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Title:  A study of the soliton solutions of the Boussinesq and other nonlinear evolution equations of fluid mechanics 
Authors:  Isa, Mukheta Bin 
Issue Date:  1988 
Publisher:  Newcastle University 
Abstract:  After introducing the nonlinear evolution equations of interest: the
finite depth fluid (FDF), the KadomtsevPetviashvili (KP), the Classical and
the ordinary Boussinesq equations, formal asymptotic derivations of the KP and
the FDF equations are given for the description of surface and interfacial
waves.
The Nsoliton solution of the FDF equation is reconstructed as a finite
sum of Wronskian type determinants. This solution is then shown to reduce to
the solutions of the KdV and the Benjamin  Ono equations under specific
limiting conditions. Interactions between two solitons of the FDF equation
are studied and their interaction properties are shown to reduce to those of
the KdV and the Benjamin  Ono equations. Computer plots of the interactions
of twosoliton solutions of the FDF and the Benjamin  Ono equations are
given.
Resonance phenomena in solitons are studied with reference to the KP
equation. After discussion of the basic concepts of these phenomena, the
Nsoliton solution is shown to reduce to the Wronskian of N/2 functions
(Neven), each of which represents a triad of solitons when the solitons
resonate in pairs. Asymptotic behaviour of the interactions between a triad
and a soliton and between two triads are examined and the phase shifts of the
triads are obtained directly from the Wronskian representation. The
interactions are analysed in detail with reference to numerical computations
of the full solutions.
After showing that the Classical Boussinesq equations are obtained from
Whitham's shallow water wave equations, the basic concept of Hirota's pq=c
reduction of the first modified KP hierarchy is outlined. The Classical
Boussinesq equations are shown as the pq=O reduction of the same hierarchy.
The solution of the hierarchy is manipulated to incorporate the pq=O
reduction. As a result of these limiting procedures applied to the problem,
Wronskian solutions of the Classical Boussinesq equations in terms of rational
functions are produced.
Finally the pq=c reduction of the KP hierarchy is applied to the ordinary
Boussinesq equation. Using this, the Nsoliton solution is expressed as a
finite sum of Wronskian type determinants. Analytic verification made for the
twosoliton solution shows that a number of Wronskian identities are needed
for this purpose. The reason for this behaviour is examined. 
Description:  PhD Thesis 
URI:  http://hdl.handle.net/10443/723 
Appears in Collections:  School of Mathematics and Statistics

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