主要研究由Blatz-Ko材料所導出的圓球對稱動態變形波方程式，並利用李群理論導出此波方程式的李群與求得三個不變解，其中兩個不變解為可分離變數解，另一個則為不可分離變數不變解，利用不可分離變數不變解對原Blatz-Ko波方程做降維後導出的是一個含有奇異性(singular)的二階non-autonomous常微分方程式。我們利用變數變換將此常微分方程式轉換成autonomous 微分方程組，並且成功的分析其相平面(phase plane)上的軌跡。根據上面所述可以完整的分析方程式的解在全域上的行為模式，而這組特殊解的分析可以明確的指出在什麼樣的問題以及初始條件下可能發生解的奇異性。另外也分析多參數子群在群轉換作用下，所得到新的不變解在相平面上的初位置與時間彼此間參數變化所對應的新的關係。本論文也研究Blatz-Ko波方程的一種對稱性-倒數時間平移對稱，此一特殊的對稱性質能讓將不同的波動方程初始邊界條件彼此連結起來，根據這個連結轉換可以從已知的解轉換到其它的解。在實驗及計算方面根據這個倒數時間轉換也可以減少Blatz-Ko材料在實驗或數值計算的時間。最後分析在逆時間轉換下的解的極限行為與導數性質。 This dissertation investigates the symmetry properties of the spherical wave equation for Blatz-Ko materials. Three invariant solutions are obtained by symmetry reduction. Two of them are in separable forms. The non-separable one is governed by a singular, non-autonomous second order ordinary differential equation. A variable transform is applied to turn this non-autonomous equation into an autonomous equation. By analyzing the phase plane of this autonomous equation we find out the condition for the solutions to develop singularity. Similar analyses are also carried out for the governing equation of an invariant solution derived from a multi-parameter symmetry. A special symmetry, the inverse time translational symmetry, of the Blatz-Ko wave equation is also studied. A correspondence between initial/boundary value problems is derived using this special symmetry. And this correspondence can be applied to reduce the duration of the experiments or numerical calculations for the dynamic behavior of the Blatz-Ko spheres. The correspondence can also be used to analyze the limiting dynamic behavior of the Blatz-Ko sphere.
