Symmetry And Exact Solutions To Nonlinear Equations

The symmetries and exact solutions to the nonlinear equations are studied in thisthesis. The nonlinear equations such as the KD equation, the NNV equation, the BKKequation and the (3+1)-dimension rotating stratified flows modle are studied.In the study of symmetries of the nonlinear equations. Applying the general CKdirect method to KD equation, NNV equation and BKK equation, both the Lie pointsymmetry and the non-Lie symmetry group are obtained. At the same time, we prove theLie algebra structure of KD equation and NNV equation has Kac-Moody-Virasoro type.To the (3+1)-dimension rotating stratified flows modle, we obtained the symmetry trans-formation group by means of classic method and the famous first fundamental theoremof Lie.In the study of the exact solutions to the nonlinear equation, group transforma-tion method is applied to KD equation, NNV equation, BKK equation and the (3+1)-dimension rotating stratified flows modle, and the nonlocalized coherent structures ofthese equations are obtained. To the BKK equation, By selecting the arbitrary functions,introduced by the transformation group, one can find complicated and abundant struc-tures of the BKK equation, such as special curve solutions, the periodic solution, themulti-dromion solution, dromion solution and so on. To the (3+1)-dimension rotatingstratified flows modle, the symmetry is a bridge between the (3+1)-dimension rotatingstratified flows modle and the steady baroclinic laminar model. Beginning with the exactsolution of the steady baroclinic laminar model, a new exact solution of (3+1)-dimensionrotating stratified flow modle is got.