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Integrable Extensions for Camassa-Holm Type Equations and the Matrix (2+1)-dimensional Integrable System

Author: HuangZuoHui
Tutor: ZengYunBo
School: Tsinghua University
Course: Mathematics
Keywords: Integrable systems with self- compatible source Lax representation dressing method Soliton
CLC: O175.29
Type: PhD thesis
Year: 2010
Downloads: 78
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Abstract


Integrable systems with self-compatible source (with source) is widely used in mathematics and physics. This article was first constructed equations with source camassa-Holm (cH) Degasperis-Procesi (DP) equation with source, with source 2 component camassa-Holm (2-cH) equations and the new extension of the matrix 21 dimensional integrable systems obtain the Lax representation and solving problems. By adding eigenvalue integrable system in the form of Hamilton variational derivative of the potential, we get cH ,2-cH with the source and the DP equation and its Lax said. Symmetrical square matrix 21-dimensional integrable systems characteristic function, we introduce a new time flow TB, construct a new extended family of matrix KP equation. Matrix KP equation tribes, the new extended family of matrix KP equation contains variables of another - family time outside than tA TB, and more by the characteristic function and the introduction of the conjugate characteristic function component. The second band source equation for the first time given family this equation, with the source of the two types of 21-dimensional AKNS equation and with the source of the two types of Ds equation and its Lax representation. Matrix KP equation family new promotion tA and TB reduction given structure with the source of the two types of 21-dimensional and 11-dimensional equation - like ways. In recent years, cH equation, DP equation and 2-cH equation has been widely studied. Through reciprocal transformation, the establishment of the first of these equations and the family of a known equation - a negative flow contact. This article was first given 2-cH equation with source cH equation with source DP equation with source promotion reciproca1 transform. The binding constant change easy method of thinking, this paper, with the source the cH equation and DP equation with source Peakon solution. Darboux transformation method, the constant variation the promotion of reciproca1 transform, we construct the the band source cH equation is, with the source of the DP equation and with the source of all types of solutions of the equations of the 2-cH, including soliton solutions, cuspon the solution, positon solution negaton solution, and so on. Dressing method is an important tool for solving the KP equation family, but already dressing method can not be directly applied to the extended family of matrix KP equation. Development of solving the extended family of matrix KP equation \The method provides - a simple, - like a way to get the extended family of solutions of the matrix KP equation. Using this method, this paper, with the source of the two types of 21-dimensional AKNS equation and two types with source Ds soliton solutions of the equation.

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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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