Exact Solutions To The (2+1) Dimensinal Nonlinear Equations And Numerical Simulation

Since nineteen sixties,there appeared the research upsurge on the nonlinear prob-lems in the felds of natural science,which made the nonliear evolutions widely ap-plied in plasams,fuid mechanics,optical communications and other felds of natural sci-ence.Exploring exact solutions to the nonlinear evolution equations and seeking the kineticprorerties of these solutions have very vital signifcance eitther in theoretical rearch or inpractial application. In recent years,there appeare new solving methods,such as homo-clinic method,which has been successfully used in solving some equations’ homoclininc or-bit. Extend homoclininc test method can be used to solve solitary wave solutions,peiodicwave solutions,solitary wave solutions,as well as a more broad class of exact solutions.This paper uses the homoclinic test method and Hirota bilinear method to fnd thebilinear form and explicit exact solutionsof the (2+1) laser equation.Based on the formerof the bilinear form and homoclinic solutions of the (1+1)-dimensional Hirota equationand Ginzburg-Landau equation, we obtained the homoclinic solutions to the (2+1)-dimensional Hirota equation and Ginzburg-Landau equation.At the same time,we usethe extended homoclinic test technique,obtained the exact solutions to (2+1)-dimensionalKP equation,and we use the Matlab software to simulate the corresponding solutions, Byanalysis the graphic visual to their solutions, we found the results corresponde with thetheory.The chapters and contents for this article are arranged as follows:The frst chapter introduced the soliton theory development history and several com-mon methods on the solutions of nonlinear partial diferential equations,especially hirotabilinear method.The second chapter introduced Hirota method,which is invented by Hirota in20th cen-tury seventies.It is a common method to solve nonlinear partial diferential equations.Weaslo introduced three transforms and some characters about bilinear operators.We use thismethod to solve (2+1)-dimension laser equation, Hirota equation and Ginzburg-Landauequation.The third chapter wrote the Hirota bilinear of KP equaation.We use4forms of so-lutions to derive the exact solutions of KP,and deduced the relationship among cof-cients.We also use the software of Matlab to refect the process of solutions.