# Journal of Nonlinear Mathematical Physics

Volume 18, Issue Supplement 1, September 2011

**Research Article**

## 2. An Invertible Transformation and Some of its Applications

M. Bruschi, F. Calogero, F. Leyvraz, M. Sommacal

Pages: 1 - 31

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained...

**Research Article**

## 3. Exact Travelling Wave Solutions of a Beam Equation

J. C. Camacho, M. S. Bruzón, J. Ramírez, M. L. Gandarias

Pages: 33 - 49

In this paper we make a full analysis of the symmetry reductions of a beam equation by using the classical Lie method of infinitesimals and the nonclassical method. We consider travelling wave reductions depending on the form of an arbitrary function. We have found several new classes of solutions that...

**Research Article**

## 4. Classical Lie Symmetries and Reductions of a Nonisospectral Lax Pair

P. G. Estévez, M. L. Gandarias, J. Lucas

Pages: 51 - 60

The classical Lie method is applied to a nonisospectral problem associated with a system of partial differential equations in 2 + 1 dimensions (Maccari A, J. Math. Phys. 12 (1998) 6547–6551.). Identification of the classical Lie symmetries provides a set of reductions that give rise to different nontrivial...

**Research Article**

## 5. A Class of Semilinear Fifth-Order Evolution Equations: Recursion Operators and Multipotentialisations

Marianna Euler, Norbert Euler

Pages: 61 - 75

We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.

**Research Article**

## 6. The Converse Problem for the Multipotentialisation of Evolution Equations and Systems

Norbert Euler, Marianna Euler

Pages: 77 - 105

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension...

**Research Article**

## 7. Invariant Solutions of Nonlinear Diffusion Equations with Maximal Symmetry Algebra

V. A. Galaktionov, S. R. Svirshchevskii

Pages: 107 - 121

Nonlinear n-dimensional second-order diffusion equations admitting maximal Lie algebras of point symmetries are considered. Examples of invariant solutions, as well as of solutions on invariant subspaces for some nonlinear operators, are constructed for arbitrary n. A complete description of all possible...

**Research Article**

## 8. Reductions for Some Ordinary Differential Equations Through Nonlocal Symmetries

M. L. Gandarias, M. S. Bruzón

Pages: 123 - 133

In [19] we derive nonlocal symmetries for ordinary differential equations by embedding the given equation in an auxiliary system. Since the nonlocal symmetries of the ODE's are local symmetries of the associated auxiliary system this result provides an algorithmic method to derive this kind of nonlocal...

**Research Article**

## 9. On the Non-Inheritance of Symmetries of Partial Differential Equations

Keshlan S. Govinder, Barbara Abraham-Shrauner

Pages: 135 - 142

The inheritance of symmetries of partial differential equations occurs in a different manner from that of ordinary differential equations. In particular, the Lie algebra of the symmetries of a partial differential equation is not sufficient to predict the symmetries that will be inherited by a resulting...

**Research Article**

## 10. Lie Group Analysis of Moffatt's Model in Metallurgical Industry

N. H. Ibragimov

Pages: 143 - 162

The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.

**Research Article**

## 11. Lie Group Analysis for Multi-Scale Plasma Dynamics

Vladimir F. Kovalev

Pages: 163 - 175

An application of approximate transformation groups to study dynamics of a system with distinct time scales is discussed. The utilization of the Krylov–Bogoliubov–Mitropolsky method of averaging to find solutions of the Lie equations is considered. Physical illustrations from the plasma kinetic theory...

**Research Article**

## 12. On Lie Group Classification of a Scalar Stochastic Differential Equation

Roman Kozlov

Pages: 177 - 187

Lie point symmetry group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. First we prove that the admitted symmetry group is at most three-dimensional. Then the classification is carried out with the help of Lie algebra realizations...

**Research Article**

## 13. Appell Bases on Sequence Spaces

M. Maldonado, J. Prada, M. J. Senosiain

Pages: 189 - 194

We study conditions for a sequence of Appell polynomials to be a basis on a sequence space.

**Research Article**

## 14. Nonisentropic Solutions of Simple Wave Type of the Gas Dynamics Equations

Sergey V. Meleshko, Vasilii P. Shapeev

Pages: 195 - 212

The manuscript is devoted to nonisentropic solutions of simple wave type of the gas dynamics equations. For an isentropic flow these equations (in one-dimensional and steady two-dimensional cases) are reduced to the equations written in the Riemann invariants. The system written in the Riemann invariants...

**Research Article**

## 15. Application of the Generalised Sundman Transformation to the Linearisation of Two Second-Order Ordinary Differential Equations

Sibusiso Moyo, Sergey V. Meleshko

Pages: 213 - 236

In the literature, the generalized Sundman transformation has been used for obtaining necessary and sufficient conditions for a single second- and third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form. As far as we are aware, the generalized Sundman transformation...

**Research Article**

## 16. Second-Order Ordinary Differential Equations with First Integrals of the Form *C*(*t*) + 1/(*A*(*t*, *x*)*ẋ* + *B*(*t*, *x*))

C. Muriel, J. L. Romero

Pages: 237 - 250

We study the class of the ordinary differential equations of the form ẍ + a2(t, x)ẋ2 + a1(t, x)ẋ + a0(t, x) = 0, that admit v = ∂x as λ-symmetry for some λ = α(t, x)ẋ + β(t, x). This class coincides with the class of the second-order equations that have first integrals of the form C(t) + 1/(A(t, x)ẋ...

**Research Article**

## 17. Lie Algebra of the Symmetries of the Multi-Point Equations in Statistical Turbulence Theory

Andreas M. Rosteck, Martin Oberlack

Pages: 251 - 264

We briefly derive the infinite set of multi-point correlation equations based on the Navier–Stokes equations for an incompressible fluid. From this we reconsider the previously derived set of Lie symmetries, i.e. those directly induced by the ones from classical mechanics and also new symmetries. The...