# Journal of Nonlinear Mathematical Physics

Volume 16, Issue Supplement 1, December 2009

**Research Article**

## 2. The Shape of Soliton-Like Solutions of a Higher-Order Kdv Equation Describing Water Waves

Kostis Andriopoulos, Tassos Bountis, K. Van Der Weele, Liana Tsigaridi

Pages: 1 - 12

We study the solitary wave solutions of a non-integrable generalized KdV equation proposed by Fokas [A. S. Fokas, Physica D87, 145 (1995)], aiming to describe unidirectional waves in shallow water with greater accuracy than the standard KdV equation. This generalized equation includes higher-order terms...

**Research Article**

## 3. Global Solvability of a Fragmentation-Coagulation Equation With Growth and Restricted Coagulation

Jacek Banasiak, Suares Clovis Oukouomi Noutchie, Ryszard Rudnicki

Pages: 13 - 26

We consider a fragmentation-coagulation equation with growth, where the nonlinear coagulation term, introduced in O. Arino and R. Rudnicki [2], is designed to model processes in which only a part of particles in the aggregates is capable of coalescence. We introduce various growth models, describing...

**Research Article**

## 4. Applications of Solvable Structures to the Nonlocal Symmetry-Reduction of Odes

Diego Catalano Ferraioli, Paola Morando

Pages: 27 - 42

An application of solvable structures to the reduction of ODEs with a lack of local symmetries is given. Solvable structures considered here are all defined in a nonlocal extension, or covering space, of a given ODE. Examples of the reduction procedure are provided.

**Research Article**

## 5. Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Giampaolo Cicogna

Pages: 43 - 60

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence...

**Research Article**

## 6. Lagrangians For Equations of Painlevé Type by Means of The Jacobi Last Multiplier

G. D’ambrosi, M. C. Nucci

Pages: 61 - 71

We apply the method of Jacobi Last Multiplier to the fifty second-order ordinary differential equations of Painlevé type as given in Ince in order to obtain a Lagrangian and consequently solve the inverse problem of Calculus of Variations for those equations. The easiness and straightforwardness of Jacobi’s...

**Research Article**

## 7. Complete Specification of Some Partial Differential Equations That Arise in Financial Mathematics

S. Dimas, K. Andriopoulos, D. Tsoubelis, P. G. L. Leach

Pages: 73 - 92

We consider some well-known partial differential equations that arise in Financial Mathematics, namely the Black–Scholes–Merton, Longstaff, Vasicek, Cox–Ingersoll–Ross and Heath equations. Our central aim is to discover any underlying connections taking into account the Lie remarkability property of...

**Research Article**

## 8. Multipotentialisations and Iterating-Solution Formulae: The Krichever–Novikov Equation

Norbert Euler, Marianna Euler

Pages: 93 - 106

We derive solution-formulae for the Krichever–Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever–Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.

**Research Article**

## 9. Twisted Symmetries of Differential Equations

Giuseppe Gaeta

Pages: 107 - 136

We review the basic ideas lying at the foundation of the recently developed theory of twisted symmetries of differential equations, and some of its developments.

**Research Article**

## 10. Integration of Systems of First-Order Equations Admitting Nonlinear Superposition

N. H. Ibragimov

Pages: 137 - 147

Systems of two nonlinear ordinary differential equations of the first order admitting nonlinear superpositions are investigated using Lie’s enumeration of groups on the plane. It is shown that the systems associated with two-dimensional Vessiot–Guldberg–Lie algebras can be integrated by quadrature upon...

**Research Article**

## 11. A *Symmetry Invariance* Analysis of the Multipliers & Conservation Laws of the Jaulent–Miodek and Some Families of Systems of KdV Type Equations

A. H. Kara

Pages: 149 - 156

In this paper, we study and classify the conservation laws of the Jaulent–Miodek equations and other systems of KdV type equations which arises in, inter alia, shallow water equations. The main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry...

**Research Article**

## 12. A Novel Riccati Sequence

P. G. L. Leach, N. Euler

Pages: 157 - 164

Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity,...

**Research Article**

## 13. Conditional Linearizability of Fourth-Order Semi-Linear Ordinary Differential Equations

F. M. Mahomed, Asghar Qadir

Pages: 165 - 178

By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally...

**Research Article**

## 14. A Group Classification of a System of Partial Differential Equations Modeling Flow in Collapsible Tubes

M. Molati, F. M. Mahomed, C. Wafo Soh

Pages: 179 - 208

The purpose of this work is to perform group classification of a coupled system of partial differential equations (PDEs) modeling a flow in collapsible tubes. This system of PDEs contains unknown functions of the dependent variables whose forms are specified via the classification with respect to subalgebras...

**Research Article**

## 15. Second-Order Ordinary Differential Equations and First Integrals of The Form *A*(*t*, *x*) ẋ + *B*(*t*, *x*)

C. Muriel, J. L. Romero

Pages: 209 - 222

We characterize the equations in the class 𝒜 of the second-order ordinary differential equations ẍ = M(t, x, ẋ) which have first integrals of the form A(t, x)ẋ + B(t, x). We give an intrinsic characterization of the equations in 𝒜 and an algorithm to calculate explicitly such first integrals. Although...