Unsteady Solutions of Euler Equations Generated by Steady Solutions

Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.     

Solving a nonlinear pde that prices real options using utility based pricing methods

We provide group invariant solutions to two nonlinear differential equations associated with the valuing of real options with utility pricing theory. We achieve these through the use of the Lie theory of continuous groups, namely, the classical Lie point symmetries. These group invariant solutions, constructed through the use of the symmetries that also leave the boundary conditions invariant, are consistent with the results in the literature. Thus it may be shown that Lie symmetry algorithms underlie many ad hoc methods that are utilised to solve differential equations in finance.     

A hierarchy of submodels of differential equations

We consider a system of differential equations admitting a group of transformations. The Lie algebra of the group generates a hierarchy of submodels. This hierarchy can be chosen so that the solutions to each of submodels are solutions to some other submodel in the same hierarchy. For this we must calculate an optimal system of subalgebras and construct a graph of embedded subalgebras and then calculate the differential invariants and invariant differentiation operators for each subalgebra. The invariants of a superalgebra are functions of the invariants of the algebra. The invariant differentiation operators of a superalgebra are linear combinations of invariant differentiation operators of a subalgebra over the field of invariants of the subalgebra. The comparison of the representations of group solutions gives a relation between the solutions to the models of the superalgebra and the subalgebra. Some examples are given of embedded submodels for the equations of gas dynamics.     

Gaudin subalgebras and wonderful models

Gaudin Hamiltonians form families of r-dimensional abelian Lie subalgebras of the holonomy Lie algebra of the arrangement of reflection hyperplanes of a Coxeter group of rank r. We consider the set of principal Gaudin subalgebras, which is the closure in the appropriate Grassmannian of the set of spans of Gaudin Hamiltonians. We show that principal Gaudin subalgebras form a smooth projective variety isomorphic to the De Concini–Procesi compactification of the projectivized complement of the arrangement of reflection hyperplanes.     

On differential equations characterized by their Lie point symmetries

We study the geometry of differential equations determined uniquely by their point symmetries, that we call Lie remarkable. We determine necessary and sufficient conditions for a differential equation to be Lie remarkable. Furthermore, we see how, in some cases, Lie remarkability is related to the existence of invariant solutions. We apply our results to minimal submanifold equations and to Monge-Ampère equations in two independent variables of various orders.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

About the classification of the holonomy algebras of lorentzian manifolds

The classification of the holonomy algebras of Lorentzian manifolds can be reduced to the classification of the irreducible subalgebras h ? so(n) that are spanned by the images of linear maps from ? ⁿ to h satisfying some identity similar to the Bianchi identity. Leistner found all these subalgebras and it turned out that the obtained list coincides with the list of irreducible holonomy algebras of Riemannian manifolds. The natural problem is to give a simple direct proof of this fact. We give such a proof for the case of semisimple not simple Lie algebras h.     

Irregular Partially Invariant Solutions of Rank 2 and Defect 1 to Equations of Gas Dynamics

We consider three-dimensional subalgebras admitted by the equations of gas dynamics having time as an invariant and containing no rotation operator. For such subalgebras we seek for irregular partially invariant solutions of rank 2 and defect 1. The representation for solutions has the form which generalizes motion of a gas with a linear velocity field. We show that partially invariant solutions exist for each subalgebra. We describe the set of these solutions. We find solutions with the indicated representation that are not partially invariant. The solutions reducible to invariant solutions are generalized to new submodels.     

On non-linear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra AN(z) in N spatial dimensions is studied. The special case A₁(2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrödinger Lie algebra sch₁ as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of AN(z) such that the well-known Monge-Ampère equation is the required additional condition. New exact solutions are found for some representatives of this class.     

Classification of Differential Invariant Submodels

Calculation of differential invariants and invariant differentiation operators of a subalgebra of the Lie algebra admitted by a system of differential equations enables us to construct differential invariant submodels. We classify submodels for every subalgebra of an optimal system of subalgebras. Classification includes the invariant submodels and partially invariant submodels considered earlier. We give examples of classification for three-dimensional subalgebras admitted by the equations of gas dynamics.     

Invariant operators of four-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group <Emphasis Type="Italic">P</Emphasis>(1,4)

The classification of four-dimensional nonconjugate subalgebras of the Lie algebra of the Poincare group P(1, 4) into classes of isomorphic subalgebras is performed. Using this classification, we construct invariant operators (generalized Casimir operators) [30] for all four-dimensional nonconjugate subalgebras of the Lie algebra of the group P(1, 4) and present them in the explicit form.     

On the normalizers of subalgebras in an infinite Lie algebra

An algorithm for calculating the normalizer of subalgebra in an infinite Lie symmetry algebra is proposed. The classification problem for a subalgebra spanned by generators that depend on arbitrary functions is formulated. This problem lies in finding the specifications of arbitrary functions and calculating the normalizers of the subalgebras so obtained. As an example, we consider the Lie symmetry algebra L admitted by the thermal diffusion equations. The first-order optimal system of subalgebras Θ₁L is constructed and the normalizers of finite subalgebras from this system are found. The classification of subalgebras depending on arbitrary functions is made.     

Group analysis of the von Kármán–Howarth equation. Part I. Submodels

We study the von Kármán–Howarth (KH) equation by group theoretical methods. This scalar partial differential equation involves two dependent variables (closure problem) and, it has been derived from the Navier–Stokes equations. The equivalence Lie algebra L has been found to be infinite-dimensional and, it is spanned by the four operators. The subalgebra of L is spanned by the three operators. Furthermore, ideal comprises one operator. Optimal systems of one-, two- and three-dimensional subalgebras have been obtained. Normalizers for the one- and two-dimensional subalgebras have been calculated. Finally we have obtained the submodels of the KH equation corresponding to optimal system of one- and two-dimensional subalgebras. This merely suggests alternative solutions to the closure problem of isotropic turbulence.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

Optimal classification,exact solutions,and wave interactions of Euler system with large friction

We derive exact solutions of one-dimensional Euler system that accounts for gravity together with large friction. Certain optimal classes of subalgebra using Lie symmetry analysis are obtained for this system. We apply the reduction procedure to reduce the Euler system to a system of ordinary differential equations in terms of new similarity variable for each class of subalgebras leading to invariant solutions. The evolution of characteristic shock and its interaction with the weak discontinuity by using one of the invariant solutions is studied. Further, the properties of reflected and transmitted waves and jump in acceleration influenced by the incident wave have been characterized.     

Einstein–Weyl structures from hyper–Kähler metrics with conformal Killing vectors

We consider four (real or complex) dimensional hyper-Kähler metrics with a conformal symmetry K. The three-dimensional space of orbits of K is shown to have an Einstein–Weyl structure which admits a shear-free geodesics congruence for which the twist is a constant multiple of the divergence. In this case the Einstein–Weyl equations reduce down to a single second order PDE for one function. The Lax representation, Lie point symmetries, hidden symmetries and the recursion operator associated with this PDE are found, and some group invariant solutions are considered.     

Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions

Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.     

Lie and Engel theorems for n-tuple lie algebras

We prove some analogs of the Lie and Engel theorems for n-tuple Lie algebras. Furthermore, we establish existence of Cartan subalgebras in the n-tuple Lie algebras.     

Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L₀ that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L₀ has finite codimension in L and there are Lie subalgebras L₀=L⁰⊂L¹⊂?⊂Lp=L such that Li+1=Li+[Li,Li+1] for all i; (2) L₀ is a Lie ideal of L and dim(L₀)=∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.