Journal of Nonlinear Mathematical Physics

Latest Articles

Research Article

Affine Ricci Solitons of Three-Dimensional Lorentzian Lie Groups

Yong Wang
In Press, Corrected Proof, Available Online: 15 February 2021
In this paper, we classify affine Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections and perturbed canonical connections and perturbed Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.
Research Article

On Ramsey Dynamical Model and Closed-Form Solutions

Gülden Gün Polat, Teoman Özer
In Press, Corrected Proof, Available Online: 21 January 2021
This study focuses on the analysis of Ramsey dynamical model with current Hamiltonian defining an optimal control problem in a neoclassical growth model by utilizing Lie group theory. Lie point symmetries of coupled nonlinear first-order ordinary differential equations corresponding to first-order conditions...
Research Article

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

Norbert Euler, Marianna Euler
Volume 16, Issue Supplement 1, March 2013, 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

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

S. Dimas, K. Andriopoulos, D. Tsoubelis, P. G. L. Leach
Volume 16, Issue Supplement 1, March 2013, 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

Integration of Systems of First-Order Equations Admitting Nonlinear Superposition

N. H. Ibragimov
Volume 16, Issue Supplement 1, March 2013, 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

Twisted Symmetries of Differential Equations

Giuseppe Gaeta
Volume 16, Issue Supplement 1, March 2013, 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

A Novel Riccati Sequence

P. G. L. Leach, N. Euler
Volume 16, Issue Supplement 1, March 2013, 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

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
Volume 16, Issue Supplement 1, March 2013, 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

Conditional Linearizability of Fourth-Order Semi-Linear Ordinary Differential Equations

F. M. Mahomed, Asghar Qadir
Volume 16, Issue Supplement 1, March 2013, 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

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

M. Molati, F. M. Mahomed, C. Wafo Soh
Volume 16, Issue Supplement 1, March 2013, 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

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
Volume 16, Issue Supplement 1, March 2013, 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

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

Jacek Banasiak, Suares Clovis Oukouomi Noutchie, Ryszard Rudnicki
Volume 16, Issue Supplement 1, March 2013, 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

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

C. Muriel, J. L. Romero
Volume 16, Issue Supplement 1, March 2013, Pages 209-222
We characterize the equations in the class ������ 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 ������ and an algorithm to calculate explicitly such first integrals....
Research Article

Preface

P. Basarab-Horwath, M. Euler, N. Euler, P. G. L. Leach
Volume 16, Issue Supplement 1, March 2013, Pages v-v
Research Article

Applications of Solvable Structures to the Nonlocal Symmetry-Reduction of Odes

Diego Catalano Ferraioli, Paola Morando
Volume 16, Issue Supplement 1, March 2013, 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

Symmetries of Hamiltonian Equations and Λ-Constants of Motion

Giampaolo Cicogna
Volume 16, Issue Supplement 1, March 2013, 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

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

G. D’ambrosi, M. C. Nucci
Volume 16, Issue Supplement 1, March 2013, 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

Nonstandard Separability on the Minkowski Plane

Giuseppe Pucacco, Kjell Rosquist
Volume 16, Issue 4, March 2013, Pages 421-430
We present examples of nonstandard separation of the natural Hamilton–Jacobi equation on the Minkowski plane ������2. By “nonstandard” we refer to the cases in which the form of the metric, when expressed in separating coordinates, does not have the usual Liouville structure. There are two possibilities:...
Research Article

Euler–Lagrange Equations for Functionals Defined On Fréchet Manifolds

José Antonio Vallejo
Volume 16, Issue 4, March 2013, Pages 443-454
We prove a version of the variational Euler–Lagrange equations valid for functionals defined on Fréchet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.
Guidelines

Author Index (Volume 16)

Volume 16, Issue 4, March 2013, Pages 517-519
Research Article

An Old Method of Jacobi to Find Lagrangians

M. C. Nucci, P. G. L. Leach
Volume 16, Issue 4, March 2013, Pages 431-441
In a recent paper by Ibragimov a method was presented in order to find Lagrangians of certain second-order ordinary differential equations admitting a two-dimensional Lie symmetry algebra. We present a method devised by Jacobi which enables one to derive (many) Lagrangians of any second-order differential...
Research Article

Averaging in Weakly Coupled Discrete Dynamical Systems

Niklas Brännström
Volume 16, Issue 4, March 2013, Pages 465-487
In [Y. Kifer, Averaging in difference equations driven by dynamical systems, Asterisque287 (2003) 103–123] a general averaging principle for slow-fast discrete dynamical systems was presented. In this paper we extend this method to weakly coupled slow-fast systems. For this setting we obtain sharper...
Research Article

On Nonlocal Symmetries, Nonlocal Conservation Laws and Nonlocal Transformations of Evolution Equations: Two Linearisable Hierarchies

Norbert Euler, Marianna Euler
Volume 16, Issue 4, March 2013, Pages 489-504
We discuss nonlocal symmetries and nonlocal conservation laws that follow from the systematic potentialisation of evolution equations. Those are the Lie point symmetries of the auxiliary systems, also known as potential symmetries. We define higher-degree potential symmetries which then lead to nonlocal...
Research Article

Global Analytic First Integrals for the Simplified Multistrain/Two-Stream Model for Tuberculosis and Dengue Fever

Jaume Llibre, Clàudia Valls
Volume 16, Issue 4, March 2013, Pages 505-516
We provide the complete classification of all global analytic first integrals of the simplified multistrain/two-stream model for tuberculosis and dengue fever that can be written as x˙=x(β1−b−γ1−β1x−(β1−ν)y),     y˙=y(β2−b−γ2−(β2−ν)x−β2y), with β1, β2, b, γ1, γ2, ν ∈ ℝ.
Research Article

Orbital Linearization in the Quadratic Lotka–Volterra Systems Around Singular Points Via Lie Symmetries

Jaume Giné, Susanna Maza
Volume 16, Issue 4, March 2013, Pages 455-464
In this paper, we consider linearizability and orbital linearizability properties of the Lotka–Volterra system in the neighborhood of a singular point with eigenvalues 1 and -q. In this paper we give the explicit smooth near-identity change of variables that linearizes or orbital linearizes such Lotka–Volterra...