123 research outputs found
Conservation laws for a generalized seventh order KdV equation
In this paper, by applying the multiplier method we obtain a complete classification of low-order local conservation laws for a generalized seventh-order KdV equation depending
on seven arbitrary nonzero parameters. We apply the Lie method in order to classify all point symmetries admitted by the equation in terms of the arbitrary parameters. We find that there are no special cases of the parameters for which the equation admits extra
symmetries, other than those that can be found by inspection (scaling symmetry and space and time translation symmetries). We consider the reduced ordinary differential equations
and we determined all integrating factors of the reduced equation from the combined x- and t- translation symmetries. Finally, we observe that all integrating factors arise by reduction of the low-order multipliers of the generalized seventh-order KdV equation.7 página
A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta
We searched integrable 2D homogeneous polynomial potential with a polynomial
first integral by using the so-called direct method of searching for first
integrals. We proved that there exist no polynomial first integrals which are
genuinely cubic or quartic in the momenta if the degree of homogeneous
polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge
On the algebraic structure of rational discrete dynamical systems
We show how singularities shape the evolution of rational discrete dynamical
systems. The stabilisation of the form of the iterates suggests a description
providing among other things generalised Hirota form, exact evaluation of the
algebraic entropy as well as remarkable polynomial factorisation properties. We
illustrate the phenomenon explicitly with examples covering a wide range of
models
Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
We consider multiple lattices and functions defined on them. We introduce
slow varying conditions for functions defined on the lattice and express the
variation of a function in terms of an asymptotic expansion with respect to the
slow varying lattices.
We use these results to perform the multiple--scale reduction of the lattice
potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur
Dark-Bright Optical Soliton and Conserved Vectors to the Biswas-Arshed Equation With Third-Order Dispersions in the Absence of Self-Phase Modulation
The form-I version of the new celebrated Biswas-Arshed equation is studied in this work with the aid of complex envelope ansatz method. The equation is considered when self-phase is absent and velocity dispersion is negligibly small. New Dark-bright optical soliton solution of the equation emerge from the integration. The acquired solution combines the features of dark and bright solitons in one expression. The solution obtained are not yet reported in the literature. Moreover, we showed that the equation possess conservation laws (Cls)
Discrete Lax pairs, reductions and hierarchies
The term `Lax pair' refers to linear systems (of various types) that are related to nonlinear equations through a compatibility condition. If a nonlinear equation possesses a Lax pair, then the Lax pair may be used to gather information about the behaviour of the solutions to the nonlinear equation. Conserved quantities, asymptotics and even explicit solutions to the nonlinear equation, amongst other information, can be calculated using a Lax pair. Importantly, the existence of a Lax pair is a signature of integrability of the associated nonlinear equation. While Lax pairs were originally devised in the context of continuous equations, Lax pairs for discrete integrable systems have risen to prominence over the last three decades or so and this thesis focuses entirely on discrete equations. Famous continuous systems such as the Korteweg de Vries equation and the Painleve equations all have integrable discrete analogues, which retrieve the original systems in the continuous limit. Links between the different types of integrable systems are well known, such as reductions from partial difference equations to ordinary difference equations. Infinite hierarchies of integrable equations can be constructed where each equation is related to adjacent members of the hierarchy and the order of the equations can be increased arbitrarily. After a literature review, the original material in this thesis is instigated by a completeness study that finds all possible Lax pairs of a certain type, including one for the lattice modified Korteweg de Vries equation. The lattice modified Korteweg de Vries equation is subsequently reduced to several q-discrete Painleve equations, and the reductions are used to form Lax pairs for those equations. The series of reductions suggests the presence of a hierarchy of equations, where each equation is obtained by applying a recursion relation to an earlier member of the hierarchy, this is confirmed using expansions within the Lax pairs for the q-Painleve equations. Lastly, some explorations are included into fake Lax pairs, as well as sets of equivalent nonlinear equations with similar Lax pairs
Continuous symmetric reductions of the Adler-Bobenko-Suris equations
Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete
integrable equations are presented. Initially defined by their invariance under
the action of both of the extended three point generalized symmetries admitted
by the corresponding equations, these solutions are shown to be determined by
an integrable system of partial differential equations. The connection of this
system to the Nijhoff-Hone-Joshi "generating partial differential equations" is
established and an auto-Backlund transformation and a Lax pair for it are
constructed. Applied to the H1 and Q1 members of the
Adler-Bobenko-Suris family, the method of continuously symmetric reductions
yields explicit solutions determined by the Painleve trancendents.Comment: 28 pages, submitted to J. Phys. A: Math. Theo
Similarity Reductions and Integrable Lattice Equations
In this thesis I extend the theory of integrable partial difference equations (PAEs)
and reductions of these systems under scaling symmetries. The main approach used is
the direct linearization method which was developed previously and forms a powerful
tool for dealing with both continuous and discrete equations. This approach is further
developed and applied to several important classes of integrable systems.
Whilst the theory of continuous integrable systems is well established, the theory of
analogous difference equations is much less advanced. In this context the study of
symmetry reductions of integrable (PAEs) which lead to ordinary difference equations
(OAEs) of Painleve type, forms a key aspect of a more general theory that is still in its
infancy.
The first part of the thesis lays down the general framework of the direct linearization
scheme and reviews previous results obtained by this method. Most results so far have
been obtained for lattice systems of KdV type. One novel result here is a new approach
for deriving Lax pairs. New results in this context start with the embedding of the
lattice KdV systems into a multi-dimensional lattice, the reduction of which leads
to both continuous and discrete Painleve hierarchies associated with the Painleve VI
equation.
The issue of multidimensional lattice equations also appears, albeit in a different way,
in the context of the lattice KP equations, which by dimensional reduction lead to new
classes of discrete equations.
This brings us in a natural way to a different class of continuous and discrete systems,
namely those which can be identified to be of Boussinesq (BSQ) type. The development
of this class by means of the direct linearization method forms one of the major parts of
the thesis. In particular, within this class we derive new differential-difference equations
and exhibit associated linear problems (Lax pairs). The consistency of initial value
problems on the multi-dimensional lattice is established. Furthermore, the similarity
constraints and their compatibility with the lattice systems guarantee the consistency
of the reductions that are considered. As such the resulting systems of lattice equations
are conjectured to be of Painleve type.
The final part of the thesis contains the general framework for lattice systems of AKNS
type for which we establish the basic equations as well as similarity constraints
- …