Denis Blackmore, Anatoliy K. Prykarpatsky
Volume 19, Issue 1, March 2012, Pages 1-15
A novel approach — based upon vertex operator representation — is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex...
Xiaoli Wang, Jian-Qin Mei
Volume 28, Issue 3, September 2021, Pages 337-343
Based on the Nambu 3-bracket and the operators of the KP hierarchy, we propose the generalized Lax equation of the Lax triple. Under the operator constraints, we construct the generalized KdV hierarchy and Boussinesq hierarchy. Moreover, we present the exact solutions of some nonlinear evolution equations.
Mingxuan Zhu, Lu Cao, Zaihong Jiang, Zhijun Qiao
Volume 28, Issue 3, September 2021, Pages 321-336
This paper devotes to present analysis work on the fifth order Camassa-Holm (FOCH) model which recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss...
Kaihua Bao, Jian Wang, Yong Wang
Volume 28, Issue 3, September 2021, Pages 309-320
In this paper, we prove a local equivariant index theorem for sub-signature operators which generalizes Weiping Zhang’s index theorem for sub-signature operators.
Sining Wei, Yong Wang
Volume 28, Issue 2, June 2021, Pages 254-275
In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.
Volume 28, Issue 3, September 2021, Pages 303-308
There are four quartic integrable Hénon-Heiles systems. Only one of them has been separated in the generic form while the other three have been solved only for particular values of the constants. We consider two of them, related by a canonical transformation, and we give their separation coordinates...
Hassan Almusawa, Ryad Ghanam, Gerard Thompson
Volume 28, Issue 2, June 2021, Pages 242-253
This paper studies the canonical symmetric connection ∇ associated to any Lie group G. The salient properties of ∇ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to ∇ in the special case where the Lie algebra...
Linjie Shi, Na Wang, Minru Chen
Volume 28, Issue 3, September 2021, Pages 292-302
In this paper, we first construct an integrable system whose solutions include the orthogonal Schur functions and the symplectic Schur functions. We find that the orthogonal Schur functions and the symplectic Schur functions can be obtained by one kind of Boson-Fermion correspondence which is slightly...
K. Sethukumarasamy, P. Vijayaraju, P. Prakash
Volume 28, Issue 2, June 2021, Pages 219-241
In this article, we explain how to extend the Lie symmetry analysis method for n-coupled system of fractional ordinary differential equations in the sense of Riemann-Liouville fractional derivative. Also, we systematically investigated how to derive Lie point symmetries of scalar and coupled fractional...
Volume 28, Issue 3, September 2021, Pages 277-291
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.
Gülden Gün Polat, Teoman Özer
Volume 28, Issue 2, June 2021, Pages 209-218
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...
Norbert Euler, Marianna Euler
Volume 16, Issue Supplement 1, December 2009, 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.
S. Dimas, K. Andriopoulos, D. Tsoubelis, P. G. L. Leach
Volume 16, Issue Supplement 1, December 2009, 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...
N. H. Ibragimov
Volume 16, Issue Supplement 1, December 2009, 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...
Volume 16, Issue Supplement 1, December 2009, 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.
P. G. L. Leach, N. Euler
Volume 16, Issue Supplement 1, December 2009, 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,...
A. H. Kara
Volume 16, Issue Supplement 1, December 2009, 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...
F. M. Mahomed, Asghar Qadir
Volume 16, Issue Supplement 1, December 2009, 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...
M. Molati, F. M. Mahomed, C. Wafo Soh
Volume 16, Issue Supplement 1, December 2009, 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...
Kostis Andriopoulos, Tassos Bountis, K. Van Der Weele, Liana Tsigaridi
Volume 16, Issue Supplement 1, December 2009, 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...
Jacek Banasiak, Suares Clovis Oukouomi Noutchie, Ryszard Rudnicki
Volume 16, Issue Supplement 1, December 2009, Pages 13-26
We consider a fragmentation-coagulation equation with growth, where the nonlinear coagulation term, introduced in O. Arino and R. Rudnicki , 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...
C. Muriel, J. L. Romero
Volume 16, Issue Supplement 1, December 2009, 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....
P. Basarab-Horwath, M. Euler, N. Euler, P. G. L. Leach
Volume 16, Issue Supplement 1, December 2009, Pages v-v
Diego Catalano Ferraioli, Paola Morando
Volume 16, Issue Supplement 1, December 2009, 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.
Volume 16, Issue Supplement 1, December 2009, 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...