Brunelli, J. C.; Gürses, M.; Zheltukhin, K. On the integrability of a class of Monge-Ampère equations. (English) Zbl 1036.35139 Rev. Math. Phys. 13, No. 4, 529-543 (2001). 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. Reviewer: John Urbas (Canberra) Cited in 5 Documents 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 Keywords:Bateman equation; Bianchi transformation; Born-Infeld equation; minimal surface equation × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] DOI: 10.1007/BF00739421 · Zbl 0814.35075 · doi:10.1007/BF00739421 [2] DOI: 10.1098/rspa.1934.0059 · Zbl 0008.42203 · doi:10.1098/rspa.1934.0059 [3] DOI: 10.1007/BF01361114 · Zbl 0055.08404 · doi:10.1007/BF01361114 [4] DOI: 10.1023/A:1007425719838 · Zbl 0966.53010 · doi:10.1023/A:1007425719838 [5] Barbishov B. M., Sov. Phys. JETP 24 pp 93– (1966) [6] DOI: 10.1016/S0370-2693(98)00265-2 · Zbl 1049.78503 · doi:10.1016/S0370-2693(98)00265-2 [7] DOI: 10.1063/1.528314 · Zbl 0850.70173 · doi:10.1063/1.528314 [8] DOI: 10.1016/0550-3213(92)90455-K · doi:10.1016/0550-3213(92)90455-K [9] DOI: 10.1142/S0129055X96000378 · Zbl 0869.35081 · doi:10.1142/S0129055X96000378 [10] DOI: 10.1016/S0375-9601(97)00708-1 · Zbl 0969.76578 · doi:10.1016/S0375-9601(97)00708-1 [11] Gürses M., Int. J. Mod. Phys. pp A6– (1991) [12] Gürses M., Lett. Math. Phys. pp 26– (1992) [13] DOI: 10.1070/RM1989v044n06ABEH002300 · Zbl 0712.58032 · doi:10.1070/RM1989v044n06ABEH002300 [14] DOI: 10.1063/1.526530 · Zbl 0563.35050 · doi:10.1063/1.526530 [15] Stanyukovich K. P., New York pp 137– (1960) [16] DOI: 10.1063/1.527909 · Zbl 0697.35084 · doi:10.1063/1.527909 [17] DOI: 10.1016/0375-9601(93)90277-7 · doi:10.1016/0375-9601(93)90277-7 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.