Invertible Point Transformation Research Articles

Point equivalence of second-order ordinary differential equations to the fifth Painlevé equation with one and two nonzero parameters

We consider the problem of the equivalence of scalar second-order ordinary differential equations under invertible point transformations. To solve this problem in the case of Painleve equations, we use previously constructed invariants of a family of equations whose right-hand sides have a cubic nonlinearity in the first derivative. We obtain the conditions for point equivalence to the fifth Painleve equation if two or three of its parameters are equal to zero.

Reconstruction of f(R) Lagrangian from a massive scalar field

We present a reconstruction of the Lagrangian for $f(R)$ gravity by using a massive scalar field. The scalar field is minimally coupled to the action of $f(R)$ gravity. We demonstrate the use of a theorem based on invertible point transformation of anharmonic oscillator equation that has only recently been applied to gravitational physics. This avenue gives us a direct way to solve the field equations for a few well-known self-interaction potential of the scalar field. A few cases which do not fall in the regime of the theorem are also discussed. The $f(R)$ models are discussed in the context of recent cosmological observations and viability issues.

Characterization of canonical classes for systems of linear ODEs

The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation to the canonical form y(n)=0 consists of copies of the same iterative scalar equation. It is also shown that contrary to the scalar case, an iterative vector equation need not be reducible to the canonical form by an invertible point transformation. Other properties of iterative linear systems are also derived, as well as a simple algebraic formula for their general solution. Copyright © 2017 John Wiley & Sons, Ltd.

Exact solution of the cylindrical Korteweg–de Vries equation for dust ion acoustic wave in unmagnetised plasma

The cylindrical Korteweg–de Vries equation is derived for dust ion acoustic waves in unmagnetized plasma by employing reductive perturbation technique. Lie group analysis is employed to obtain exact solution of the cylindrical Korteweg–de Vries equation. The original non-planar system is mapped by invertible point transformations to the equivalent autonomous planar form to obtain exact solutions. It is found that the non-planar cylindrical geometry differs from that of the planar one-dimensional geometry.

Invariant linearization criteria for a three-dimensional dynamical system of second-order ordinary differential equations and applications

Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results.

Nonlinear self-adjointness: a criterion for linearization of PDEs

In this communication we show that the property of nonlinear self-adjointness is conserved by an invertible point transformation. We use this property to give necessary conditions for the existence of an invertible point transformation mapping of a given partial differential equation (PDE) to a linear PDE.

Exact solutions of one-dimensional nonlinear shallow water equations over even and sloping bottoms

We establish an equivalence of two systems of equations of one-dimensional shallow water models describing the propagation of surface waves over even and sloping bottoms. For each of these systems, we obtain formulas for the general form of their nondegenerate solutions, which are expressible in terms of solutions of the Darboux equation. The invariant solutions of the Darboux equation that we find are simplest representatives of its essentially different exact solutions (those not related by invertible point transformations). They depend on 21 arbitrary real constants; after “proliferation” formulas derived by methods of group theory analysis are applied, they generate a 27-parameter family of essentially different exact solutions. Subsequently using the derived infinitesimal “proliferation” formulas for the solutions in this family generates a denumerable set of exact solutions, whose linear span constitutes an infinite-dimensional vector space of solutions of the Darboux equation. This vector space of solutions of the Darboux equation and the general formulas for nondegenerate solutions of systems of shallow water equations with even and sloping bottoms give an infinite set of their solutions. The “proliferation” formulas for these systems determine their additional nondegenerate solutions. We also find all degenerate solutions of these systems and thus construct a database of an infinite set of exact solutions of systems of equations of the one-dimensional nonlinear shallow water model with even and sloping bottoms.

General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form

Within the framework of Lie group analysis of differential equations, a theorem is determined stating necessary and sufficient conditions allowing one to recover an invertible point transformation mapping a general dynamical system described by nonhomogeneous and nonautonomous first order quasilinear partial differential equations to homogeneous and autonomous form. The proof of the theorem is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated to a suitable subalgebra of the Lie algebra of point symmetries admitted by the source system. The theorem is applied by considering some examples of physical interest arising from different contexts.

Linearization criteria for a system of two second-order ordinary differential equations

For a system of second-order ordinary differential equations conditions of linearizability to the form x″ = 0 are well known. However, an arbitrary linear system need not be equivalent via an invertible point transformation to this simple form. We provide the criteria for a system of two second-order equations to be mapped to the linear system of the general form. Necessary and sufficient conditions for linearization by means of a point transformation are given in terms of coefficients of the system. These results are illustrated with a number of examples.

Nonlocal transformations and linearization of second-order ordinary differential equations

The class of nonlinear second-order equations that are linearizable by means of generalized Sundman transformations (S-transformations) is identified as the class of equations admitting first integrals that are polynomials of first degree in the first-order derivative. This class is also characterized in terms of the coefficients of the equations and constructive methods to derive the linearizing S-transformations are presented. Only the equations of a well-defined subclass can also be linearized by invertible point transformations. These invertible point transformations can be constructed by using the algorithms for the calculation of linearizing S-transformations. Several examples illustrate that both types of linearization are strictly different.

Conditionally invariant solutions of the rotating shallow water wave equations

This paper is devoted to the extension of the recently proposed conditional symmetry method to first-order nonhomogeneous quasilinear systems which are equivalent to homogeneous systems through a locally invertible point transformation. We perform a systematic analysis of the rank-1 and rank-2 solutions admitted by the shallow water wave equations in (2 + 1) dimensions and construct the corresponding solutions of the rotating shallow water wave equations. These solutions involve in general arbitrary functions depending on Riemann invariants, which allow us to construct new interesting classes of solutions.

Reduction of nonhomogeneous quasilinear 2×2 systems to homogeneous and autonomous form

By using the invariance with respect to suitable Lie groups of point transformations, necessary and sufficient conditions are determined allowing one to map through an invertible point transformation nonhomogeneous and nonautonomous quasilinear 2×2 systems to homogeneous and autonomous form. The procedure is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated with the Lie groups of point symmetries admitted by the source system. Various examples of physical interest arising from different contexts are considered.

A unification in the theory of linearization of second-order nonlinear ordinary differential equations

In this letter, we introduce a new generalized linearizing transformation (GLT) for second order nonlinear ordinary differential equations (SNODEs). The well known invertible point (IPT) and non-point transformations (NPT) can be derived as sub-cases of the GLT. A wider class of nonlinear ODEs that cannot be linearized through NPT and IPT can be linearized by this GLT. We also illustrate how to construct GLTs and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this GLT for the SNODEs. We demonstrate the theory with two examples which are of contemporary interest.

Concerning the action of an invertible point transformation on a class of ordinary differential equations

In the category of the group theoretic methods (direct methods) using invertible point transformation, we give a useful recursive relation from one step to the next, which is performed without any further computations. This approach shows that the method is straightforward to implement. Also, the applications of the method for the general nth order differential equations are given.

Exact solutions to the equations of perfect gases through Lie group analysis and substitution principles

In this paper we consider the equations that govern the motion of perfect gases. We explicitly characterize some classes of steady solutions in two and three space dimensions, by introducing invertible point transformations suggested by Lie group analysis; moreover, by using various transformations known as substitution principles, new steady and unsteady solutions are constructed.

Linearizable Second Order Monge–Ampère Equations

In this paper we consider the second order Monge–Ampère equations in (1+1), (2+1), and (3+1) dimensions. We face the problem of reducing to linear form the first order systems that are equivalent to the given Monge–Ampère equations. When the coefficients are assumed constant there are introduced invertible point transformations, suggested by the invariance with respect to one-parameter Lie groups of point symmetries, allowing us to get linear equations. Moreover, the case of nonconstant coefficients is discussed and some linearizable classes of Monge–Ampère equations with coefficients depending on first order derivatives are characterized.

Invertible point transformations, Painlev� test, and the second Painlev� transcendent

The techniques of invertible point transformations and the Painleve analysis can be used to construct integrable ordinary differential equations. We compare both techniques for the second Painleve transcendent.

Lie symmetries and invariants for the time-dependent generalizations of the equation R+C1RnL+C2RmR=0

In this comment the authors use an invertible point transformation for the study of time-dependent equations derived from the equation R+C1Rn L=C2RmR=0. This procedure generalizes some results obtained recently by Leach and Gorringe (1990).

Invertible point transformations, Lie symmetries and the Painlevé test for the equation d2x/dt2 + f1(t)dx/dt + f2(t)x + f3(t)xn = 0

The anharmonic oscillator d2x/dt2 + f1(t)dx/dt + f2(t)x + f3(t)xn = 0 (n ≠ 0, 1) is investigated applying the invertible point transformation, Lie symmetries and the Painlevé test.

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.