Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions - Mathematics > Differential GeometryReport as inadecuate




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Abstract: We study the geometry of multidimensional scalar $2^{nd}$ order PDEs i.e.PDEs with $n$ independent variables with one unknown function, viewed ashypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{1}$ over a$2n+1$-dimensional contact manifold $M,\mathcal{C}$. We develop the theoryof characteristics of the equation $\mathcal{E}$ in terms of contact geometryand of the geometry of Lagrangian Grassmannian and study their relationshipwith intermediate integrals of $\mathcal{E}$. After specifying the results togeneral Monge-Amp\`ere equations MAEs, we focus our attention to MAEs of typeintroduced by Goursat, i.e. MAEs of the form $$ \det|\frac{\partial^2f}{\partial x^i\partial x^j}-b {ij}x,f, abla f\|=0. $$ We show that any MAEof the aforementioned class is associated with an $n$-dimensionalsubdistribution $\mathcal{D}$ of the contact distribution $\mathcal{C}$, andviceversa. We characterize this Goursat-type equations together with itsintermediate integrals in terms of their characteristics and give a criterionof local contact equivalence. Finally, we develop a method of solutions of aCauchy problem, provided the existence of a suitable number of intermediateintegrals.



Author: Dmitri Alekseevsky, Ricardo Alonso-Blanco, Gianni Manno, Fabrizio Pugliese

Source: https://arxiv.org/







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