The generalized dressing method and algebra curves method and their applications to integrable equations
廣義穿衣方法與代數曲綫方法及其應用於可積方程
Student thesis: Doctoral Thesis
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Award date  2 Oct 2009 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(f460dcacfbc24fb198173c4844294864).html 

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Abstract
The thesis is mainly divided into two parts. In the first part, a hierarchy of integrable
variablecoefficient nonlinear SchrÄodinger equations, integrable variablecoefficient
Dirac system and Toda Lattice equations are discussed by using a generalized version
of the dressing method. In the second part, some 2+1dimensional integrable discrete
systems are studied by utilizing algebra curves approaches.
The generalized version of the dressing method was presented by Dai and Jeffrey
in 1990s, which is an extension of the original dressing method. The generalization
provides a procedure not only for construction of integrable variablecoefficient nonlinear
evolution equations, but also giving their explicit solutions and Lax pairs. It was
based on the problem of factorization of an integral operator F on the line into the
product of two Volterra type integral operators K§, from which the Gel’fandLevitan
Marchenko (GLM) equation is obtained. These Volterra operators are then used to
construct dressed operators (N1;N2) starting from a pair of initial variablecoefficient
operators (M1;M2). Integrable variablecoefficient nonlinear evolution equations are
obtained from the compatibility condition of the dressed operators. In order to derive
the solutions of these equations, it is necessary to explicitly construct the kernel F of the
integral operator F from the commutative relation between the integral operator F and
initial operators (M1;M2). Then the kernel K of Volterra operators is obtained from
the GLM equation, that is, the solution of the integrable nonlinear evolution equation
can be described. As an application, two problems are discussed in this thesis.
Firstly, with the aid of n £ n AKNS matrix isospectrum (n=2, 3, N+1, 2N+1) and
Dirac system, integrable variablecoefficient coupled cylindrical NLS equations and
mKdV equation, integrable variablecoefficient coupled Hirota equation and Manakov equation, a hierarchy of integrable variablecoefficient Ncoupled NLS equations and
integrable variablecoefficient defocusing NLS equation, are discussed respectively.
Secondly, the generalized dressing method is applied to the discrete system, from
which integrable variablecoefficient Toda lattice equations are presented. Further,
some solutions and Lax pairs of the equations are given explicitly.
The nonlinearization approach of eigenvalue problem was presented by Cao in
1988, which is linked to algebrageometric curves. The new scheme is further shown to
be a very powerful tool, through which quasiperiodic solutions of multidimensional
continuous and discrete soliton equations can be obtained by using decomposition
technique. With the help of the constraint between potential and characteristic function,
the continuous and discrete spectrums are nonlinearized. In the 6th chapter of
this thesis, two discrete spectrums are studied by using the nonlinearization scheme.
Firstly, a new discrete spectrum is proposed, and nonlinear differential difference equations
of the corresponding hierarchy are obtained. We derive an interesting 2+1
dimensional discrete NLS model. Then, under the Bargmann constraint, the soliton
equations are decomposed into certain finitedimensional systems and a new integrable
symplectic map. And then, the generating function method is applied to the study of
integrability, by which it is easy to prove the involutivity and independence of integrals
of motion. Introducing the elliptic coordinates and AbelJacobi coordinates, discrete
flow and continuous flow are straightened. Finally, quasiperiodic solutions of the soliton
equations in the original coordinates are obtained by Riemanntheta function and
AbelJacobi inversion. Secondly, in a similar way, a new 2+1dimensional discrete
model is proposed, and some interesting conclusions are drawn.
Finiteorder expansion of the solution matrix of Lax equation is derived by Geng,
which is also a strong powerful tool for obtaining the solutions of multidimensional
soliton equations. This can be realized through three steps: decomposition!straightening
!inversion. A 2+1dimensional discrete model is decomposed into two compatible
ordinary differential equations and discrete flow inversion. With the aid of Lax
equation matrix of characteristic function, elliptic variables are introduced. And using
algebrageometric curves, it is easy to construct the Riemann surface. Infinite dimensional integrable system and discrete system are straightened by introducing the Abel
Jacobi coordinate. In the 5th chapter, two semidiscrete systems are studied. In the first
section, semidiscrete KaupNewell system is discussed. It is interesting that the continuous
limit of a 2+1dimensional discrete model is exactly a 2+1dimensional Chen
LeeLiu equation. Furthermore, quasiperiodic solution of the equation is described
by introducing the elliptic coordinates and AbelJacobi inversion. In the second section,
semidiscrete ChenLeeLiu system is studied in detail. With the help of Lenard’s
gradient sequence, a hierarchy of nonlinear differentialdifference equation is given.
Moreover, a wellknown 2+1dimensional derivation Toda lattice equation is derived.
Similarly, we obtain the quasiperiodic solution of the corresponding equation.
Keywords: Dressing method, the nonlinearizaton of Lax pair, quasiperiodic solution,
Lax equation, the generating function.
 Integral equations, Curves, Algebraic, Evolution equations, Nonlinear