Formal solutions of nonlinear first order totally characteristic type PDE with irregular singularity.

*(English)*Zbl 0993.35003Consider a holomorphic function \(F(t,x,u,v)\) defined on an open polydisk \( \Delta \) centered at the origin of \(\mathbb{C}_{t}\times \mathbb{C}_{x}\times \mathbb{C}_{u}\times \mathbb{C}_{v}\) and which satisfies \(F(0,x,0,0) \equiv 0\) on \( \Delta _{0}= \Delta \cap \{t=0,u=0,v=0\}\). The authors study the first order scalar nonlinear singular partial differential equation
\[
t( \partial u / \partial t)=F(t,x,u, \partial u/\partial x).
\]
From the assumptions it follows that \(F(t,x,u,v)= a(x)t +b(x)u + \gamma (x) v + \sum _{i+j+\alpha } a _{i,j,\alpha }(x) t ^{i} u ^{j} v ^{\alpha }\) where \(a(x),b(x)\), \(\gamma (x)\) and the \( a _{i,j,\alpha }(x)\) are holomorphic in \(x\) on \( \Delta _{0}\). The case when \( \gamma (x) \equiv 0\) has been studied in a series of papers by R. Gerard and H. Tahara and when \( \gamma (0)\neq 0\), one can solve for \( (\partial u/\partial x)\) and reduce oneself to the Cauchy-Kowalewsky theorem. Accordingly, the authors assume in this paper that \( \gamma (x)= x ^{p } c(x)\), with \(p\) some natural number and \(c(0)\neq 0\). The main result of the authors now refers to the case \( p \geq 2\), the case \(p=1\) having been already studied in a paper by Chen and Tahara. It states that when \( b(0) \neq 0\), then the above equation admits a formal solution \( u \in \mathbb{C}[[t,x]]\) with \(u(0,x) \equiv 0\), which belongs to some formal Gevrey class. For the discussion of the relation of formal solutions to true solutions, the authors refer to a forthcoming paper.

Reviewer: Otto Liess (Bologna)

##### MSC:

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |

35C10 | Series solutions to PDEs |

35A20 | Analyticity in context of PDEs |

35A10 | Cauchy-Kovalevskaya theorems |

PDF
BibTeX
XML
Cite

\textit{H. Chen} et al., Ann. Inst. Fourier 51, No. 6, 1599--1620 (2001; Zbl 0993.35003)

**OpenURL**

##### References:

[1] | Invariant varieties through singularities of holomorphic vector fields, Annals of Math., 115, (1982) · Zbl 0503.32007 |

[2] | On the holomorphic solution of non-linear totally characteristic equations · Zbl 1017.35006 |

[3] | On totally characteristic type non-linear partial differential equations in the complex domain, Publ. RIMS, Kyoto Univ., 26, 621-636, (1999) · Zbl 0961.35002 |

[4] | On the holomorphic solution of nonlinear totally characteristic equations with several space variables · Zbl 1003.35005 |

[5] | Nonlinear singular first order partial differential equations of briot-bouquet type, Proc. Japan Acad., 66, 72-74, (1990) · Zbl 0711.35034 |

[6] | Holomorphic and singular solution of nonlinear singular first order partial differential equations, Publ. RIMS, Kyoto Univ., 26, 979-1000, (1990) · Zbl 0736.35022 |

[7] | Singular nonlinear partial differential equations, E 28, (1996), Vieweg · Zbl 0874.35001 |

[8] | Formal power series solutions of nonlinear first order partial differential equations, Funkcial. Ekvac., 41, 133-166, (1998) · Zbl 1142.35310 |

[9] | Formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, 1, 205-237, (1994) · Zbl 0810.35006 |

[10] | Maillet type theorems for nonlinear partial differential equations and the Newton polygons · Zbl 0995.35002 |

[11] | A course of modern analysis, (1958), Cambridge Univ. Press |

[12] | Newton polyhedrons and a formal Gevrey space of double indices for linear partial differential operators, Funkcial. Ekvac., 41, 337-345, (1998) · Zbl 1140.35575 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.