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• W1510365635 abstract "This thesis is concerned with solutions to nonlinear evolution equations. In particularwe examine two soliton equations, namely the Novikov- Veselov-Nithzik (NVN) equationsand the modified Novikov-Veselov-Nithzik (mNVN) equations. We are interestedin the role that determinants and pfaffians play in determining new solutions to varioussoliton equations. The thesis is organised as follows.In chapter 1 we give an introduction and historical background to the soliton theoryand recall John Scott Russell's observation of a solitary wave, made in 1844. Weexplain the Lax method and Hirota method and discuss the relevant basic topics ofsoliton theory that are used throughout this thesis. We also discuss different types ofsolutions that are applicable to nonlinear evolution equations in soliton theory. Theseare wronskians, grammians and pfaffians.In chapter 2 we give an introduction to pfaffians which are the main elements ofthis thesis. We give the definition of a pfaffian and a classical notation for the pfaffiansis also introduced. We discuss the identities of pfaffians which correspond to the Jacobiidentity of determinants. We also discuss the differentiation of pfaffians which is usefulin pfaffian technique. By applying the pfaffian technique to the BKP equation, anexample 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 Davey-Stewartson(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. Summariesof these asymptotic properties are given. As an application, we apply the generalresults obtained for the (M, N)-dromion solution to the (2,2)-dromion solution andto the (1, 2)-dromion solution and show the asymptotic calculations explicitly for eachdromion. In the last section we give a number of plots which show various kind ofdromion scattering. These illustrate that dromion interaction properties are differentthan the usual soliton interactions.In chapter 4 we exploit the algebraic structure of the soliton equations and find solutionsin terms of fermion particles [54]. We show how determinants and pfaffians arisenaturally in the fermionic approach to soliton equations. We write the r-function forcharged 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 toget soliton and dromion solutions from r-functions for the various soliton equations aregiven,In chapter 5 we use some results from [61] and [62]. We study two nonlinear evolutionequations, namely the Konopelchenko-Rogers (KR) equations and the modifiedNovikov- Veselov-Nithzik (mNVN) equations. We derive a new Lax pair for the mNVNequations which is gauge equivalent to a pair of operators. We apply the pfaffian techniqueto the KR and mNVN equations and show that these equations in the bilinearform reduce to a pfaffian identity.In this thesis, chapter 1 is a general introduction to soliton theory and chapter 2 isan introduction to the main elements of this thesis. The contents of these chapters aretaken from various references as indicated throughout the chapters. Chapters 3, 4, 5are the author's own work with some results used from other references also indicatedin the chapters." @default.
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• W1510365635 date "1998-01-01" @default.
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• W1510365635 title "Applications of pfaffians to soliton theory" @default.
• W1510365635 hasPublicationYear "1998" @default.
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