Reduction of the KP Hierarchy

This thesis will delve into the Kadomtsev-Petviashvili equation or KP equation and it\u27s hierarchy. More specifically, the solition theory around it. To do so, we first explore the soliton theory for the Korteweg de-Vries equation or KdV equation by analysing it through the inverse scattering transform method and presenting it\u27s soliton solutions. Second, we will introduce, Hirota\u27s bilinear form, and understand its main idea. Third, introduce Sato Theory, and use it to formulate the KP hierarchy, via using pseudo-differential operators, presenting the lax operator, the dressing operator, Sato’s equation, and the zero curvature equation (Zakharov-Shabat Equation). Fourth, find the general solution and one-soliton solution to the KP hierarchy and perform a 2-reduction and 3-reduction on the KP hierarchy. Finally, use Hirota\u27s bilinear method (direct method) to find the multiple solition solutions for the KP hierarchy

Integrable Theory of the Perturbation Equations

An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations: hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures etc. and provides us a method to generate hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.Comment: 27 pages, latex, to appear in Chaos, Soliton & Fractal

The direct algorithm to construct the Davey-Stewartson hierarchy by two scalar pseudo-differential operators

The infinite many symmetries of DS(Davey-Stewartson) system are closely connected to the integrable deformations of surfaces in $\mathbb{R}^{4}$. In this paper, we give a direct algorithm to construct the expression of the DS(Davey-Stewartson) hierarchy by two scalar pseudo-differential operators involved with $\partial$ and $\hat{\partial}$.Comment: 16 page

Asymptotic dynamics of higher-order lumps in the Davey-Stewartson II equation

A family of higher-order rational lumps on non-zero constant background of Davey–Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of which is markedly distinct from the necessary orthogonal scattering for the higher-order lumps on zero background. Additionally, it is illustrated that the higher-order lumps containing multi-peaked n-lumps can be regarded as a nonlinear superposition of n first-order ones as