Extended shockwave decomposability related to boundaries of holomorphic 1-chains within $\mathbb{CP}^2$ Ronald Walker 32C3035Q5335N10boundaries of holomorphic chainsBurgers equationsums of shockwavesoverdetermined systems We consider the notion of meromorphic Whitney multifunction solutions to $ff_{\xi} = f_{\eta}$, which yields an enhanced version of the Dolbeault Henkin characterization of boundaries ofholomorphic 1-chains within $\mathbb{C}\mathbb{P}^2$. By analyzing the equations describing meromorphic Whitney multifunction solutions to $ff_{\xi} = f_{\eta}$ and by creating some generalizationsof certain linear dependence results, we show that a function $G$ may be decomposed into a sum of such solutions, modulo $\xi$-affine functions and with a selected bound on the degree of such sum, if and only if $G_{\xi\xi}$ satisfies a finite set of explicitly constructible partial differential equations. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3221 10.1512/iumj.2008.57.3221 en Indiana Univ. Math. J. 57 (2008) 1133 - 1172 state-of-the-art mathematics http://iumj.org/access/