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The Application of Some Methods of Solving Nonlinear Models and the Research on Interactions between Solitons for Some High Dimensional Models

Author: LiZhiFang
Tutor: RuanHangYu
School: Ningbo University
Course: Theoretical Physics
Keywords: nonlinear evolution equations the methods for solving nonlinear models interactions between solitons
CLC: O175.29
Type: Master's thesis
Year: 2006
Downloads: 24
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Abstract


Firstly, the paper makes a simple introduction and review on some methods of solving nonlinear models: (1) the improving hyperbola function method; (2) Hirota bilinear direct method; (3) variable separation approach; (4) conformal invariant Painlevéasymptotic expansion approach; (5) multiple scales method.Secondly, the paper applies the above methods in some nonlinear models: (1) by improving the hyperbola function method and then applying it in one of the examples of the original reference much more abundant solutions are obtained; (2) basing on Hirota bilinear direct method, the interactions between y–periodic soliton and algebraic soliton are studied for the (2+1)-dimensional Sawada-Kotera equation, the interactions between two y–periodic solitons for which haave ever been studied; (3) the interactions between line soliton and algebraic soliton are studied for (2+1)-dimensional ANNV equation in detail both analytically and graphically with the help of the variable separation approach, and two kinds of singular interactions are got: the resonant interaction and the extremely repulsive interaction; (4) by applying the conformal invariant Painlevéasymptotic expansion approach in the (3+1)-dimensional Jimbo-Miwa equation and trying to fiding other solutions besides the kink-type of plane solitary waves solutions for the new (3+1)-dimensional Painlevéintegrable models which are conformal invariant, the multi-soliton solutions are obtained when taking parameters properly and then much more abundant approximate solutions are got for the JM equation, containing some periodic solutions when taking some real parameters as complex; (5) by use of the method of multiple scales combined with a quasidiscreteness approximation, the two-dimensional monatomic lattice with nearest-neighbor interaction is investigated and the Davey-Stewartson II equation is obtained from the original 2D differential-difference system, the explicit periodic solutions, soliton solutions and rational function solutions for DS-II equation are obtained and then the leading order approximated solutions of the 2D monatomic lattice are got constructed by the explicit solutions of the DS-II equation.At last, the paper presents some problems which are left to be solved.

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Nonlinear partial differential equations
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