[Paper Review] Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions

Dmitri V. Alekseevsky, Ricardo J. Alonso-Blanco+2 more|arXiv (Cornell University)|Mar 26, 2010

Geometry and complex manifolds4 citations

TL;DR

This paper develops a contact geometric framework for multidimensional Monge-Ampère equations (MAEs), introducing a characterization of their characteristics and intermediate integrals via subdistributions of the contact distribution. It establishes a local contact equivalence criterion for Goursat-type MAEs and provides a generalized Monge method to solve Cauchy problems using intermediate integrals, yielding explicit solutions through Hamiltonian flow propagation.

ABSTRACT

We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a $(2n+1)$-dimensional contact manifold $(M,\mathcal{C})$. We develop the theory of characteristics of the equation $\mathcal{E}$ in terms of contact geometry and of the geometry of Lagrangian Grassmannian and study their relationship with intermediate integrals of $\mathcal{E}$. After specifying the results to general Monge-Ampère equations (MAEs), we focus our attention to MAEs of type introduced by Goursat, i.e. MAEs of the form $$ \det|\frac{\partial^2 f}{\partial x^i\partial x^j}-b_{ij}(x,f, abla f)\|=0. $$ We show that any MAE of the aforementioned class is associated with an $n$-dimensional subdistribution $\mathcal{D}$ of the contact distribution $\mathcal{C}$, and viceversa. We characterize this Goursat-type equations together with its intermediate integrals in terms of their characteristics and give a criterion of local contact equivalence. Finally, we develop a method of solutions of a Cauchy problem, provided the existence of a suitable number of intermediate integrals.

Motivation & Objective

• To develop a contact geometric theory for multidimensional second-order PDEs, particularly Monge-Ampère equations, by analyzing their characteristics and intermediate integrals.
• To characterize Goursat-type MAEs through associated n-dimensional subdistributions of the contact distribution.
• To establish a local contact equivalence criterion for such MAEs based on their geometric invariants.
• To generalize the classical Monge method for solving Cauchy problems of MAEs by leveraging intermediate integrals and Hamiltonian flows.
• To provide a constructive method for solving Cauchy problems when sufficient intermediate integrals exist, ensuring local uniqueness of solutions.

Proposed method

• The authors model second-order PDEs as hypersurfaces in the Lagrangian Grassmann bundle over a (2n+1)-dimensional contact manifold, using contact geometry to analyze their structure.
• They define the characteristic cone and conformal metric of a PDE using the geometry of the Lagrangian Grassmannian and relate these to intermediate integrals.
• For Goursat-type MAEs, they associate each equation with an n-dimensional subdistribution D of the contact distribution C, and vice versa, establishing a duality.
• The generalized Monge method is implemented by constructing intermediate integrals φ(x^i, p_i) and φ(overline{x}^i, overline{p}_i), which are used to define Hamiltonian vector fields whose integral curves propagate the Cauchy datum.
• Solutions are constructed by integrating the Hamiltonian vector field associated with an intermediate integral, using first integrals to eliminate variables and reconstruct the solution surface.
• Explicit solution formulas are derived by eliminating coordinates using relations among first integrals, as demonstrated in a 2D example with z = x^1 e^{x^2} + e^{x^1 + overline{x}^1} - x^1 overline{x}^2.

Experimental results

Research questions

• RQ1 How can the characteristics of multidimensional Monge-Ampère equations be described using contact geometry and the Lagrangian Grassmannian?
• RQ2 What is the geometric relationship between intermediate integrals and the characteristic structure of a second-order PDE?
• RQ3 How can Goursat-type Monge-Ampère equations be characterized via subdistributions of the contact distribution?
• RQ4 What criterion ensures local contact equivalence between two such MAEs?
• RQ5 What conditions allow the generalized Monge method to solve a Cauchy problem for a multidimensional MAE?

Key findings

• Any Goursat-type Monge-Ampère equation is locally contact equivalent to a canonical form associated with an n-dimensional subdistribution D of the contact distribution C.
• The intermediate integrals of such MAEs are functions φ(x^1,…,x^k,p_1,…,p_k) and φ(overline{x}^1,…,overline{x}^k,overline{p}_1,…,overline{p}_k), where k = n/2.
• The generalized Monge method successfully constructs a unique local solution to a Cauchy problem when sufficient intermediate integrals exist, as demonstrated by explicit integration of the Hamiltonian vector field.
• The solution to the example MAE det(∂²z/∂overline{x}^i ∂x^j) = 0 is z = x^1 e^{x^2} + e^{x^1 + overline{x}^1} - x^1 overline{x}^2, derived via propagation along the Hamiltonian flow of an intermediate integral.
• The method relies on identifying five independent relations among eight first integrals of the Hamiltonian vector field, which are used to eliminate variables and reconstruct the solution surface.
• The existence of intermediate integrals ensures the formal integrability of the PDE and enables the unique extension of non-characteristic Cauchy data to a local solution.