Study Of Partial Differential Equation Method

Partial differential equations plays an important role in real life, a wide range ofapplications in biological, physical, and many other subjects, thus solving partial differentialequations become utmost importance. Partial differential equations have a lot of methods inthe field of mathematics, while different of methods to get a variety of solutions of equations,including traveling wave solution, multiple track solitary wave solutions, periodic solutions,Jacobi elliptic function solutions and so on. In this paper, as an example of (2+1)dimensional dispersive long water waves equation, Broer-Kaup equation, HBK equation, usethe methods of the hyperbolic tangent, the projective Riccati, the Jacobi elliptic function,homogeneous balance, Backlund transformations, Lie point symmetries and Darbouxtransformations can be calculated soliton solution,, similarity solution and invariant solutionof partial differential equations.The first chapter is concerned with the historical background of the soliton theory andthe exposition of the development about the exact solutions of Partial differential equations,the main results of this dissertation are introduced at the end.The second chapter by a simple transformation,(2+1) dimensional dispersive long waterwaves equation are turned to a simple equation. Some new solitary wave solutions areobtained by combining the transformation with the hyperbolic tangent method、the projectiveRiccati method、the Jacobi elliptic function method、the homogeneous balance method.The third chapter a Backlund transformation and the similarity solution of BK equationare derived, then a systematic investigation to derive the Lie point symmetries of BK equationis presented. Using the obtained point symmetries, similarity reductions are derived, andinvariant solutions are obtained.The forth chapter three basic Darboux transformations of a spectral problem and thegauge transformation associated with equation of HBK are considered by the lax pair of HBKequation, they are used to generate new soliton solutions of HBK equation.