Study On Integrable Properties And Solutions For Some Nonlinear Partial Diferential Equations

With the development of modern science and technology, nonlinear science has become a new disciplines, which including more and more research fields. In the research of typhoon, tsunami, mining, fluid, mechanics, plasma, plasma chemistry, optics, communications, biology, there has been a great deal of nonlinear partial differential equations. Therefore a variety of the nature of these equations play a important role in explaining physical phenomenon, such as explicit and exact solutions, Hamilton structure, conservation laws, symmetries, integrable property of partial differential equation.The thesis is arranged as follows:Chapter1. We first introduce the history and development of the solitons; the topics and work in this article.Chapter2. We introduce the well-known Painleve test. Through generalizing Painleve test for ordinary differential equations, In the1980s, J. Weiss, M. Tabor and G. Carnevale advanced Painleve test for partial differential equations, which provides a necessary condition to estimate the integrability of the equation. Next, through Painleve test for the variable coefficient Hirota equation, we know that the variable coefficient Hirota equation has Painleve property under certain constraints. By zero curvature equation, we will show that the variable coefficient Hirota equation is lax integrable under certain constraints; Through the similarity transformation, we obtain the solutions of the variable coefficient Hirota equation.Chapter3. The Darboux transformation is a powerful method for constructing the solutions for nonlinear partial differential equations. By one or several times Darboux transformation, one-soliton or multi-soliton solutions could be obtained from trivial seed solution. In Chapter3, by Darboux transformation, we gained the solutions of the Gerdjikov-Ivanov equation from non-zero seed, such as dark soliton solution, breathers solution.