Gröbner bases and involutive methods for algebraic and differential equations
The paper illustrates computer algebra methods for analyzing systems of (non)linear algebraic partial differential equations. Sections 2 and 3 present background to and the algebraic Gröbner bases (GB-)algorithm and corresponding differential analogs; difficulties occurring in the latter are frequently non-terminating algorithms in the case of generation of differential ideals by nonlinear polynomials \(f_l=f_l(x,u,\ldots, D^\alpha u)\), \(l=1,\ldots, k\). Section 4 discusses involutive algebraic bases. For definiteness assume the monomials of \(K[x_1,\ldots, x_n]\) compared by degree reverse lex ordering, as customary in GB-theory. Then \(x_1\succ x_2 \succ \ldots \succ x_n\) and we have the notion of a leading term lt\((p)\) of a polynomial \(p\). For a term \(u=c\cdot x_1^{k_1} \cdots x_i^{k_i}\), in which all the first \(i\) variables genuinely occur, the set of \textit{multiplicative} variables is \({\mathcal M}(u)= \{x_i, x_{i+1},\ldots, x_n\}.\) Given polynomials \(p\) and \(q\), assume \(p\) has a term \(t\) such that \(t=\text{lt}(q)\times u\) with \(u\) variables in \({\mathcal M}({\text lt}(q))\). Then \(p\) can be involutively reduced by \(q\) yielding a reduction \(p\rightarrow p'=p-q\times u\). Similarly as in GB-theory one can define this way the involutive normal form \(N_I(p,F)\) of a polynomial \(p\) modulo a finite polynomial set \(F\) and develop in favorable cases (e.g. zero dimensional ideals) a theory of involutive bases [see \textit{A. Yu. Zharkov} and \textit{Yu. A. Blinkov} Math. Comput. Simulation 42, 323-332 (1996) and \textit{A. Yu. Zharkov} in: Algorithms in Algebraic Geometry and Applications, Proc. MEGA-94 Conf., Santander 1994, Prog. Math. 143, 389-399 (1996; Zbl 0878.13019)]. Section 5: Involutive bases are (if existent) special cases of GBs with particular properties. These make them attractive for studies of linear PDE-systems by means of the correspondence \(\frac{\partial^{\alpha_1+\ldots +\alpha_n} u_j} {\partial x_1^{\alpha_1} \ldots \partial x_n^{\alpha_n}} \Leftrightarrow x_1^{\alpha_1} \cdots x_n^{\alpha_n} \) and by Lie analysis also for nonlinear systems. Qualitative insights as e.g. the number of constants or arbitrary functions in the general solution of the related PDEs are obtainable. In section 6 the system of three linear PDEs obtained from \(u_{xxx}-u_{yy}+u_z-u=0\) upon cyclic permutation of \(x,y,z\) is analyzed and completely solved via corresponding algebraic GB; involutive basis is also given. Section 7 exemplifies via the nonlinear evolution PDE of diffusion type, \(u_t+u u_x -t u_{xx}=0\), how to use involutive bases to calculate the Lie symmetry algebra [see also \textit{F. Schwarz}, Computing 49, No. 2, 95-115 (1992; Zbl 0759.68042)]. The integrability analysis of systems of nonlinear PDE's by the inverse spectral transform [cf. \textit{V. P. Gerdt}, Int. J. Modern Phys. C 4, No. 2, 2-79-286 (1993)] is also briefly discussed.
scientific article; zbMATH DE number 1067235
scientific article