Application Of Hirota Method To Soliton Equations

To look for exact solutions to soliton equations and find new integrable systems or new coupled integrable systems are two important aspect in soliton theory. In this thesis, based on the Hirota bilinear method, we mainly discuss how to solve various forms of exact solutions. Moreover, two type of coupled equations are derived using Hirota bilinear method. The thesis proceeds as followPart 1 is devoted to study and construction of two type of coupled higher order Ito equation on the basis of bilinear formalism.First, the Hirota bilinear method is generalized and investigated, where the novel N-soliton solutions with singular slowly decaying at infinity for the higher order Ito equation are obtained.Second, a multi-component higher order Ito equation is proposed and N-soliton solution expressed by Pfaffians are obtained.Third, we give a new type of coupled higher order Ito equation, in which the phase shifts induced by collisions of solitions depend on the mutual positions of solitons at initial time. 2-soliton and 3-soliton are given for the purpose of illustrating the resonance of solitons in the coupled higher order Ito equation and N-soliton solution is constructed in the form of the pfaffians.Part 2 is mainly focused on discussing of various tpye of explicit exact solutions which include soliton, positon, negaton, complexiton and rational solutions for integrable variable-coefficient Korteweg-de Vries equation.Starting from Schrodinger eigenvalue problem with the constant and variable coefficients, we deduce the hierarchy of the isospectral and nonisospectral variable-coefficient Korteweg-de Vries equation.Furthermore, we give the bilinear form of the two integrable variable-coefficient Korteweg-de Vries equation and solve it by means of Hirota bilinear method and Wronskian technique.At last, a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented which guarantee that the Wronskian determinant is the solutions of the integrable variable-coefficient Korteweg-de Vries equation. The generalized Wronskian solutions provide us with a comprehensive approach to construct many exact solutions including soliton, hegaton, positon, complexiton and rational solutions.