26 research outputs found
Symbolic computation of exact solutions for fractional differential-difference equation models
The aim of the present study is to extend the G'/G-expansion method to fractional differential-difference equations of rational type. Particular time-fractional models are considered to show the strength of the method. Three types of exact solutions are observed: hyperbolic, trigonometric and rational. Exact solutions in terms of topological solitons and singular periodic functions are also obtained. As far as we are aware, our results have not been published elsewhere previously
Invariant subspace method to the initial and boundary value problem of the higher dimensional nonlinear time-fractional PDEs
This paper systematically explains how to apply the invariant subspace method
using variable transformation for finding the exact solutions of the
(k+1)-dimensional nonlinear time-fractional PDEs in detail. More precisely, we
have shown how to transform the given (k+1)-dimensional nonlinear
time-fractional PDEs into (1+1)-dimensional nonlinear time-fractional PDEs
using the variable transformation procedure. Also, we explain how to derive the
exact solutions for the reduced equations using the invariant subspace method.
Additionally, in this careful and systematic study, we will investigate how to
find the various types of exact solutions of the (3+1)-dimensional nonlinear
time-fractional convection-diffusion-reaction equation along with appropriate
initial and boundary conditions for the first time. Moreover, the obtained
exact solutions of the equation as mentioned above can be written in terms of
polynomial, exponential, trigonometric, hyperbolic, and Mittag-Leffler
functions. Finally, the discussed method is extended for the (k+1)-dimensional
nonlinear time-fractional PDEs with several linear time delays, and the exact
solution of the (3+1)-dimensional nonlinear time-fractional delay
convection-diffusion-reaction equation is derived.Comment: 45 page
Affine Toda solitons and fusing rules
This thesis is concerned with various soliton solutions to some of the affine Toda field theories. These are field theories in 1+1 dimensions that possess a rich underlying Lie algebraic structure and they are known to be integrable. The soliton solutions occur as a result of the multi-vacua that appear in the field theory when the coupling constant is taken to be purely imaginary. In chapter one a review of the affine Toda field theories is undertaken. This is meant to be a relatively complete and exhaustive survey of the literature that has appeared on the subject in recent years. A brief introduction to the theory of solitons and the methods of obtaining such solutions in field theory is given in chapter two, resulting in the construction of the relevant machinery for the Toda theories. In chapter three, Hi rota's method is used to construct single and double soliton solutions to these theories. As a consequence of these explicit formulae the fusing structure of the solitons may be investigated and shown to be equivalent to that found in the classical particle regime, supplemented by further 'annihilations' of 'soliton-antisoliton'. The calculations of the double soliton solutions are claimed to be original in this context. The fusing has also been examined by Olive, Turok and Underwood(^16) through an abstract group-theoretical approach to the affine Toda field theories, however very few explicit formulae are given by them, and hence all the solutions given here are important in their own right. An algebra-independent analysis of such phenomena is undertaken in chapter four where a vertex operator construction is given for the relevant interaction functions. Some properties of these functions are noted; (some of these facts correspond with those in [16] concerning the fusing structure of the solitons)
Quasilocal charges in integrable lattice systems
We review recent progress in understanding the notion of locality in
integrable quantum lattice systems. The central concept are the so-called
quasilocal conserved quantities, which go beyond the standard perception of
locality. Two systematic procedures to rigorously construct families of
quasilocal conserved operators based on quantum transfer matrices are outlined,
specializing on anisotropic Heisenberg XXZ spin-1/2 chain. Quasilocal conserved
operators stem from two distinct classes of representations of the auxiliary
space algebra, comprised of unitary (compact) representations, which can be
naturally linked to the fusion algebra and quasiparticle content of the model,
and non-unitary (non-compact) representations giving rise to charges,
manifestly orthogonal to the unitary ones. Various condensed matter
applications in which quasilocal conservation laws play an essential role are
presented, with special emphasis on their implications for anomalous transport
properties (finite Drude weight) and relaxation to non-thermal steady states in
the quantum quench scenario.Comment: 51 pages, 3 figures; review article for special issue of JSTAT on
non-equilibrium dynamics in integrable systems; revised version to appear in
JSTA
Geometric Numerical Integration
The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods
Analytical and Numerical Methods for Differential Equations and Applications
The book is a printed version of the Special issue Analytical and Numerical Methods for Differential Equations and Applications, published in Frontiers in Applied Mathematics and Statistic
Algebraic Approaches to Partial Differential Equations
Partial differential equations are fundamental tools in mathematics,sciences
and engineering. This book is mainly an exposition of the various algebraic
techniques of solving partial differential equations for exact solutions
developed by the author in recent years, with emphasis on physical equations
such as: the Calogero-Sutherland model of quantum many-body system in
one-dimension, the Maxwell equations, the free Dirac equations, the generalized
acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and
Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave
equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the
equation of geopotential forecast, the nonlinear Schrodinger equation and
coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson
equations of three-dimensional packets of surface waves, the equation of the
dynamic convection in a sea, the Boussinesq equations in geophysics, the
incompressible Navier-Stokes equations and the classical boundary layer
equations.
In linear partial differential equations, we focus on finding all the
polynomial solutions and solving the initial-value problems. Intuitive
derivations of Lie symmetry of nonlinear partial differential equations are
given. These symmetry transformations generate sophisticated solutions with
more parameters from relatively simple ones. They are also used to simplify our
process of finding exact solutions. We have extensively used moving frames,
asymmetric conditions, stable ranges of nonlinear terms, special functions and
linearizations in our approaches to nonlinear partial differential equations.
The exact solutions we obtained usually contain multiple parameter functions
and most of them are not of traveling-wave type.Comment: This is part of the monograph to be published by Springe