Convergence of formal solutions of first order singular nonlinear partial differential equations in the complex domain

Tom 74 / 2000

Masatake Miyake, Akira Shirai Annales Polonici Mathematici 74 (2000), 215-228 DOI: 10.4064/ap-74-1-215-228


We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,D_x u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,ξ^{0}) ∈ ℂ^{n}_{x} × ℂ_{u} × ℂ^{n}_{ξ} (ξ^{0} = D_{x}u(0))$ and $f(0,0,ξ^{0}) = 0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $(ξ ∈ ℂ^{n})$. The criterion of convergence of a formal solution $u(x) = ∑_{|α| ≥ 1} u_{α}x^{α}$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal solution diverges a precise rate of divergence or the formal Gevrey order is specified which can be interpreted in terms of the Newton polygon as in the case of linear equations but for nonlinear equations it depends on the individual formal solution.


  • Masatake Miyake
  • Akira Shirai

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