Metin Ünal

Applications of Pfaffians to Soliton theory

1998

Supervisor: Jon Nimmo

Abstract:

This thesis is concerned with solutions to nonlinear evolution equations. In particular we examine two soliton equations: the Novikov-Veselov-Nithzik (NVN) equations and the modified Novikov-Veselov-Nithzik (mNVN) equations. We are interested in the role that determinants 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 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, Gammians 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 whcih is useful in the 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 dromion solutions written in terms of Pfaffians. We apply the technique that is used in [Gilson, C. and Nimmo, JJC., Proc. Roy. Soc. Lon. A 435 (1991) 339-357; MathSciNet Review] for the Davey-Stewartson (DS) equtaions to the NVN equations. We study the asymptotic properties of the (1,1)-dromion solution and generalize them to the (2,2)-dromion solution and to the (2,1)-dromion solution and show the asymptotic calculations explicitly for each dromion. In the last section we give a number of plots which show various kinds of dromion scattering. These illustrate that dromion interaction properties are different from the usual soliton interactions.

In chapter 4 we exploit the algebraic structure of the soliton equations and find solutions in terms of fermion particles [Jimbo, M. and Miwa, T. RIMS, Kyoto Univ. 19 (1983) 943-1001; MathSciNet Review]. We show how determinants and Pfaffians arise naturally in the fermionic approach to soliton equations. We write the tau function 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 solurions from tau functions for the various soliton equations are given.

In chapter 5 we use some results from [Nimmo, JJC., in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Ed. P.A.Clarkson, pp 183-92 (1993), Kluwer; MathSciNet Review] and [Nimmo, JJC., Phys. Lett. A 168 (1992) 113-9; MathSciNet Review]. We study two nonlinear evolution equations: the Konopelchenko-Rogers (KR) equations and the modified Novikov-Veselov-Nithzik (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 KR and mNVN equations and show that these equations in the bilinear form reduce to a Pfaffian identity.

In this thesis, chpater 1 is a general introduction to soliton theory and chapter 2 is an introduction to the main element of this thesis. The contents of these chpaters are taken from various references as indicated throughout the chapters. Chapters 3, 4 and 5 are the author's own work with some results used from other references also indicated in the chapters.