Ünal, Metin
(1998)
Applications of pfaffians to soliton theory.
PhD thesis, University of Glasgow.
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Abstract
This thesis is concerned with solutions to nonlinear evolution equations. In particular
we examine two soliton equations, namely the Novikov VeselovNithzik (NVN) equations
and the modified NovikovVeselovNithzik (mNVN) equations. We are interested
in the role that determinants and pfaffians play in determining new solutions to various
soliton equations. The thesis is organised as follows.
In chapter 1 we give an introduction and historical background to the soliton theory
and recall John Scott Russell's observation of a solitary wave, made in 1844. We
explain the Lax method and Hirota method and discuss the relevant basic topics of
soliton theory that are used throughout this thesis. We also discuss different types of
solutions that are applicable to nonlinear evolution equations in soliton theory. These
are wronskians, grammians and pfaffians.
In chapter 2 we give an introduction to pfaffians which are the main elements of
this thesis. We give the definition of a pfaffian and a classical notation for the pfaffians
is also introduced. We discuss the identities of pfaffians which correspond to the Jacobi
identity of determinants. We also discuss the differentiation of pfaffians which is useful
in pfaffian technique. By applying the pfaffian technique to the BKP equation, an
example of soliton solutions to the BKP equation is also given.
In chapter 3 we study the asymptotic properties of terms of pfaffians. We apply the technique that is used in [35] for the DaveyStewartson
(DS) equations to the NVN equations. We study the asymptotic properties of the (1, 1) dromion solutions written in dromion solution and generalize them to the (M, N)dromion solution. Summaries
of these asymptotic properties are given. As an application, we apply the general
results obtained for the (M, N)dromion solution to the (2,2)dromion solution and
to the (1, 2)dromion solution and show the asymptotic calculations explicitly for each
dromion. In the last section we give a number of plots which show various kind of
dromion scattering. These illustrate that dromion interaction properties are different
than the usual soliton interactions.
In chapter 4 we exploit the algebraic structure of the soliton equations and find solutions
in terms of fermion particles [54]. We show how determinants and pfaffians arise
naturally in the fermionic approach to soliton equations. We write the rfunction for
charged and neutral free fermions in terms of determinants and pfaffians respectively,
and show that these two concepts are analogous to one another. Examples of how to
get soliton and dromion solutions from rfunctions for the various soliton equations are
given,
In chapter 5 we use some results from [61] and [62]. We study two nonlinear evolution
equations, namely the KonopelchenkoRogers (KR) equations and the modified
Novikov VeselovNithzik (mNVN) equations. We derive a new Lax pair for the mNVN
equations which is gauge equivalent to a pair of operators. We apply the pfaffian technique
to the KR and mNVN equations and show that these equations in the bilinear
form reduce to a pfaffian identity.
In this thesis, chapter 1 is a general introduction to soliton theory and chapter 2 is
an introduction to the main elements of this thesis. The contents of these chapters are
taken from various references as indicated throughout the chapters. Chapters 3, 4, 5
are the author's own work with some results used from other references also indicated
in the chapters.
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