The Periodic Solutions And Point Transformation Application For The Soliton Equations

This dissertation amasses my main research results obtained in pursuit of my master’s degree, mainly involving periodic wave solutions and asymptotic behavior for variable-coefficient KdV equation, the equivalence transformations and its applications for variable-coefficient KdV and variable-coefficient modified KdV equations.The first chapter introduces the background, significance of the soliton theory.The second chapter introduces some related concepts and properties in this paper, such as Hirota operator and properties, Riemann theta function definition and properties.The third chapter focus on constructing one-and two-periodic wave solutions and presenting to analyze asymptotic behavior for the variable-coefficient KdV equation. We study the Backlund transformation of the variable-coefficient KdV equation, via the ex-tend Riemann theta function method to explicitly construct periodic solutions. At the same time, we show that the solitons can be reduced form the periodic wave solutions.The fourth chapter we mainly consider the equivalence transformations of the variable-coefficient KdV and variable-coefficient modified KdV equations. Then we can obtain the soliton solutions, Lie symmetries and the group-invariant solutions for the variable-coefficient KdV and variable-coefficient modified KdV by the equivalence transformations.