subset \mathfrak{G}$ and the commutant [$\mathfrak{G},\mathfrak{G}]$ of algebra $\mathfrak{G}$. The following condition for the closedness of the subgroup $\mathcal{H}$ in the group $\mathcal{G}$ is proved. If $ℌ\cap \left(ℨ+\left[\mathfrak{G};\mathfrak{G}\right]\right)=ℌ\cap [\mathfrak{G};\mathfrak{G}$], then $\mathcal{H}$ is closed in $\mathcal{G}$. To prove it, an arbitrary group $\mathcal{G}$ is considered as a group of transformations of the set of left cosets $\mathcal{G}/\mathcal{H}$, where $\mathcal{H}$ is an arbitrary subgroup that does not contain normal subgroups of the group $\mathcal{G}$. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group$\mathcal{G}$. However, it can contain the right multiplication by element $\overline{\mathcal{𝒽}}$, belonging to normalizator of subgroup $\mathcal{H}$ and not belonging to the center of a group $\mathcal{G}$. In the case when $\mathcal{G}$ is in the Lie group, corresponding to the algebra $\mathfrak{G}$ of all infinitesimal affine transformations of the affine space $\mathcal{M}$ and its subgroup $\mathcal{H}$ corresponding to its stationary subalgebra $ℌ\subset \mathfrak{G}$, we prove that such element $\overline{{\displaystyle \mathcal{𝒽}}}$ exists if subgroup $\mathcal{H}$ is not closed in $\mathcal{G}$. Moreover $\overline{\mathcal{𝒽}}$ belongs to the closures $\overline{\mathcal{H}}$ of subgroup $\mathcal{H}$ in $\mathcal{G}$ and does not belong to commutant $\left(\mathcal{G},\mathcal{G}\right)$ of group $\mathcal{G}$. It is also proved that $\mathcal{H}$ is closed in $\mathcal{G}$ if $\left(\mathfrak{P}+ℨ\right)\cap ℌ=\mathfrak{P}\cap ℌ$ for any semisimple algebra $\mathfrak{P}\in \mathfrak{G}$ for which $\mathfrak{P}+\Re =\mathfrak{G}$. Full article