Study Of Several Nonlinear Equations And Its Extend In Meteorology

The exact solutions to the nonlinear Broer-Kaup-Kupershmidt (BKK) equations, theapproximate solutions to5order perturbation KdV equations and Lie point symmetryalgebra and finite transmission group of BP model in meteorology, are studied in thisthesis.In the research of finding exact solution to (2+1) dimension Broer-Kaup-Kupershmidt(BKK) equations, the modified CK direct method is applied in this thesis. It can beused for dimension reductions of the model. Then the symmetry algebras and group areobtained to solve group invariable solutions of the nonlinear system. The fundamentalproperty of the system can be obtained from the new solutions. Particularly, a certain caseof compaction solution without elastic property and another case of compaction solutionwith elastic property are token as initial solution. Finally, check whether the new solutionswhich are obtained from group transformation can keep their original property.Two methods derived from perturbation method are used to solve the approximatesolutions of5order perturbation KdV equation. One is approximate direct method and theother is similarity reductions. The advantage of these two methods is that the approximatesolutions can be used to approach the exact solutions. Approximate direct method is anapproximate method from CK direct method. The main idea of this method is dimensionreductions to equations. Similarity reductions is very close to approximate direct methodin the sense of solving process. However, the solutions of these two methods are diferent.Therefore, diferent types of solutions can be used to study the physical meanings of5order perturbation KdV equations.The traditional method is applied in the solving of BP model in meteorology. First, Kac-Moody-Virasoro (KMV) symmetry algebras is obtained from the solutions of sym-metry and symmetry algebras. It reveals that the BP Model contains well structuredproperty. It is difcult to solve the symmetry group due to the arbitrary function. Newmethod is proposed to solve this problem. The group invariable solution reveals that ageneral hypothesis in meteorology is wrong.