Research On Exact Solutions Of Some Nonlinear Equations And Their Local Structures

With the inhomogeneities of media taken into account,a generalized variable-coefficient Kadomtsev-Petviashvili(vcKP)equation is proposed to model nonlinear waves in fluid mechanics and plasma physics.Based on Hirota bilinear method and symbolic com-putation,we present lump and lump-soliton interaction solutions of the vcKP equation These local solutions are derived by taking the auxiliary function as the positive quadratic function or the linear combination of the positive quadratic function and the exponential function.Compared with the results allowed by the constant coefficient KP equation,lump and lump-soliton solutions for the vcKP equation possess more abundant proper-ties.It is shown that the velocity,moving path and maximum height of the lump are completely characterized by the time functions rather than the constant parameters.The interaction between a lump and one line soliton are still nonelastic but the track of the lump obeys the controllable function of time.The lump interacting with resonance soliton pairs exhibits a kind of special rogue wave in which the peak emerges and evolves with the varying path.The detailed analysis and discussion of these solutions are provided and illustrated.The third part introduces the Alice-Bob Sawada-Kotera system.It is well known that events occurring in different space-time may be closely related and/or entangled.If event.A(Alice)occurs,then event B(Bob)is determined immediately by the relevant condition B=fA,for the appropriate operator f.To describe two-place events,Alice-Bob systems have been established by means of the shifted parity and delayed time reversal.In this paper,we mainly study exact solutions of the integrable Alice-Bob Sawada-Kotera system.Some types of solutions including one-soliton,two-soliton,and breather solutions are explicitly obtained.