Exact Solutions Of Two(2+1)-dimensional Nonlinear Evolution Equations

In this thesis,the solutions of the Kadomtsev-Petviashvili I(KPI)equation and the(2+1)-dimensional complex modified Korteweg-de Vries(cmKdV)equation are stud-ied by using the Hirota bilinear method and the Darboux Transformation method.The following results are obtained:In the second chapter,by using the Hirota bilinear method and the complexication method,high-order breathers of the Kadomtsev-Petviashvili I equation are obtained.The period,extremum,trajectory of the order-1 breather solution are analyzed.Fur-thermore,the full degeneration process of the high-order breather solution to high-order lump solution is studied by the long wave limit,which means that one peak of breather is a very good approximation of lump.In addition,the partial degeneration process of high-order breathers is discussed,which generates the hybrid solutions consisting of soliton,breather,or lump.In the third chapter,we first construct the determinant form of the n-fold Dar-boux transformation for these(2+1)-dimensional cmKdV equation.The order-n reg-ular soliton solutions and the order-n deformed soliton solutions are obtained by use of zero seeds.Three types of order-1 deformed solitons,namely,the polynomial type,the trigonometric type,and the hyperbolic type,are derived.Meanwhile,their dynami-cal behaviours,including amplitude,velocity,direction,periodicity and symmetry,are also investigated in detail.In particular,the formulas of |q[1]| and trajectories are pro-vided analytically.For order-2 cases,the analytical expressions of deformed solitons are obtained.In the fourth chapter,starting with a plane wave seed,the order-n periodic solutions for the(2+1)-dimensional cmKdV equation are obtained by using the determinant form of the Darboux transformation.To the order-1 periodic solutions,we get two types of periodic solutions that are respectively induced by different properties of the parameter l2,namely,the periodic line wave solutions and the breather solutions.The detailed dy-namical characteristics of these solutions are also analyzed,including period,velocity,extreme values,amputations,the dynamical behavior,and so on.To the order-2 peri-odic solutions,we obtain the order-2 periodic line wave solution,the second order-2 breather solution,and the order-2 mixed periodic solution composed of a periodic line wave and a breather by choosing different lk.In the fifth chapter,based on the periodic solutions given in chapter 4,two kinds of rational solutions for the(2+1)-dimensional cmKdV equation,namely,line rogue wave solutions and lump solutions,are constructed by use of the taylor expansion and the long wave limit.And the property of the order-1 rational solutions are analyzed,including the distribution of extreme points,amplitude,and the time evolution process.