Poincare-Lighthill-Kuo Method and Symbolic Computation

This paper elucidates the effectiveness of combining the Poincare_Lighthill_Kuo method(PLK method, for short) and symbolic computation. Firstly, the idea and history of the PLK method are briefly introduced. Then, the difficulty of intermediate expression swell, often encountered in symbolic computation, is outlined. For overcoming the difficulty, a semi_inverse algorithm was proposed by the author, with which the lengthy parts of intermediate expressions are first frozen in the form of symbols till the final stage of seeking perturbation solutions. To discuss the applications of the above algorithm, the related work of the author and his research group on nonlinear oscillations and waves is concisely reviewed. The computer_extended perturbation solution of the Duffing equation shows that the asymptotic solution obtained with the PLK method possesses the convergence radius of 1 and thus the range of validity of the solution is considerably enlarged. The studies on internal solitary waves in stratified fluid and on the head_on collision between two solitary waves in a hyperelastic rod indicate that by means of the presented methods, very complicated manipulation, unconceivable in hand calculation, can be conducted and thus result in higher_order evolution equations and asymptotic solutions. The examples illustrate that the algorithm helps to realize the symbolic computation on micro_commputers. Finally, it is concluded that with the aid of symbolic computation, the vitality of the PLK method is greatly strengthened and at least for the solutions to conservative systems of oscillations and waves, it is a powerful tool.