Localization Of The Nonlocal Symmetries And Symmetry Reductions Of The Burgers Equation

In the thesis,the Backlund transformation and group-invariant solutions concerned with symmetry group for a nonlinear partial different equations(PDE)are studies through the symmetry theory.Chapter 1 is a brief review of the background and development of symmetry theory and group theory with explanation of some concepts related to symmetry group.In Chapter 2,by using the WTC method of the Painlev?e analytics,the integrability of Burgers equation is proved.Chapter 3 is mainly focused on the Schwarzian and the nonlocal symmetries of the Burgers equation.Using the truncated Painlev?e expansion,the Schwarzian and the nonlocal symmetry of the Burgers equation are obtained.In consideration of the localization procedure for the nonlocal symmetries,the auto-B¨acklund transformation and group invariant solutions are obtained.At last,the above method is generalized.The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlev?e method.Because of the complexity of reduction equations,we only study the situation with two nonlocal symmetries.The auto-B¨acklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries.Moreover,the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.