We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

Differential-algebraic equations (DAE) and partial differential-algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics. A DAE has differential index r if a minimum of r + 1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties.Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan-Kuranishi algorithm for completing a system of differential equations to involutive form. The CartanKuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier-Stokes equations are given. Discussion of computer algebra implementations is also provided.

We present Existence ant1 Uniqueness Theorems for formal pO\Wr series solutions Of ilnd~t.i(' s\lSteIlls Of PDF. in il cmtain form. 'This form can be obtained by it finil.c number of differediations and cliulinat.ic)us of the original systen~~ and allows its formal power series solut,ious t.0 lx coniput~ed in a11 alg0rithmic fashion.The result.ing reduced involutiw form (rif' form) produced by our rif' algorit,liui is a generalizitt.ion of the ClassiCal fornl of Riquier and .Janet; and that of CauchKOV~l.lC?\?h~iL I;(: waken the assumpt.ion of linearity iu the highest dermdves iu t~hosc approaches t.O allow for systcrns which are n0nlineiK in their highest deriva.t.ives.A new fornml developn~cx~t. of Riqnicr's theory is given: with proofs. n~otleled after t,how in Griilmcr Basis Theory. For the uonlincar tllcoryz the concept of rclatiw Riquiel Bases is introduced.This allows for t.he easy esteusion of ideas from the linear t0 tlw nonlinear t,hrory. Tile essent.ial idea is that an arbitrary noulincar system can Ix writ.teu (aft.cr tliffcrcutiatiou if necessary), as il syst.cmi which is liw ear in its highw, dcrivat.ivcs, and a constraint syst,em. which is n0nlinear in its highclst, derivatives. Our t,heorems iwe applied t,o S6T~rill eximplcs.

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