Differential equations with soliton behaviour

Abstract

Various non-linear wave equations are found to possess solitons - stable solitary waves which only undergo a change of position on collision with each other. It is shown in chapter 1, how the various soliton properties of the sine-Gordon equation, u(_xy) = F(u) sin u, May be derived from its Backlund Transformation. Most of the rest of the thesis consists of several attempts to find Backlund Transformations for other equations of the form u = F(u) by generalizing the usual form of the Backlund Transformation. The only exception to this is in chapter 2 where equations of the form u(_xy) = A(x,y,u).u(_x) + B(x,y,u).u(_y) + C(x,y,u) are considered. The rest of chapter 2 considers the effect of allowing the Backlund Transformation to depend explicitly on the independent variables or on integrals of the dependent variables. The rest of this thesis concentrates on allowing the Backlund Transformation to depend on derivatives only of the "old" and "new" variables, u and u'. It is found that if u and u' satisfy u(_xy) = F(u) where F'''(u) = K.F''(u) and F''(u) = K.F(u) then there are no Backlund Transformations of the following form. Chapter 3. u’(_x) = P(u,u';p(_1),.., ,P(_N);q(_1),...,q(M)) u;(_y)= Q(u,u';p(_1),...,P(_N);q(_1),...,q(_M)) except possibly when M = 1 , N > 7 and F(u) = A(_1).e(^cu) + A(_2).e(^-2cu). Chapter 4. u’(_xx) = P(u, u’, u(_x), u’(_x), x(_y), u(_xx), u(_yy) u’(_y) = Q(u, u’, u(_x), u’(_x), x(_y), u(_xx), u(_yy) Chapter 5 (^1)/(_2) (p’(_N+1) p(_N+1) – P(P(_0), P(_1),…,P(_N);P’(_0),…,P’(_N) N <5 (^1)/(_2)(q’ +q) = Q(P(_0), P(_1),…,P(_N);P(_0),…,P’(_N)