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On the integrability of a class of Monge-Ampère equations. (English) Zbl 1036.35139

The authors consider the Monge-Ampère equation in \(1+1\) dimensions, \[ U_{tt}U_{xx}-U_{tx}^2 = -k (*) \] where \(k=1\), \(-1\) or \(0\). These choices of \(k\) give the hyperbolic, elliptic and homogeneous Monge-Ampère equations which can be transformed to the Born-Infeld, minimal surface, and Bateman equations, respectively, by using the Bianchi transformation. The last three equations can be written simultaneously as \[ (k^2+\phi_x^2)\phi_{tt} - 2\phi_x\phi_t\phi_{xt} + (k^2\alpha+\phi_t^2)\phi_{xx} = 0 (**) \] where \(\alpha=k^2-k-1\).
The authors use this close relation between \((*)\) and \((**)\) to obtain a scalar dispersionless Lax representation for \((*)\). A matrix dispersive Lax representation for \((*)\) is also derived using the correspondence between sigma models, a two parameter equation for minimal surfaces and the Monge-Ampère equation. Local as well as nonlocal conserved quantities are obtained.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35J60 Nonlinear elliptic equations
35M10 PDEs of mixed type

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