scholarly journals Classification of singular differential invariants in (1+3)-dimensional space and integrability

2021 ◽  
Vol 104 (4) ◽  
pp. 003685042110542
Author(s):  
Muhammad Ayub ◽  
Zahida Sultan ◽  
Muhammad Naeem Qureshi ◽  
Fazal Mahmood Mahomed
Keyword(s):  
Differential Equations ◽  
Lie Algebra ◽  
Dimensional Space ◽  
Classification Problem ◽  
Second Order ◽  
Differential Invariants ◽  
Physical Aspect ◽  
Canonical Forms ◽  
3 Dimensional ◽  
Dimension Formula

Singularity is one of the important features in invariant structures in several physical phenomena reflected often in the associated invariant differential equations. The classification problem for singular differential invariants in (1+3)-dimensional space associated with Lie algebras of dimension 4 is investigated. The formulation of singular invariants for a Lie algebra of dimension [Formula: see text] possessed by the underlying system of three second-order ordinary differential equations is studied in detail and the corresponding canonical forms for these systems are deduced. Furthermore, the categorization of singular invariants on the basis of conditional singularity, weak uncoupling, weak linearization, partial uncoupling and partial linearization are described for the underlying canonical forms. In addition, those cases of classified canonical forms are also mentioned which do not lead to singular invariant systems of three second-order ODEs for a Lie algebra of dimension 4. The integrability aspect of these classified singular-invariant systems in (1+3)-dimensional space is discussed in a detailed manner for a Lie algebra of dimension 4. Finally, two physical systems from mechanics are presented to illustrate the utilization of the physical aspect of these singular invariants.

scholarly journals GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

2019 ◽  
Vol 4 (1) ◽  
pp. 149-155
Author(s):  
Kholmatzhon Imomnazarov ◽  
Ravshanbek Yusupov ◽  
Ilham Iskandarov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Group Classification ◽  
Second Order ◽  
Partial Differential ◽  
One Dimensional ◽  
Preliminary Group

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.

A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

1993 ◽  
Vol 48 (4) ◽  
pp. 535-550 ◽  
Author(s):  
H. Kötz
Keyword(s):  
Differential Equations ◽  
Symmetry Group ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Classification Problem ◽  
Similarity Solutions ◽  
Point Symmetry ◽  
Point Symmetry Group ◽  
Relativistic Case ◽  
Lie Point Symmetry

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

scholarly journals Second-Order Systems of ODEs Admitting Three-Dimensional Lie Algebras and Integrability

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Muhammad Ayub ◽  
Masood Khan ◽  
F. M. Mahomed
Keyword(s):  
Lie Algebras ◽  
Kepler Problem ◽  
Natural Extension ◽  
Three Dimensional ◽  
Second Order ◽  
Differential Invariants ◽  
Canonical Forms ◽  
Solvable Lie Algebra ◽  
Complete Set ◽  
Invariant Representations

We present a systematic procedure for the determination of a complete set ofkth-order (k≥2) differential invariants corresponding to vector fields in three variables for three-dimensional Lie algebras. In addition, we give a procedure for the construction of a system of twokth-order ODEs admitting three-dimensional Lie algebras from the associated complete set of invariants and show that there are 29 classes for the case ofk= 2 and 31 classes for the case ofk≥3. We discuss the singular invariant representations of canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras. Furthermore, we give an integration procedure for canonical forms for systems of two second-order ODEs admitting three-dimensional Lie algebras which comprises of two approaches, namely, division into four types I, II, III, and IV and that of integrability of the invariant representations. We prove that if a system of two second-order ODEs has a three-dimensional solvable Lie algebra, then, its general solution can be obtained from a partially linear, partially coupled or reduced invariantly represented system of equations. A natural extension of this result is provided for a system of twokth-order (k≥3) ODEs. We present illustrative examples of familiar integrable physical systems which admit three-dimensional Lie algebras such as the classical Kepler problem and the generalized Ermakov systems that give rise to closed trajectories.

scholarly journals Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160461 ◽  
Author(s):  
A. A. Gainetdinova ◽  
R. K. Gazizov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Algebras ◽  
Transformation Group ◽  
Second Order ◽  
Differential Invariants ◽  
Invariant Representation ◽  
Invariant Differentiation

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

scholarly journals Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 94
Author(s):  
Dimitris M. Christodoulou ◽  
Eric Kehoe ◽  
Qutaibeh D. Katatbeh
Keyword(s):  
Differential Equations ◽  
Fundamental Equation ◽  
Mathematical Physics ◽  
Analytic Solutions ◽  
Second Order ◽  
Canonical Forms ◽  
Order Linear ◽  
Physical Problems ◽  
Radial Schrödinger Equation ◽  
Linear Homogeneous Differential Equation

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

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